A decoding circuit 54 for a simultaneous operation non-volatile memory device with a flexible bank partition architecture comprises an X-decoder 44, a lower bank decoder 58, an upper bank decoder 56, and a plurality of flexibly partitioned conductive lines coupled between the upper and lower bank decoders 56 and 58. The flexibly partitioned conductive lines 60, 62, 64, . . . 74 provide a plurality of bank address pre-decoding bits for the X-decoder 44 to row decode the memory cells along the respective word lines in the memory array 20. The memory array 20 includes a plurality of flexibly partitioned bit lines comprising first and second bit line segments to partition the memory array into upper and lower memory banks. The bit line segments in the upper and lower memory banks are coupled to two Y-decoders 32 and 34 which provide column decoding for the memory cells in the upper and lower memory banks.

Non-volatile memory devices have been developed by the semiconductor integrated circuit industry for various applications such as computers and digital communications. A conventional non-simultaneous operation flash memory device typically includes a single fixed memory bank. A conventional simultaneous operation flash memory device typically includes two fixed memory banks each comprising a fixed number of sectors of memory cells. Each sector of memory cells has a fixed amount of memory storage, for example, 0.5 megabytes (MB) of memory and consists of a fixed number pages of memory cells, for example, sixteen pages of memory cells. A page is typically defined as one word of memory stored in the memory cells on a single word line.
In a conventional simultaneous operation flash memory device, such as a conventional simultaneous operation NOR flash memory array, the memory cells are divided into an upper memory bank and a lower memory bank. The upper and lower memory banks each have a predetermined fixed size of memory storage. The upper and lower memory banks are typically used for different functions in simultaneous reading and writing operations. For example, the lower memory bank may be used for code storage, whereas the upper memory bank may be used for data storage.
Since the upper and lower memory bank partitions are fixed in conventional simultaneous operation flash memory devices, different flash memory devices with different integrated circuit designs are required for different memory partitions. Therefore, in order to customize the memory circuit for a variety of applications which require different partitions of the upper and lower memory banks for code and data storage, a different circuit design would be required for each of the fixed upper and lower memory bank partitions in a conventional simultaneous operation flash memory device.
Because the size of memory storage in each of the upper and lower memory banks is not variable in a conventional simultaneous operation non-volatile memory device, a single circuit design may be suitable for only one of several different applications in which different partitions of the upper and lower memory banks are required. In order to produce simultaneous operation flash memory devices with different partitions of the upper and lower memory banks, a different circuit design and a different set of associated masks are required for each of the devices.
A disadvantage of having to design a different integrated circuit and a full set of different masks for each of the simultaneous operation flash memory devices with different upper and lower memory bank partitions is that the design, fabrication and testing processes can be very costly and time-consuming. Therefore, there is a need for a simultaneous operation flash memory device with a flexible bank partition architecture. Furthermore, there is a need for a simplified decoding architecture for the simultaneous operation flash memory device with a flexible bank partition architecture.

Memory address decoding circuit for a simultaneous operation flash memory device with a flexible bank partition architecture


Complexity of information of a video signal is detected to control a quantizing step based on the above complexity. Difference of the said video signal is quantized with different quantizing steps in response to the control. Generated code lengths of a plurality of resultant quantized outputs are compared with a target code length. A quantization index indicative of a quantizing step is determined based on a result of the comparison. The above video signal is quantized with the quantizing step based on the quantization index. The above quantized output is subjected to a variable length encoding processing. Further, when the quantization index is determined, a value of them target quantization index indicative of a target quantizing step size allowing the target code length to be obtained is searched only between two quantization indexes respectively indicating two quantizing steps which allow two generated code lengths most approximate to a value of the target code length.

In a picture compression system employing combination of motion compensation and DCT (Discrete Cosine Transform) represented by an MPEG (Moving Pictures Expert Group) standard, a code amount is controlled so that a bit stream should be transferred to a transmission line at a desired bit rate. A conventional control of the encoding amount is a feed-back control of a quantizing step based on a relationship between a previous quantizing step and a previous code amount and also on a present mean bit rate.
FIG. 7 shows an example of a picture compression and encoding apparatus employing combination of motion compensation and DCT. A digital input video signal is supplied through an input terminal T1 to a scan converter circuit 1. The scan converter circuit divides a picture of one frame into a large number of macroblocks. According to the MPEG standard, a macroblock of a luminance signal has a size of (16.times.16) formed of four DCT blocks. Each of macroblocks of color difference signals Cr, Cb each has a size of (8.times.8) formed of two DCT blocks in the case of (4:2:2). One macroblock is formed of total 8 DCT blocks.
Data converted into data of macroblocks by the scan converter circuit 1 are supplied to a subtracting circuit 2 and to a motion detection unit 3 for calculating a motion vector. The motion detection unit 3 calculates a motion vector with respect to the present macroblock with reference to a reference picture. The motion vector is supplied to a motion compensation unit 4 which carries out motion compensation using the motion vector. In this operation for the motion compensation, a locally coded data described later on is employed as the reference picture for carrying out the same motion compensation as that carried out by the decoder side.
If the data of the macroblock is to be processed for an intraframe coding, then the subtracting circuit 2 supplies the data to a DCT processing unit 5 as it is without carrying out the subtraction processing. If on the other hand the data of the macroblock is to be processed for an interframe coding, the subtracting circuit 2 calculates difference between the data and a picture data from the motion compensation unit 4 and supplies the calculated difference to the DCT processing unit 5.
The DCT processing unit 5 effects the DCT on every DCT block of (8.times.8) to generate a DCT coefficient data. A quantizer 6 quantizes the coefficient data therefrom with a quantizing step. A variable-length encoder 7 variable-length encodes data quantized by the quantizer 6 (referred to as a quantization level). The variable length encoder 7 outputs data to a buffer 8. A bit stream output from the buffer 8 is supplied to a transmission line through an output terminal T2. The quantizing step of the quantizer 6 is controlled in order to set a bit rate of the output from the buffer 8 constant in response to the bit rate of the transmission line.
The data output from the quantizer 6 is also supplied to an inverse quantizer 9. Data (representative value) output from the inverse quantizer 9 is supplied to an inverse DCT processing unit 10. The inverse DCT processing unit 10 provides a decoded video data of a sample picture. If the video data is one for the intraframe coding, then the video data for the sample picture becomes the data of a local decoded picture as it is. If on the other hand the data is one for the interframe coding, then an adder circuit 11 adds the video data to the video data supplied from the motion-compensated picture unit 4, thereby a decoded picture data being obtained.
When the code amount is controlled, a quantization index indicative of the quantizing step of the quantizer 5 is controlled. A code amount control proposed as a test model in an MPEG2 standard is a code amount control that is a feed back control employing a remaining amount of a virtual buffer and a relationship between a quantization index used upon a previous encoding and an amount of codes generated thereby.
The above conventional code amount control encounters the following problems.
First, since the code amount control is the feed back control, if a damping amount is set smaller, then response becomes faster but the processing becomes oscillatory, and if on the other hand the damping amount is set larger, then the oscillation of the processing is reduced but the response becomes slower.
Secondarily, since the bit rate instantaneously becomes large when a scene of a picture is changed, a reproduced picture becomes erroneous or extremely deteriorated in picture quality.
Thirdly, it is difficult to control frames of a predetermined number to be transferred at a predetermined bit rate.
To solve the above problems, a system of controlling the code amount by a feed forward system is considered. According to this system, an amount of codes generated in an equal-length unit is previously calculated with respect to a plurality of quantizing steps and a proper quantizing step is set within the range in which the amount of the generated codes does not exceed a target amount of codes.
A GOP (Group Of Picture), a frame, a macroblock and so on are considered as the equal-length unit used in such code-amount control of the feed-forward system. The GOP is a picture data formed of one frame or greater. The MPEG2 is established to control the code amount at a GOP unit.
As described above, when the code amount is controlled, a plurality of quantizing steps have different fixed values and one of these quantizing steps is selected for the GOP, which brings the problem that a picture quality of a decoded picture is different depending upon the macroblocks. For example, a picture quality of a macroblock of a minute picture is deteriorated, and conversely that of a macroblock of a flat picture is improved. If picture qualities of the adjacent macroblocks are considerably different from each other as described above, then a block distortion is caused.
Therefore, an object of the present invention is to provide a method of encoding a video signal and an apparatus therefor which can improve a picture quality of a decoded picture by employing a code amount control of a feed-forward system for allowing the control corresponding to a local characteristic of a video signal.

Adaptive quantization of video based on target code length


An image forming apparatus includes an apparatus main body, a paper sheet feeder, an image forming circuit, a paper sheet discharge unit, and a high voltage circuit board. The paper sheet feeder is located in the apparatus main body to feed a sheet. The image forming circuit is located in the apparatus main body to form an image on the sheet fed by the paper sheet feeder. The image formed sheet with is discharged to the paper sheet discharge unit. The high voltage circuit board is located along a horizontal direction over the image forming circuit in the apparatus main body to supply a high voltage to the image forming circuit.

Unless otherwise indicated herein, the description in this section is not prior art to the claims in this application and is not admitted to be prior art by inclusion in this section.
An image forming apparatus that forms an image on a sheet has been proposed. The image forming apparatus includes a paper sheet feeder, an image forming unit, and a paper sheet discharge unit. The image forming unit forms an image on a sheet sent out from the paper sheet feeder in the image forming unit. Then, after a fixing process of the image is performed on the sheet, the sheet is discharged in a paper sheet discharge unit.
There is proposed a constitution where a high voltage circuit board that supplies a high voltage to an image forming unit is located on a sidewall of an apparatus main body of an image forming apparatus. Further, there is proposed a constitution where a high voltage circuit board and an electric power supply circuit board are located on a sidewall or bottom portion of an apparatus main body of an image forming apparatus.

Image forming apparatus that prevents occurrence of failure in association with temperature rise of circuit board


A newly proposed articulating invention, each of whose four constituent embodiments is designed to trisect any of a multitude of suitably described angles by means of becoming properly set to its designated magnitude; thus automatically portraying a motion related solution for the trisection of an angle that discloses complete routing details of a pathway that leads from such designated magnitude all the way back to its trisector; thereby discerning the whereabouts of certain intersection points which evade detection when attempting to otherwise locate them by means of applying only a straightedge and compass to an angle of such designated magnitude; furthermore projecting finite lengths of any trisector that bears cubic irrational trigonometric properties, being those that cannot be duplicated, but only approximated, when applying a straightedge and compass to a given length of unity; and being of a unique design that could be adapted to function as a level.

By now, it most certainly should be recognized that a singular angle of virtually any designated size cannot be divided into three parts when acted upon only by a straightedge and a compass.
Such protracted problem is considered to be so famous that herein it formally shall be referred to as the classical problem of the trisection of an angle!
Unfortunately, many other descriptions of such problem also exist, each conflicting in some rather subtle manner, but nevertheless having profound effect upon its interpretation. Leading examples of such differences are presented below where it is found that in certain cases throughout the twentieth century such problem of the trisection of an angle:                is not specified, but its solution nevertheless is posed;        is considered to involve the use of a ruler; and        is algebraically, rather than geometrically resolved.
A looming trisection mystery, steeped in controversy for millennia, persists right up until present day; merely because it never was examined from the proper perspective!
In order to crack such conundrum, not only do correct questions need to asked, but proper answers also need to be supplied.
In this regard, the very first of four fundamental questions is about to be posed.
The first riddle is: How can the classical problem of the trisection of an angle actually be solved?
The answer is: It cannot!
Granted, it is commonly known that such response has been hotly contested by many pundits over the years.
But, to its credit, after bearing the brunt of a constant barrage of brutal assaults by noted protagonists, such contention endures; whereby it appears that to date no solution for the classical problem of the trisection of an angle ever has been solicited that has withstood the test of time!
Because it is far more difficult to prove an impossibility than something that is thought to be possible, fewer proofs exist which attempt to validate that the classical problem of the trisection of an angle truly is unsolvable!
To this effect, page 246 appearing in A Dictionary of Mathematics by T. A. Millington, Barnes & Noble; 1966 stated, "The classical problem of trisecting an angle by Euclidean constructions (use of straight lines and circles only) was proved to be insoluble by Wantzel in 1847".
Below, my express intent is to acknowledge Wantzel's outright conclusion as being absolutely correct in such regard, in hopes of thereby identifying some distinct difficulty which prevails that otherwise prevents the classical problem of the trisection of an angle from being solved!
This is to be accomplished by means of delving into what before were considered to be impenetrable depths of a great enigma in order to finally unravel the very mystery that today engulfs trisection.
Such trisection enigma specifically shall state: If the classical problem of the trisection of an angle truly cannot be solved, then geometry, itself, must be the culprit; thereby being imperfect!
This author well understands that if it were learned that geometry actually is limited, marred, or otherwise flawed in some strange, presently unknown fashion, then the very perception of such form of mathematics inevitably would become altered, even to the point where it possibly might become tarnished forevermore!
The second riddle is: What specific, well known thermodynamics limitation now should be considered to furthermore apply to geometry?
The answer is: Irreversibility!
A commentary, as presented below, should settle any doubts, or quell any lingering suspicions, that the classical problem of the trisection of an angle might become solved.
A fair number of rather elaborate proofs that have gained notoriety over the years have done so by means of incorrectly claiming that the classical problem of the trisection of an angle actually can be solved.
Such proofs either are quite faulty in their logic, or else rely upon a basic deception; whereby their geometric solutions, although complying with a primary requirement of applying only a straightedge and compass, achieve trisection instead by having them act upon other than a singular angle of virtually any designated size!
In such manner, geometric solutions actually do achieve trisection, but only to their detriment by means of violating a remaining requirement that the classical problem of the trisection of an angle furthermore imposes!
As such, for any and all geometric solutions which might become specified in the near future claiming to succeed at trisecting an angle, perhaps it is best to offer the rather simple-minded reservation that in order to solve the classical problem of the trisection of an angle, all of its imposed requirements explicitly must be complied with!
In other words, the very reason why any of such actual geometric solutions should not be misinterpreted as a solution for the classical problem of the trisection of an angle is because they violate the second of its requirements, fundamentally specifying that geometric activity must proceed only from a singular angle of virtually any designated size.
Instead of complying with such second requirement, geometric solutions otherwise input extraneous information into such classical problem, thereby serving to corrupt it!
In particular, extraneous information consists of that which is not germane to such classical problem. Hence it consists of any geometric information other than that which can be derived by applying only a straightedge and compass exclusively to a singular angle of designated magnitude which is intended to be trisected.
Moreover, extraneous information is considered to be relevant whenever it independently can be acted upon solely by a straightedge and compass in order to pose an actual solution for some corrupted version of the classical problem of the trisection of an angle; thereby solving an entirely different problem!
Hence, it becomes rather obvious that such very controversy undeniably has been fueled over the years by an unrelenting confusion which has been stirred over nothing more than various cases of mistaken identity!
A rather elementary example of how to administer such overall theory is presented directly below; whereby for the unsolvable classical problem of the trisection of an angle of 135° designated magnitude, it is desired to specify relevant information that enables a geometric solution to be obtained.
For such example, the approach to obtain such relevant information is given as follows:                extraneous information quite readily can be determined simply by dividing such 135° angle by a factor of three. Such resulting 45° size qualifies as extraneous information because it clearly cannot be derived in any way by means of applying only a compass and straightedge to such 135° designated angle; and        such calculated 45° magnitude furthermore is relevant because it can be drawn solely by application of a straightedge and compass as one of the diagonals in a square of virtually any size, thereby posing a geometric solution for such corrupted classical trisection problem.
Such types of mishap, assuming the form of mistaken identifies, very easily can be avoided whenever any geometric solution becomes posed in the future that falsely alleges that the classical problem of the trisection of an angle actually can be solved.
Quite simply, such approach consists of determining whether a posed geometric solution is linked to a formally specified problem. If a corresponding problem becomes located, it then should be examined to verify that it:                is entirely consistent with how such classical problem of the trisection of an angle is specified above, in which case such geometric solution therefore must be in error; or        incorporates extraneous information that is relevant, thereby otherwise solving some corrupted version of such classical trisection problem instead.
On the other hand, if it is found that such geometric solution is not associated to any formally specified problem, this means that it cannot be ascertained to any certainty what such posed geometric solution even applies to!
Regrettably, such type mishap is well documented, thereby being known to truly have occurred in the past!
As such, for any literature that claims to have solved the classical problem of the trisection of an angle without indicating exactly what particular issue it has remedied, it might be a good idea to examine:                whether such erroneous claim was met by a great fanfare that credited its author for an outstanding discovery.        whether a geometric solution for some corrupted version of the classical problem of the trisection of an angle was provided instead, without expressing the exact nature of extraneous information which such solution was based upon; or        whether some totally unrelated type of trisection solution was identified due to the discovery of some invention; thereby only serving to expand upon the overall scope of aforementioned trisection matters as cited in connection with such attendant trisection mystery.
The third riddle is: What other types of trisection solutions, besides geometric solutions, are there?
The answer is: Motion related solutions!
These consist of various events, as opposed to activities where singular geometric patterns otherwise become drawn, whereby an invention becomes set to a singular angle of virtually any designated size and its associated trisector automatically becomes portrayed forthwith.
Such type of invention cannot solve the classical problem of the trisection of an angle because its features are quite different from a mere straightedge and compass that instead individually must be applied to a single piece of paper without otherwise violating any imposed requirements. Hence, any trisection portrayal of this nature clearly would qualify as a corrupted motion related solution of the classical problem of the trisection of an angle.
The fourth riddle is: Can an iteration process of successive angular bisections, as presented in complete detail later, which clearly complies with all of the requirements imposed by the classical problem of the trisection of an angle actually solve it?
The answer again emphatically is no!
In particular, this is because:                an infinite number of iterations would have to be performed in order to generate an exact geometric solution; thereby qualifying as a task that could never be fully completed; and        the resolution of the naked eye to distinguish actual drawing separations would become impaired very shortly after commencing upon such iteration process, whereby successive bisectors then would appear to run over themselves, requiring larger arcs to be drawn in order to avail added viewing relief. Eventually, the very straightedge and compass instruments themselves, along with the paper needed to avail such precision could not be manufactured due to the massive increases in their sizes which would be needed to maintain such viability.
In conclusion, it is impossible to solve the classical problem of the trisection of an angle when explicitly complying with all of its requirements!
Important forerunners of trisection, hereinafter listed chronologically by the dates of their conception include:                geometric construction, dating all the way back to ancient Babylonian times around the year 3000 BC, in which only a straightedge and compass are permitted for use when describing straight lines, angles, and circular arcs;        intersection points, as established during the same time period, in which discrete positions become completely distinguishable wherever either straight lines, arcs of circles, or straight lines in combination with the arcs of circles cross one another. Center points of circles also qualify as intersection points because they describe common locations where geometrically constructed perpendicular bisectors of respective chords of such circles meet;        the Pythagorean Theorem, as developed in or about the year 500 BC, in which the square of the length of the hypotenuse of a right triangle is proven to be equal to the sum of the squares of the lengths of its two sides;        conventional Euclidean practice, as established prior to the year 265 BC, in which definitions and rules describe the very manner in which geometric construction may be administered, and axioms identify certain geometric relationships which become evident after conducting specific geometric construction operations. Directly below, two principal definitions which further characterize such practice are specified, followed by three distinct examples of its rules:                    its most basic rule principally states that a given set of previously defined geometric data must be furnished that specifies the locations of initial positions from which geometric construction may be launched. A designated angle of sixty degree magnitude very well could be expressed as such given data. Moreover, such definition furthermore can apply to locations that are not entirely distinguishable solely by geometric construction. For example, given data might specify a twenty degree angle; one whose magnitude only could be approximated by means of geometric construction;            its cardinal rule essentially stipulates that geometric construction must proceed either from a given set of previously defined geometric data, or from other locations which become distinguishable with respect to such positions by means of applying only a straightedge and compass to them;            at least two points must be specified in order to draw a straight line;            at least two points must be specified in order to draw a circle when one of those points designates its center point; and            at least three points must be specified in order to draw a circle when none of those points denotes its center point;                        conventional Euclidean means, whose terminology became commonplace shortly thereafter, in which geometric construction is to be implemented in strict accordance with the specific definitions and rules stipulated by conventional Euclidean practice;
Euclidean commands, also taking effect during the same time period, in which various instructions describe exactly how various straight lines and circular arcs are to be geometrically constructed with respect to identified positions;
a sequence of Euclidean operations, as conceived during the same timeframe, in which a specific set of Euclidean commands, enumerated as a series of discrete steps become executed in consecutive order with respect to a given set of previously defined geometric data in order to distinguish various rendered positions;
a geometric construction pattern, as introduced during that time, in which the specific features of a given set of previously defined geometric data, in combination with any rendered positions which become established by means of executing an attendant sequence of Euclidean operations with respect to it, as well as any additional straight lines drawn between such established locations or circular extensions made to them become depicted within a single drawing;
an Archimedes proposition, as devised prior to 212 BC, introduced as Proposition 8 in the Book of Lemmas, then later translated from Arabic into Latin in 1661, and finally published in the English language in 1897, in which it is stated on page 309 in The Works of Archimedes that if AB be any chord of a circle whose centre is 0, and if AB be produced to C so that BC is equal to the radius; if further CO meets the circle in D and be produced to meet the circle a second time in E, the arc AE will be equal to three times the arc BD;
an Archimedes proof for such Proposition 8, as appears on page 310 in The Works of Archimedes, in which it is stated:                draw the chord EF parallel to AB, and join radius OB, OF;∠COF=2∠OEF =2∠BCO, by parallels,=2∠BOD, since BC=BO;         therefore, ∠BOF=3∠BOD, so that the arc BF is equal to three times the arc BD; and        hence the arc AE, which is equal to the arc BF, must be equal to three times the arc BD;        an Archimedes formulation, as evidenced only on a few sporadic occasions in the past, as typically accompanied by only partial documentation, in which a multitude of distinct Archimedes geometric construction patterns, qualifying as such because they conform to all requirements posed in such Archimedes proposition, become represented upon a single drawing. Such representation is made possible by strategically placing a Greek letter either within or alongside what later will be shown to be the given angle of such sole diagram, thereby distinguishing it to be an entire formulation in itself, instead of a mere singular geometric construction pattern. Once ascribing a specific designation, such as θ or φ at such location, it is to mean that such given angle is allowed to vary infinitesimally in size over some prescribed range of values. By means of administering a specific sequence of Euclidean operations to each of such given angles—one whose commands account for all of the individual requirements posed in such aforementioned Archimedes proposition, all of the varying shapes which emerge thereby must qualify as legitimate Archimedes geometric construction patterns; and        an Al-Mahani categorization, as derived prior to the year 900 AD, in which square root quantities become classified as quadratic irrational numbers, thereby distinguishing them apart from rational numbers.
Over the years, both mathematicians, as well as inventors alike have been somewhat awed by the spectacle of an incredibly perplexing trisection mystery whose vital secrets evidently have escaped detection.
Nevertheless, both parties are acutely aware that the diminution of an angle to one-third its actual size, although being truly indicative of trisection, cannot be obtained simply by reversing the sequence of Euclidean operations which governs the geometric construction of a pattern that complies with all of the requirements imposed by such Archimedes proposition.
Rabid speculation concerning such unexplainable dichotomy eventually gave rise to contrasting interpretations, reflective of the particular leanings of various involved personages, outlined as follows:                on the one hand, mathematicians even today remain splintered over how to explain trisection in terms of conventional Euclidean practice, whereby their discordant positions concerning such geometrical matters are expressed as follows:                    one traditionalist camp contends that trisection solely via conventional Euclidean means is entirely unsolvable; whereby it becomes utterly impossible to divide an angle into three parts solely by means of applying only a straightedge and compass to it; whereas            another non-patronizing faction instead advocates that certain angles, in fact, can be trisected solely via straightedge and compass; whereas                        on the other hand, inventors were the first members in society to demonstrate that trisection could be achieved by imparting certain prescribed motions.
The latter of such two geometrical claims, essentially alleging that certain angles can be trisected solely via straightedge and compass unfortunately has managed to gain widespread notoriety throughout the world today, thereby flourishing in the form of independent airings by a coterie of indulging sources and journalists, as well as being supplemented by a rampant proliferation of publications by self-proclaimed mathematical experts who already have accepted such premise as being generally established fact.
Be that as it may, nothing could be further from the truth!
This becomes evident by paraphrasing such latter stated mathematical claim into its only possible correct interpretation. This is accomplished by means of inserting the following bracketed italicized words to its initial substance, thereby asserting that certain angles, in fact, can be trisected solely via straightedge and compass [so long as the very magnitudes of their respective trisecting angles become disclosed beforehand]—thereby essentially solving a corrupted version of the classical problem of the trisection of an angle.
Such above stated clarification achieves two specific objectives, itemized as follows:                it completely eliminates the potential for misconstruing such latter stated mathematical claim to mean that in certain cases, trisection solely via conventional Euclidean means is entirely possible Obviously, such faulty presumption might be harmful because it very easily could be considered to refute, or contradict such former factually correct claim, as asserted above by such aforementioned traditionalist camp of mathematicians; and        it evidences that such latter mathematical clause really has nothing at all to do with trisection solely via conventional Euclidean means; thereby removing the intended stigma out of such statement entirely. As such, its true interpretation with such bracketed input included thereby becomes reduced to the rather insignificant equivalent meaning that all angles which can be described solely via straightedge and compass furthermore constitute respective trisecting angles of other angles whose respective magnitudes amount to exactly three times their size. Such result is of little consequence too considering that it furthermore can be broken down into the mere definition of a trisector, in concert with the understanding that certain angles effectively can be reproduced solely via straightedge and compass once having knowledge of their trigonometric properties.
It should be emphasized that the only way to input outside information which is relevant to the classical problem of the trisection of an angle is by means of violating the cardinal rule of conventional Euclidean practice!
By definition, that is because such input must consist of data that cannot be derived by means of launching geometric construction operations exclusively from a singular angle of designated magnitude which is intended to be trisected. Hence, such corrupting input must constitute that which cannot be distinguished by conventional Euclidean means.
As such, it is concluded that the classical problem of the trisection of an angle cannot be solved by conventional Euclidean practice; thereby exposing its very limitations!
Furthermore, calculating the size of a trisector merely by dividing the designated magnitude of an angle that is intended to be trisected by a factor of three is not permitted because such action cannot be duplicated by a geometric construction process which is governed by Euclidean rules.
Below, a composite history describing the very first important trisecting events recorded in the English speaking language is afforded; whereby trisection was observed to occur on three separate occasions as unique articulating mechanisms became invented.
During such first of such documented incidents, having taken place sometime during the late 1870's, Alfred Kempe discovered that so-called anti-parallelograms could be used for purposes of regulating motion!
One of such Kempe masterful designs, truly considered to be capable of performing trisection, is depicted as prior art in FIG. 1A.
Therein, an overall formation which could have been fashioned to have either rounded or pointed corners is not displayed. Only the longitudinal centerlines of its eight linkages and radial centerlines of its eight interconnecting pivot pins are depicted instead because only these portions of such device govern trisection!
Such Kempe prior art features a basic fan array whose movement is controlled by three independent anti-parallelograms; the complete breakdowns of which are described as follows:                its fan array portion consists of four separate linkages, modeled as straight lines BA, BD, BE, and BC in FIG. 1A; all of which are hinged together by an interconnecting pivot pin whose shank passes through an end portion of each. In particular, the longitudinal centerlines of such four basic fan array linkages overlap one another along the radial centerline of such interconnecting pivot pin, as illustrated by a very small circle drawn located at their juncture about axis B; and        its control section features anti-parallelograms BFGH, BGJI, and BJLK, so denoted by identifying their diagonal compositions, rather than by enumerating their respective corners in consecutive order.
Such control section serves to maintain the three angles interposed between adjacent longitudinal centerlines of the four linkages, where they more particularly radiate about the hub of such basic fan array portion, at a constant magnitude; even as such sizes vary during device flexure.
Accordingly, such three interposed angles relate to one another according to the algebraically expressed equality ∠ABD=∠DBE=∠EBC=θ.
Whereas all three subtended angles in combination constitute angle ABC, its magnitude must amount to 3θ.
In order to trisect angle ABC, such Kempe device first must be set to a designated magnitude. In this particular case, the size of such setting is shown to be that of angle ABC, as it actually appears in FIG. 1A. Considering that a varying magnitude of such angle ABC might have been selected instead, such drawing would have had to assume an entirely different overall shape in order to compensate for such change.
In effect, every time such Kempe device becomes set to a different preselected magnitude, each of such three subtended angles, nonetheless, automatically portray its actual trisector; whereby the very process of portraying an angle whose magnitude amounts to exactly one-third the size of an angle of designated magnitude is indicative of trisection!
In total, such Kempe device is comprised of eight linkages whose longitudinal centerlines are modeled as straight lines in FIG. 1A, in addition to eight interconnecting pivot pins whose radial centerlines are denoted by very small circles therein.
Extensions made to anti-parallelogram linkage members BF and BL therein enable easier access for flexing device arm BC with respect to linkage member BA during articulation; whereas extensions made to anti-parallelogram linkages members BG and BJ enable a better viewing of trisecting members BD and BE.
All told, FIG. 1A depicts longitudinal centerlines of linkages and radial centerlines of interconnecting pivot pins which collectively constitute such Kempe mechanism.
Of note, various other trisecting devices exist which also are contrived of a four linkage and interconnecting pivot pin arrangement which are considered to conform to that of such basic fan array, as described in detail above. However, they each apply control mechanisms that are entirely different than the anti-parallelogram linkage arrangements, as previously developed by Mr. Kempe.
By means of classifying mechanisms such as these into a singular category, it becomes possible to validate the striking geometrical resemblance which exists between them. Once grouped together, their uniqueness thereby can be substantiated by means of describing how their control mechanisms differ from each other.
Such categorization is necessary for the very same reason that it previously has been applied in biology; namely, to suitably characterize the very diversity which exists between living things that exhibit common physical traits.
Likewise, by means of comparing trisecting mechanisms which exhibit similar geometries, proper conclusions can be drawn concerning both how and why they relate to one another, as well as how they fundamentally differ!
What should bond trisecting mechanisms together is a common geometry which they all share.
For example, the geometry of such basic fan array design is perhaps the simplest in all of mathematics to comprehend because it simply consists of a given angle that becomes duplicated twice, such that all three angles become grouped together at a common vertex in order to eventually geometrically construct a rendered angle whose magnitude amounts to exactly three times the size of such given angle.
A singular geometric construction pattern, as represented in FIG. 1A, furthermore could qualify as an entire formulation because the sequence of Euclidean operations from which such distinctive geometric construction pattern stems furthermore could be applied to virtually any sized given angle, as algebraically denoted therein by the Greek letter θ; thereby governing the various positions which the longitudinal centerlines of linkages and radial centerlines of interconnecting pins which collectively constitute such Kempe device thereby would assume as it becomes articulated.
In support of such logic, it is recommended that any articulating trisecting mechanism which exhibits distinctive fan shape features should be classified as a CATEGORY I type device. Its complete inventory shall consist of:                CATEGORY I, sub-classification A articulating trisection devices which feature four linkages of equal length, hinged together by an interconnecting pivot pin that is passed through one end portion of each, thereby collectively constituting the array of a fan whose two inner linkages become regulated in some fashion so that their longitudinal centerlines divide the angle subtended by the longitudinal centerlines of its two outer linkages into three equal portions throughout device flexure; and        CATEGORY I, sub-classification B articulating trisection devices which feature three linkages of equal length, hinged together by an interconnecting pivot pin that is passed through one end portion of each, either of whose outer linkages instead could be represented by a straight line that is impressed upon a piece of paper or board, thereby collectively constituting the array of a fan whose single inner linkage becomes regulated in some fashion so that its longitudinal centerline trisects the angle subtended by the longitudinal centerlines of its two outer linkages throughout device flexure.
With particular regard to FIG. 1A, each of the distinct Euclidean commands required to suitably locate straight line BC with respect to given angle ABD therein, solely by Euclidean means, is specified below, thereby together comprising the complete twenty-one steps of its sequence of Euclidean operations:                step 1—given angle ABD, of arbitrarily selected magnitude θ, is drawn such that its side BA is constructed to be of equal length to its other side BD;        step 2—point F is arbitrarily selected somewhere along side BA of given angle ABD;        step 3—a circle is drawn about vertex B whose radius is arbitrarily selected to be less than length BF, whereby a portion of its circumference becomes designated as the FIRST CIRCULAR ARC in FIG. 1A;        step 4—The intersection between such FIRST CIRCULAR ARC and side BD of given angle ABD becomes designated as point G;        step 5—a second circle is drawn about point F whose radius is set equal in length to straight line segment BG, a portion of whose circumference is designated as the SECOND CIRCULAR ARC in FIG. 1A;        step 6—a third circle is drawn about point G whose radius is set equal in length to straight line segment BF, a portion of whose circumference is designated as the THIRD CIRCULAR ARC in FIG. 1A;        step 7—the intersection point between such SECOND CIRCULAR ARC and THIRD CIRCULAR ARC is designated as point H;        step 8—diagonal GH and side segment FH of anti-parallelogram BFGH are drawn in order to complete its geometry;        step 9—an angle whose magnitude is equal to that of angle FBH is geometrically constructed with its vertex placed at point B such that its counterclockwise side becomes aligned along straight line BD, whereby the intersection of its clockwise side with diagonal GH becomes designated as point I;        step 10—a fourth circle is drawn about point I, whose radius is set equal in length to line segment BG, a portion of whose circumference becomes designated as the FOURTH CIRCULAR ARC in FIG. 1A;        step 11—a fifth circle is drawn about point B, whose radius is set equal in length to line segment IG, a portion of whose circumference is designated as the FIFTH CIRCULAR ARC in FIG. 1A;        step 12—the intersection point between such FOURTH CIRCULAR ARC and FIFTH CIRCULAR ARC becomes designated as point J;        step 13—straight line diagonal IJ and side segment BJ are drawn in order to complete an additional anti-parallelogram BGJI;        step 14—straight line BE is geometrically constructed to be equal in length to side BA of given angle ABD such that it aligns with side segment BJ of such additional anti-parallelogram BGJI by means of passing through point J, whereby it serves as an extension to it;        step 15—an angle whose magnitude is equal to that of angle GBI is geometrically constructed with its vertex placed at point B such that its counterclockwise side becomes aligned along straight line BE whereby the intersection of its clockwise side with diagonal JI becomes designated as point K;        step 16—a sixth circle is drawn about point K, whose radius is set equal in length to line segment BJ, a portion of whose circumference is designated as the SIXTH CIRCULAR ARC in FIG. 1A;        step 17—a seventh circle is drawn about vertex B of given angle ABD whose radius is set equal in length to line segment KJ, a portion of whose circumference is designated as the SEVENTH CIRCULAR ARC in FIG. 1A;        step 18—the intersection point between such SIXTH CIRCULAR ARC and SEVENTH CIRCULAR ARC is designated as point L;        step 19—straight line diagonal KL and side segment BL are drawn in order to complete the third and last of such anti-parallelograms, being duly notated as BJLK;        step 20—straight line BC is geometrically constructed to be equal in length to side BA of given angle ABD such that it aligns with side segment BL of anti-parallelogram BJLK by means of passing through point L, thereby serving as an extension to it; and        step 21—whereas side BA is constructed to be of equal length to side BD of given angle ABD, and straight lines BE and BC are geometrically constructed to be equal in length to such side BA, all four straight lines furthermore represent radii of a circle that all emanate from center point B.
Verification that angle ABC, as depicted in FIG. 1A, is equal to three times the size of given angle ABD is provided in the following twenty-five step proof:                step 1—by construction, straight line HG is equal in length to straight line BF, and straight line GB is equal in length to straight line FH;        step 2—by identity, straight line BH is equal in length to straight line HB;        step 3—then, since the three sides represented in triangle HGB are of equal length to the corresponding sides of triangle BFH, such triangles must be congruent to each other;        step 4—by construction, straight line IJ is equal in length to straight line BG, and straight line JB is equal in length to straight line GI;        step 5—by identity, straight line BI is equal in length to straight line IB;        step 6—then, since the three sides represented in triangle IJB are of equal length to the corresponding sides of triangle BGI, such triangles must be congruent;        step 7—by construction, straight lines KL and LB respectively are equal in length to straight lines BJ and JK, and by identity, straight line BK is equal in length to straight line KB;        step 8—then, since the three sides represented in triangle KLB are of equal length to the corresponding sides of triangle BJK, such triangles must be congruent to each other;        step 9—since triangle BFH is congruent to triangle HGB, each of their corresponding angles must be of equal magnitudes, such that ∠FBH=∠GHB;        step 10—but, by construction ∠GBI=∠FBH;        step 11—then, by substitution ∠GBI=∠GHB;        step 12—by identity, ∠BGI=∠BGH;        step 13—since ∠GBI=∠GHB, in addition to the fact that ∠BGI=∠BGH, triangle BGI and triangle HGB contain two sets of angles whose respective magnitudes are of equal values, whereby such triangles must be similar to one another;        step 14—because triangle IJB is congruent to a triangle BGI which, in turn, is similar to triangle HGB, triangle IJB also must be similar to triangle HGB;        step 15—whereby angle IBJ must be equal to corresponding angle HBG;        step 16—because triangle BGI is congruent to triangle IJB, each of their corresponding angles must be of equal magnitudes, such that ∠GBI=∠JIB;        step 17—but, by construction ∠JBK=∠GBI;        step 18—then, by substitution ∠JBK=∠JIB;        step 19—by identity, ∠BJK=∠BJI;        step 20—since ∠JBK=∠JIB, in addition to the fact that ∠BJK=∠BJI, triangle BJK and triangle IJB contain two sets of angles whose respective magnitudes are of equal values, whereby such triangles must be similar to one another;        step 21—because triangle KLB is congruent to a triangle BJK which, in turn, is similar to triangle IJB, triangle KLB also must be similar to triangle IJB;        step 22—whereby angle KBL must be equal to corresponding angle IBJ;        step 23—since the whole is equal to the sum of its parts,∠HBA+∠ABD=∠HBD ∠ABD=∠HBD−∠HBA ∠ABD=∠HBD−∠HBF ∠ABD=∠HBD−∠FBH;         step 24—whereby the following expression is obtained by substituting relevant previously determined equalities and identities ∠HBD=∠HBG, ∠IBJ=∠IBE, ∠KBJ=∠KBE, ∠KBL=∠KBC, and ∠IBG=∠IBD, and by reversing the order of the three letter designators of certain angles without influencing the values of their respective magnitudes, such that:
                              ∠                                                              ABD                =                                                                ∠                                                                          HBD                    -                      ∠                                                                          FBH                                                  =                              ∠                                                                          HBG                    -                      ∠                                                                          FBH                                                  =                              ∠                                                                          IBJ                    -                      ∠                                                                          GBI                                                  =                              ∠                                                                          IBE                    -                      ∠                                                                          IBG                                                  =                                            ∠                                                                                IBE                        -                          ∠                                                                                IBD                                =                      ∠                                                                          DBE                                                  =                              ∠                                                                          IBJ                    -                      ∠                                                                          GBI                                                  =                              ∠                                                                          KBL                    -                      ∠                                                                          JBK                                                  =                              ∠                                                                          KBC                    -                      ∠                                                                          KBJ                                                            =                                                    ∠                                                                                KBC                            -                              ∠                                                                                KBE                                      =                          ∠                                                                                EBC                                      ;            and                step 25—since angle ABC is comprised of angle ABD, angle DBE, and angle EBC, the following expression demonstrates that angle ABC amounts to exactly three times the size of given angle ABD:
                              ∠                                                              ABC                =                              ∠                                                                          ABD                    +                      ∠                                                                          DBE                    +                      ∠                                                                          EBC                                                  =                              ∠                                                                          ABD                    +                      ∠                                                                          ABD                    +                      ∠                                                                          ABD                                                  =                  3                                                              ∠                                                              ABD                                        =                  3                                θ            .
Notice that such above digression applies equally as well to varying magnitudes of given angle ABD since each successive step listed in such sequence of Euclidean operations and accompanying proof acts upon prior information that commences directly from such angle ABD, no matter what its initial size. That is to say, according to such proof, angle ABC still would be of magnitude 3θ, even if given angle ABD were to become varied to some degree in overall size. During such conditions, FIG. 1A would assume reconstituted shapes that depict actual Kempe device reconfigurations after being articulated to different positions.
In other words, by defining the magnitude of given angle ABD algebraically, instead of just assigning a singular value to it, such FIG. 1A thereby describes all of the various attitudes which such invention could assume during articulation. Such multiple number of actual drawings, thereby collectively comprising an entire formulation, essentially consists of an amalgamation of renderings which are generated solely by administering a specific sequence of Euclidean operations to a given angle ABD which is allowed to vary in size while, in each case, ultimately rendering an angle ABC which amounts to exactly three times the size of such given angle ABD.
As such, the three Kempe anti-parallelograms incorporated into such mechanism, as represented in FIG. 1A, thereby serve to regulate, or strictly control the overall movement of such device, whereby trisecting angles thereby become portrayed as angle ABC becomes altered to an infinite variety of angles that are in need of being trisected.
A second significant trisection development also was reported in such 1897 publication entitled The Works of Archimedes wherein it was claimed that a marked ruler arrangement placed upon a particular geometric drawing in a prescribed manner could achieve trisection. In order to account for the enormous lapse of time which had preceded such publication, page cvi therein stipulates that some such similar operation might have contributed to the development of the conchoid; a curve considered to have been discovered by Nicomedes somewhere between 200 BC and 70 BC.
Such unusual application is represented as prior art in FIG. 1B such that straight line MR describes the longitudinal centerline of such marked ruler device. The mechanism clearly qualifies as a ruler, as opposed to just a straightedge, simply because it contains a notch etched a specific distance away from one of its tips. This is denoted by the very small circle appearing upon such drawing at point N.
Therein, angle QPS represents an angle that is to become trisected, suitably drawn upon a separate piece of paper and specifically sized to be of algebraically expressed 3θ designed magnitude. Its sides QP and SP both are drawn to be equal in length to the distance which such notch resides away from the left endpoint of such marked ruler device. Thereafter, a circle is drawn whose center point is located at vertex P of angle QPS such that its circumference passes both through point Q and point S. Lastly, side SP is extended a sufficient length to the left, thereby completing such geometric construction pattern.
In order to trisect angle QPS, such marked ruler device thereby becomes indiscriminately jockeyed above such piece of paper until such time that all three of the following listed events occur:                its longitudinal centerline overlaps point Q;        its notch aligns upon the circumference of such drawn circle; and        its left endpoint hovers directly over straight line SP extended.
Therein, point N designates the position where such notch hovers above such previously drawn circle.
As validated below, once such device becomes set in this particular manner, angle RMP constitutes an actual trisector of angle QPS. Its respective sides consist of the longitudinal centerline of such repositioned marked ruler device, as represented by straight line MR in FIG. 1B, along with straight line SP extended, as previously drawn upon a piece of paper that such marked ruler device now rests upon. Therein, trisector RMP is designated to be of magnitude θ, amounting to exactly one-third the size of previously drawn angle QPS.
As depicted in FIG. 1B, angle RMP can be proven to be an actual trisector for angle QPS, by considering that both angles constitute respective given and rendered angles belonging to such famous Archimedes proposition.
The proof that FIG. 1B furthermore constitutes an Archimedes proposition is predicated upon the fact that it meets the description afforded in Proposition 8 in the Book of Lemmas, properly rephrased to state that if QN be any chord of a circle whose center is designated as point P, and if QN be produced to M so that NM is equal to the radius; if further MP produced meets the circle at point S, the arc QS will be equal to three times the arc subtended between straight lines PN and PM. The accompanying proof thereby is given as:
                                                                        ext                .                                                                                ∠                                                                                PNQ                        =                          2                            ∠                                                                                NMP                                ,                                    by                                                                                isos                .                                                                                Δ                                                                                NMP                        ;                                                                    =                          ∠                                                                                PQN                                ,                                    by                                                                                isos                .                                                                                ∠                                                                                PQN                        ;                                                                                ext                .                                                                                ∠                                                                                QPS                        =                                          ∠                                                                                QMP                            +                              ∠                                                                                PQM                                              ,                                    by                                                                                Δ                                                                                MPQ                        ;                                                        =                                    ∠                                                                                NMP                        +                          ∠                                                                                PQN                                                                    =                                    ∠                                                                                NMP                        +                          2                            ∠                                                                                NMP                                                                    =                      3                        ∠                                                                          NMP                                                        =                      3                        ∠                                                                          RMP                                                                    =                          3                            ∠                                                                                NPM                                ,                                                    since                                                                                NM                            =              NP                        ;                              and                hence, the arc QS must be equal to three times the arc subtended between straight lines PN and PM.
Since the specific sequence of Euclidean operations which governs such famous Archimedes formulation furthermore distinguishes the very same geometry as now is represented in FIG. 1B, angle RMP and angle QPS, as posed therein, also must describe its respective given and rendered angles.
With particular respect to FIG. 1B, this is demonstrated as follows, wherein each of the seven steps comprising such specific sequence of Euclidean operations is stipulated directly below:                step 1—given acute ∠RMP, of arbitrarily selected angular size, designated therein as being of magnitude θ, and exhibiting sides of arbitrarily selected lengths, so long as RM is sufficiently longer than MP is first geometrically constructed on a piece of paper, or upon a drawing board;        step 2—given ∠RMP is duplicated such that its vertex is positioned at point P; whereby straight line PM denotes one of its sides, with its other side geometrically constructed so that it resides upon the same side of straight line PM as does remaining side RM of given angle RMP, such that the duplicated angle faces given angle RMP, or opens up towards it;        step 3—point N becomes designated as the intersection point between straight line RM and the additional side drawn by such duplicated angle, thereby completing the geometric construction of isosceles triangle MNP;        step 4—a circle is drawn whose origin is placed at point P which is of radius equal in length to straight line PN;        step 5—straight line MP is extended to point S lying upon the circumference of such formed circle;        step 6—the other intersection point which straight line MR makes with the circumference of such formed circle is designated as point Q; and        step 7—straight line PQ is drawn.
A three step algebraic proof, serving to verify that such geometrically constructed angle QPS is exactly three times the magnitude of given acute angle RMP, for any and all magnitudes which it otherwise might assume is provided as follows:                step 1—because ∠PNQ is as an exterior angle of isosceles triangle MNP, it must be equal to the sum of such triangle's equally sized included angles, denoted therein as ∠NMP and ∠NPM, which algebraically can be summed to θ+θ=2θ;        step 2—since angle NQP and angle PNQ reside opposite the equal length sides of isosceles triangle NPQ, with such sides furthermore representing equal length radii of such drawn circle, they must be of equal magnitude, such that ∠NQP=∠PNQ=2θ; and        step 3—because ∠QPS qualifies as an exterior angle of triangle MPQ, it thereby must be equal to the sum of such triangle's included angles QMP and MQP which, by furthermore being related in the two identities ∠QMP=∠NMP and ∠MQP=∠NQP, can be algebraically summed to amount to θ+2θ=3θ.
Above, it has just been proven that the magnitude of rendered angle QPS always must amount to exactly three times that of given acute angle RMP, even as such given acute angle becomes varied in size
Since the actual value of given acute angle RMP is not specified anywhere in the sequence of Euclidean operations that FIG. 1B is derived from; it can have no bearing, or direct influence upon implementing it. Hence, introducing algebraic nomenclature, as is posed in FIG. 1B, cannot conflict in any way with its administration; thereby enabling such sequence of Euclidean operations to be applied many times over in order to generate a wide range of patterns that result as given acute angle RMP varies in size.
Along with such marked ruler arrangement, other trisection mechanisms whose designs could be arranged into geometric shapes that either are indicative of such famous Archimedes proposition, or particular adjuncts thereof, such as the configuration depicted in FIG. 1B, hereinafter shall constitute CATEGORY II articulating trisection devices.
A third significant evolution took root during the early 1900's when an ample supply of hinged linkage assemblies, replete with interconnecting pivot pins, summarily became incorporated into a broad spectrum of up-and-coming applications. Such design practice, extending all the way up until present day, remains of paramount importance because it enables an innumerable variety of meaningful motions to become portrayed.
As a consequence of such progress, modern day trisection devices came into existence shortly after inventors figured out how to shape the longitudinal centerlines of linkages and radial centerlines of interconnecting pivot pins which collectively constitute their mechanisms into configurations that are indicative of geometric construction patterns of rendered angles whose magnitudes amount to exactly three times the sizes of their respective given angles.
The very process of trisection became more sophisticated thereafter by slotting such linkages, thereby affording added degrees of freedom. This enabled such designs to travel over an ever increasing range of distinct trajectories that otherwise simply couldn't be duplicated by solid linkage designs of comparable shapes, the latter of which proved also to be both heavier and more costly.
A distinct example of such improved design is presented in FIG. 1C, wherein a pair of slotted linkages are featured which can be used to actuate such device by means of rotating linkage MR in either a clockwise or counterclockwise direction relative to member MS about a hinge located at axis M.
FIG. 1C, was chosen to represent a typical example of such slotted linkage design for the principal reason that the very artwork which is expressed in FIG. 1B precisely pinpoints the particular placements of the longitudinal centerlines of linkages and radial centerlines of interconnecting pivot pins which collectively comprise it.
Because of such association, notice that both FIGS. 1B and 1C denote the very same position letters, being a convenience which should permit for their easy comparison.
Whereas a virtually unlimited number of other geometric construction patterns which also serve to constitute such Archimedes proposition, as represented FIG. 1B, exactly describe a series of repositioned placements of the longitudinal centerlines of linkages and radial centerlines of interconnecting pivot pins which collectively comprise such mechanism, as depicted in FIG. 1C as it becomes actuated, the angular notations algebraically denoted as θ and 3θ in FIG. 1B thereby also are shown to carry over into FIG. 1C.
Except for these angular notations, such FIG. 1C merely is a truncated rendition of prior art. Because its overall geometry can be related in such manner directly to that of FIG. 1B, and thereby furthermore can be associated to such famous Archimedes proposition, it is said to qualify as a CATEGORY II articulating trisection device.
This truncated phenomena becomes quite evident once realizing that such basic Archimedes proposition, as featured in FIG. 1B, furthermore could be supplemented, or added to, merely by means of incorporating successive isosceles triangles onto it, each of whose angles of equal size thereby always would amount to the sum of the magnitudes of the non adjacent included angles of the triangle whose perimeter represents the outer envelope of the combined triangles which preceded it, thereby establishing the following progression:θ+θ=2θ;θ+2θ=3θ;θ+3θ=4θ; andθ+4θ=5θ; etc.
The manner in which such progression could be introduced into such prior art, as posed in FIG. 1C, is indicated by the addition of the link, as depicted by phantom lines, which extends from axis Q therein.
Aside from such superfluous phantom link, upon viewing FIG. 1C, it becomes obvious that it consists of both solid, as well as slotted linkages, along with interconnecting pivot pins, which can described in greater detail as follows:                solid linkages NP and PQ, each of the same length, have circular holes of the exactly the same size drilled through them located very close to each of their ends along their respective longitudinal centerlines. All four holes are located such that the distance between the respective radial centerlines of the holes drilled through linkage NP is the same as that afforded between the respective radial centerlines of the holes drilled through linkage PQ, thereby describing the same bolt hole footprints;        a pivot pin whose radial centerline is located at axis Q is inserted through one of the holes drilled through linkage PQ and then through the slot afforded by linkage MR which thereby constrains its movement such that axis Q always lies along the longitudinal centerline of linkage MR during device articulation; thereby furthermore conforming to the distinct geometric pattern posed in FIG. 1B wherein point Q is shown to reside upon straight line MR;        the longitudinal centerline of extended slotted linkage SP is aligned with that of linkage MR by means of drilling matching circular shaped holes at axis M, located very close to the respective endpoints of such linkages; whereby a second pivot pin then becomes inserted through such matching circular shaped holes. Hence, such axis M conforms to the location of point M, as represented in FIG. 1B, designated as the intersection point between straight lines MR and SP extended; and        another hole is drilled through a portion of linkage MR which does not contain a slot, whose radial centerline both is aligned with the longitudinal centerline of linkage MR, and is offset a fixed distance away from axis M that is equal to the length between the radial centerlines of respective holes which previously were drilled through the respective ends of linkage PQ. A third pivot pin is inserted through one of the holes drilled through linkage NP, and then through the vacant hole drilled through linkage MR. Lastly a fourth pivot pin is inserted through the vacant hole drilled through linkage NP, then through the vacant hole drilled through linkage PQ, and thereafter through the slot of linkage MP.
Based upon such design, lengths MN, NP, and PQ all must be equal; axis N must reside along the longitudinal centerline of linkage MR and upon the circumference of a circle described about axis P of radius PN, and axis P must reside along the longitudinal centerline of slotted linkage MS; thereby conforming to the geometry posed in FIG. 1B, wherein it is shown that corresponding straight lines MN, NP, and PQ again are all of equal length, point N resides along straight line MR and upon the circumference of a circle drawn about point P of radius PN, and point P is situated upon straight line MS.
In conclusion, such device, as represented in FIG. 1C, is considered to be fully capable of portraying angle RMS as a trisector for a wide range of angle QPS designated magnitudes since each discrete setting furthermore could be fully described by a singular geometric construction pattern, such as that which is afforded in such FIG. 1B; as thereby generated by means of imposing a distinct sequence of Euclidean operations that conforms to that which governs such famous Archimedes proposition.
Therefore, once such device becomes set by means of rotating linkage MR with respect to linkage MS so that angle QPS, as posed in FIG. 1C, becomes of particular size 3θ, its associated trisector, represented as an angle RMP that becomes interposed about axis M between the respective longitudinal centerlines of linkages MR and MS, automatically becomes portrayed, being of size θ.
Once a phantom linkage, as depicted in FIG. 1C, becomes incorporated into such prior art, a one-to-four angular amplification thereby becomes realized with respect to angle RMS as forecasted by the progression expressed above.
Due to such design intricacy, such device as depicted in FIG. 1C becomes capable of trisecting angles of various sizes in rapid succession. Such distinct advantage clearly cannot be matched by a marked ruler arrangement that otherwise must perform the repetitious act of reproducing all of the cumbersome motions considered necessary to achieve trisection each and every time an angle of different magnitude becomes slated for trisection.
Mechanisms which fall under the grouping entitled, CATEGORY I, sub-classification B articulating trisection devices feature new designs, and thereby shall be fully described at a later time.
For the particular contingency that other trisection methods yet might become identified in the near future, consisting of different approaches than those which govern such proposed CATEGORY I and CATEGORY II groupings, it is recommended that they too should be classified into suitable categories. For example, one yet to be related method for trisecting angles consists of portraying specific contours that represent a composite of trisecting angles, or aggregate of previously established trisection points for angles whose magnitudes amount to exactly three times their respective size. It would seem only fitting, then, to group together such types of mechanisms into an entirely new CATEGORY III articulating trisection device classification.
All told then, just one final important question still remains largely unaddressed, being that: If a unique motion related solution is required for each and every angle of different designated magnitude that intended to be trisected, then what distinct proofs, or perhaps interrelated set of proofs, would need to be specified in order to substantiate that some newly proposed mechanism could perform trisection accurately throughout its entire range of device settings? As soon will become evident, in order to suitably answer such looming question, a novel methodology, as predicated upon an extension to a limited conventional Euclidean practice that alone is incapable of solving such famous classical problem of the trisection of an angle would need to be established; thereby making what will appear in the following pages revolutionary, rather than merely evolutionary, in nature!

Angle trisector, as validated to perform accurately over a wide range of device settings by a novel geometric forming process; also capable of portraying finite lengths that only could be approximated by means of otherwise applying a compass and straightedge to a given length of unity; that furthermore functions as a level whose inherent geometry could be adapted for many other uses such as being incorporated into the design of a hydraulic car lift


There is provided a method for producing a carboxyl group-containing polymer composition that is excellent in all of the following: dispersibility in water or the like immediately after the preparation, storage stability, and the transparency of a neutral viscous solution thereof. A method for producing a carboxyl group-containing polymer composition, the carboxyl group-containing polymer composition comprising a carboxyl group-containing polymer (A) that is a copolymer of monomers comprising an α,β-unsaturated carboxylic acid (a-1) and a compound (a-2) having at least two ethylenically unsaturated groups per molecule; and a nonionic surfactant (B), the method comprising the step of copolymerizing the monomers, wherein in the step, the nonionic surfactant (B) is added into the system when the polymerization degree of the α,β-unsaturated carboxylic acid (a-1) has reached 70 to 100%.

Conventionally, carboxyl group-containing polymers are used as, for example, thickeners for cosmetics, moisturizers for cataplasms, emulsifiers, suspension stabilizers, or gel bases for batteries and the like. When these carboxyl group-containing polymers are used for these purposes, they are, for example, added to water or the like to prepare homogeneous dispersions, and then neutralized with alkalis. In general, however, because these carboxyl group-containing polymers are powders, they tend to form masses (lumps) when they are dispersed in water or the like. Once lumps are formed, a gel layer is formed on the lump surface, which slows down the rate of penetration of water into the lumps, which makes it difficult to obtain a homogeneous dispersion.
To prevent the formation of lumps, it is known that in the preparation of a carboxyl group-containing polymer by polymerizing an α,β-unsaturated carboxylic acid and the like, a specific amount of at least one compound of polyhydric alcohol fatty acid ester alkylene oxide adducts and polyhydric alcohol fatty acid esters is added, and the polymerization is carried out to obtain a carboxyl group-containing polymer composition having excellent dispersibility in water (see Patent Literature 1).

Method for producing carboxyl-group-containing polymer composition


A shiftable disk clutch or disk brake having two clutch parts which are rotatable relative to one another about a theoretical axis of rotation. Each of the two clutch parts has at least one clutch disk which is stiff against rotation and is rigidly connected with the clutch part so as to be fixed against movement in every direction. A remotely actuatable pressure supply is divided into at least two separate pressure systems, each of which is, by itself, capable of coupling or separating the clutch disks.

1. Field of the Invention
The invention is directed to a shiftable disk clutch or disk brake. The shiftable disk clutch is preferably used in the drivetrain of a ship between a prime mover and a ship's propeller shaft.
2. Discussion of the Prior Art
Known ship drive systems, especially for shuttle tankers, include a diesel engine, especially a two-cycle diesel engine, which is in driving connection with a ship's propeller shaft by means of a flexible, non-shiftable clutch and a tooth clutch. The ship's propeller shaft is provided with an axial pressure bearing for receiving axial thrust or shear forces. The shaft connection arranged between the flexible clutch and the tooth clutch can be provided with a power take-off gear unit (PTO or power take-off drive) for driving an electric generator and/or other units in the ship. During driving operation of the ship, the ship's engine drives the propeller shaft and, by way of the power take-off, also the onboard generator. The ship's propeller shaft must be disconnected from the ship's drivetrain so that the onboard generator can also be driven when the ship is stopped, for example, when the ship is docked. This disconnection is carried out by opening the tooth clutch. However, the tooth clutch can only be opened or closed when both the engine shaft and ship's propeller shaft are at a complete stop.
In the prior art, this opening and closing of the tooth clutch is carried out manually, wherein it is very time-consuming and difficult when closing the tooth coupling to align the teeth of one clutch part with the tooth spaces of the other clutch part and bring them into engagement. For this purpose, the ship's propeller shaft must often be blocked so that it will not be turned by the movement of the water. The ship's driving engine must be driven at a slow rate of rotation and switched off in an exact manner while taking into account the slow-down or run-out path such that the engine shaft aligns the teeth of one clutch part with the tooth spaces of the other clutch part. A further disadvantage of the tooth clutch is that it cannot be remotely operated, but must be actuated manually in situ. The engagement and disengagement of the tooth clutch must be carried out a number of times in a day, for example, if the ship in question is a shuttle tanker.
Another disadvantage of the tooth clutch is that the teeth of one clutch part must have some clearance or play in the tooth spaces of the other clutch part so that they can engage. However, this results in vibration problems and the teeth will strike against one another, leading to material wear and loud noise. When a remotely operated shiftable multiple-disk clutch is used instead of the manually shiftable tooth clutch, there is a danger that the loading of the ship's propeller shaft by the ship's driving engine will suddenly be interrupted in the event of a failure of the clutch closing pressure, so that the ship's driving engine overspeeds and is damaged. Another problem in all drive systems is that there is only limited space available for its installation in the ship or other engine compartment.


A document copy-relationship management system for managing a copy relationship between documents based on document copy operations performed by a document processing device. The system maintains a data base including an ID table for each document recorded in the system according to the identifier of each document and the location and update date and time of each document, and it also maintains a link information table storing data associating the identifier of an original document and its update date and time and the identifier of a copy document and its update date and time. Also, based on copy operations performed by the document processing device, an ID issue routine is executed to provide a new identifier for the copy document, an ID table update routine writes in the ID table according to the newly issued identifier the location and update date and time of the copy document, and a link information table update routine writes in the link information table data associating the identifier of the copy document and its copy date and time and the identifier of the original document and its most recent update date and time.

1. Field of the Invention
The present invention relates to a system for managing a record of documents and of the copy relationship between documents, and specifically to a system for recording the location and the identity of documents that have a copy relationship.
2. Description of the Related Art
As a system for managing the copy relationship between documents, a management apparatus is known that is described in the Japanese Patent Unexamined Publication No. Hei. 5-324438.
In this management apparatus, a link information table is provided for storing the copy relationship between documents by the use of mutual identifiers; when a copy operation has been performed, the copy relationship between documents is recorded in the link information table, and the document names of the respective documents are stored in an ID table corresponding to the document identifiers.
Therefore, the copy relationship between documents is managed in the link information table, and no matter how many times documents have been copied and recopied, by retrieving the link information table, one can easily obtain information about the copy relationship. Also, by retrieving the ID table based on the identifier of a document, the corresponding document name can also be easily obtained.
Here, when considering the actual use environment of documents, from time to time the contents of documents are updated, or their management location is changed. If such updates of document contents and changes of location could be managed, document management would be made more convenient.
However, with the conventional management apparatus, the link information table is simply for storing the identifiers of documents that have a copy relationship, and the ID table is also simply for storing the document names; one cannot perform such management as determining whether the contents of the original document and the copy document are identical to each other, and where each of the documents that have a copy relationship is saved.
Furthermore, in an environment where documents are actually used, a document processing apparatus that comprises a personal computer and so forth, does not only perform the copying of one electronic-data document to another electronic-data document; it performs various types of copy processing, such as copying an electronic-data document to a document printed on a paper medium, copying a document printed on a paper medium to a document printed on another paper medium, and copying a document printed on a paper medium to an electronic-data document.
However, with the conventional management apparatus, the copy relationship between documents with respect to these types of copy processing cannot be managed, and management that is especially suited to the actual use environment of documents is highly desirable.

System for maintaining a record of documents including document copies


An operation for an operation unit is detected. If the operation is detected and a communication unit executes close proximity wireless communication with an external apparatus, a confirmation screen for confirming execution of processing corresponding to the operation for the operation unit is displayed.

Field of the Invention
The present invention relates to a communication technique using an operation unit and a communication unit.
Description of the Related Art
In an information processing apparatus, there is known a function of performing communication using a contactless IC as a short distance wireless communication function. In an apparatus having such function, there is provided a technique of avoiding an input error by limiting input while the user rotates the apparatus (Japanese Patent Laid-Open No. 2012-069045).
As a short distance wireless communication function, for example, there is provided a system for implementing printing and scanning by bringing a terminal with an NFC (Near Field Communication) tag closer to an image forming apparatus with an NFC tag. In this system, the position of the NFC tag mounted on the image forming apparatus may be close to a specific operation key of the image forming apparatus. Therefore, depending on the model of an image forming apparatus, if a terminal is brought closer to the image forming apparatus to use the function, an operation key may respond unintentionally. If, for example, the operation key is a cancel key, an operation error for the cancel key may occur during execution of a job such as a print job or scan job, thereby canceling the job against the user's intention. This results in a disadvantage to the user, and it is thus necessary to take measures against this problem.
In Japanese Patent Laid-Open No. 2012-069045, execution of communication by the communication unit is not a condition for limiting key input. For example, it may be impossible to avoid an operation error when an external apparatus is brought closer to the apparatus for close proximity wireless communication.
The present invention has been made to solve the above problem, and provides a technique capable of preventing processing unintended by the user from being executed in processing involving communication.

Communication apparatus, control method therefor, and storage medium


A method of manufacturing a coil spring and a coil spring manufacturing apparatus are provided that enable precise forming of a coil spring even when various kinds of wire rods are used as a wire rod. This is based on the premise that the wire material being fed out is serially pressed against a rotating roller outer circumferential surface to form the wire rod into a coil shape. Under this premise, as the wire rod is fed out, a rotating roller is rotationally driven by a rotary drive force of a servomotor such that a portion pressed against the wire rod on the rotating roller outer circumferential surface moves toward the advancing side of the wire rod.

In a proposed coil spring manufacturing apparatus, as described in Patent Document 1, a rotating body is rotatably supported as a coil forming tool via a support pin by a support tool, and a wire rod being fed out is serially pressed against an outer circumferential surface of the rotating body so as to form the wire rod into a coil shape while rotating the rotating body by the movement of the wire rod.
This makes it possible to reduce a frictional resistance of the wire rod with respect to the rotating body outer circumferential surface against which the wire rod is pressed and causes a problem of frictional force at the time of forming the wire rod into a coil shape, so that a deterioration in quality can be suppressed even without plating, or applying a lubrication oil to, the wire rod, in forming of a coil spring.

Method of manufacturing coil spring and coil spring manufacturing apparatus


A method for monitoring process parameters in a multiple step process for an article molded from a plastic material containing at least two constituents to determine propensity for adhesion of a subsequently applied coating includes imaging at least one region of the article to examine surface morphology and analyzing the surface morphology to identify one of the constituents. In a preferred embodiment, the method is applied to monitor a manufacturing process for a molded plastic bumper to determine adhesion propensity of a subsequently applied adhesion promoter or paint based on surface morphology of the component. Analysis of the surface morphology includes identifying the presence of rubber globules near the surface layer or skin.

Manufacture of automotive components is continually plagued by seemingly conflicting requirements for engineering materials. For example, various components may be required to possess certain structural characteristics while also being lightweight, durable, aesthetic, and easily manufactured and assembled. Compromises are often made to optimize the most important properties of each component. Alternatively, new materials and processes are continually developed to enhance desirable properties while reducing or eliminating less desirable characteristics.
Plastics have emerged as generally possessing desirable attributes for a variety of automotive applications. Unfortunately, properties which make certain plastics desirable for use in automotive components, such as body panels, fascia, wheel covers, and the like, also result in poor adhesion characteristics for subsequently applied coatings, such as paint or metallic films. The material properties may be altered by introducing additives to enhance adhesion characteristics, varying the coating process, or both.
Once an acceptable material formulation and coating process are developed, it is desirable to implement an appropriate quality assurance procedure to consistently maintain the desirable component characteristics. The quality assurance procedure should identify any nonconforming components as early as possible in the manufacturing process to minimize the expense of time and materials which may be incurred during subsequent manufacturing, or due to customer dissatisfaction manifested in warranty repairs. Preferably, the quality assurance procedure is capable of identifying a particular step or process parameter responsible for a nonconforming component so that prompt corrective action may be instituted whether by the compounder, the part manufacturer (i.e. the molder) or the processor (applying the coating).
Material properties of plastic components may be monitored using any of a number of destructive or nondestructive testing techniques. Destructive testing techniques, such as scanning electron microscopy, are not amenable to use in high-volume manufacturing applications because they are generally time consuming and destroy the part during testing. However, such techniques are useful for gathering various sample data which may be used to investigate material properties or verify on-line testing techniques.
Nondestructive testing of coated or painted surfaces is often performed via visual inspection, either manually or automatically by computer vision/image processing systems. These systems detect defects or flaws which may be present in the coating or on the surface of the substrate. However, changes in plastic material formulation or processing may not be manifested in defects readily apparent from traditional examination of the coating. Undetected changes may then result in subsequent chipping or peeling of the coating from the substrate due to improper adhesion when the component is subjected to impact, abrasion, heat, or the like.

Method for monitoring coating adhesion propensity based on surface morphology


A memory system, memory module and memory device are described. The memory system includes a plurality of the memory modules connected in a series configuration on a first signal path. The first signal path and a second signal path carry memory control and data signals between the memory modules and a memory controller. The memory controller transmits and receives the control signals and data signals on the first and second signal paths. The first and second signal paths are connected together such that the memory modules are connected in a ring configuration. The control signals and data signals travel in opposite directions on the first and second signal paths. The first and second signal paths are shared by both the data signals and the control signals. The memory modules include multi-functional ports, each of which can receive both the control signals and the data signals and output the signals onto the connected signal paths. The memory device in accordance with the invention can include multi-functional conductors or pins which can both receive and output both the control signals and the data signals.

A variety of bus topologies capable of reducing signal distortion have been developed. One of these topologies includes a point-to-point connection. The memory module for a point-to-point connection typically has two ports. One of the ports is used exclusively as an input port, and the other port is used exclusively as an output port. This configuration increases the pin count and, therefore, the size and complexity, of the memory module.

Memory system having two-way ring topology and memory device and memory module for ring-topology memory system