directed distance from a line to a point
perpendicular distance
13690251
Distance from a point to a line The distance (or perpendicular distance) from a point to a line is the shortest distance from a fixed point to any point on a fixed infinite line in Euclidean geometry. It is the length of the line segment which joins the point to the line and is perpendicular to the line. The formula for calculating it can be derived and expressed in several ways. Knowing the shortest distance from a point to a line can be useful in various situations—for example, finding the shortest distance to reach a road, quantifying the scatter on
Distance from a point to a line
13690251
Distance from a point to a line The distance (or perpendicular distance) from a point to a line is the shortest distance from a fixed point to any point on a fixed infinite line in Euclidean geometry. It is the length of the line segment which joins the point to the line and is perpendicular to the line. The formula for calculating it can be derived and expressed in several ways. Knowing the shortest distance from a point to a line can be useful in various situations—for example, finding the shortest distance to reach a road, quantifying the scatter on
Distance from a point to a line
13690252
a graph, etc. In Deming regression, a type of linear curve fitting, if the dependent and independent variables have equal variance this results in orthogonal regression in which the degree of imperfection of the fit is measured for each data point as the perpendicular distance of the point from the regression line. In the case of a line in the plane given by the equation where "a", "b" and "c" are real constants with "a" and "b" not both zero, the distance from the line to a point ("x","y") is The point on this line which is closest to ("x","y")
Distance from a point to a line
17147069
Distance between two straight lines The distance between two straight lines in the plane is the minimum distance between any two points lying on the lines. In case of intersecting lines, the distance between them is zero, because the minimum distance between them is zero (at the point of intersection); whereas in case of two parallel lines, it is the perpendicular distance from a point on one line to the other line. Because the lines are parallel, the perpendicular distance between them is a constant, so it does not matter which point is chosen to measure the distance. Given the
Distance between two straight lines
5082253
by , and , as in (2). In (3), the axes are rotated to give an isometric view. The triangle, viewed face-on, appears equilateral. In (4), the distances of from lines , and are denoted by , and , respectively. For any line in vector form ( is a unit vector) and a point , the perpendicular distance from to is In this case, point is at Line has Using the perpendicular distance formula, Similar calculation on lines and gives This shows that the distance of the point from the respective lines is linearly proportional to the original values ,
Ternary plot
14849014
Pedal equation For a plane curve "C" and a given fixed point "O", the pedal equation of the curve is a relation between "r" and "p" where "r" is the distance from "O" to a point on "C" and "p" is the perpendicular distance from "O" to the tangent line to "C" at the point. The point "O" is called the "pedal point" and the values "r" and "p" are sometimes called the "pedal coordinates" of a point relative to the curve and the pedal point. It is also useful to measure the distance of "O" to the normal formula_1
Pedal equation
14849019
was discovered by P. Blaschke in 2017. As an example consider the so-called Kepler problem, i.e. central force problem, where the force varies inversely as a square of the distance: Pedal equation For a plane curve "C" and a given fixed point "O", the pedal equation of the curve is a relation between "r" and "p" where "r" is the distance from "O" to a point on "C" and "p" is the perpendicular distance from "O" to the tangent line to "C" at the point. The point "O" is called the "pedal point" and the values "r" and "p" are
Pedal equation
516633
from a point "A" to a point "B" should be distinguished from the straight-line distance from "A" to "B". For example, whatever the distance covered during a round trip from "A" to "B" and back to "A", the displacement is zero as start and end points coincide. In general the straight-line distance does not equal distance travelled, except for journeys in a straight line. Directed distances can be determined along straight lines and along curved lines. Directed distances along straight lines are vectors that give the distance and direction between a starting point and an ending point. A directed distance
Distance
516634
of a point "C" from point "A" in the direction of "B" on a line "AB" in a Euclidean vector space is the distance from "A" to "C" if "C" falls on the ray "AB", but is the negative of that distance if "C" falls on the ray "BA" (I.e., if "C" is not on the same side of "A" as "B" is). I.e. the directed distance from the New York City Main Library flag pole to the Statue of Liberty flag pole has: Another kind of directed distance is that between two different particles or point masses at a
Distance
516636
and "B" can indicate the sense, if the ordered sequence ("A", "B") is assumed, which implies that "A" is the starting point. A displacement (see above) is a special kind of directed distance defined in mechanics. A directed distance is called displacement when it is the distance along a straight line (minimum distance) from "A" and "B", and when "A" and "B" are positions occupied by the "same particle" at two "different instants" of time. This implies motion of the particle. The distance traveled by a particle must always be greater than or equal to its displacement, with equality occurring
Distance
445612
the vector represents a directed distance or displacement from a point "A" to a point "B" (see figure), it can also be denoted as formula_7 or "AB". Especially in literature in German it was common to represent vectors with small fraktur letters as formula_8. Vectors are usually shown in graphs or other diagrams as arrows (directed line segments), as illustrated in the figure. Here the point "A" is called the "origin", "tail", "base", or "initial point"; point "B" is called the "head", "tip", "endpoint", "terminal point" or "final point". The length of the arrow is proportional to the vector's magnitude,
Euclidean vector
386624
a straight line, such that the acceleration is directed towards the center of the motion and also proportional to the displacement from that point. The motion of a particle moving along a straight line with an acceleration whose direction is always towards a fixed point on the line and whose magnitude is proportional to the distance from the variable point is called simple harmonic motion [SHM]. In the diagram, a simple harmonic oscillator, consisting of a weight attached to one end of a spring, is shown. The other end of the spring is connected to a rigid support such as
Simple harmonic motion
825480
the distance to the nearest point on that line. That is the point at which a segment from it to the given point is perpendicular to the line. Likewise, the distance from a point to a curve is measured by a line segment that is perpendicular to a tangent line to the curve at the nearest point on the curve. Perpendicular regression fits a line to data points by minimizing the sum of squared perpendicular distances from the data points to the line. The distance from a point to a plane is measured as the length from the point along
Perpendicular
13690263
formula_39 perpendicular to the line. The distance from the point to the line is then just the norm of that vector. This more general formula is not restricted to two dimensions. If the vector space is orthonormal and if the line ("l" ) goes through point A and has a direction vector formula_45, the distance between point P and line ("l") is where formula_47 is the cross product of the vectors formula_48 and formula_45 and where formula_50 is the vector norm of formula_45. Note that cross products only exist in dimensions 3 and 7. Distance from a point to a
Distance from a point to a line
651309
"n"-space, then the distance (d) from p to q, or from q to p is given by the Pythagorean formula: The position of a point in a Euclidean "n"-space is a Euclidean vector. So, p and q may be represented as Euclidean vectors, starting from the origin of the space (initial point) with their tips (terminal points) ending at the two points. The Euclidean norm, or Euclidean length, or magnitude of a vector measures the length of the vector: where the last expression involves the dot product. Describing a vector as a directed line segment from the origin of the
Euclidean distance
2788841
point "A". Mark point "B" at a distance "r" from it, and join to form a line segment. Mark point "C" in a second, orthogonal, dimension at a distance "r" from both, and join to "A" and "B" to form an equilateral triangle. Mark point "D" in a third, orthogonal, dimension a distance "r" from all three, and join to form a regular tetrahedron. And so on for higher dimensions. These are the regular simplices or simplexes. Their names are, in order of dimensionality: Begin with a point "A". Extend a line to point "B" at distance "r", and join
Regular polytope
19315493
directed line (the "x"-axis) and after that many choices exist. Axial coordinates "x" and "y" are found by constructing a "y"-axis perpendicular to the "x"-axis through the origin. Like in the Cartesian coordinate system, the coordinates are found by dropping perpendiculars from the point onto the "x" and "y"-axes. "x" is the distance from the foot of the perpendicular on the "x"-axis to the origin (regarded as positive on one side and negative on the other); "y" is the distance from the foot of the perpendicular on the "y"-axis to the origin. Every point and most ideal points have axial
Coordinate systems for the hyperbolic plane
2760501
possible to the real view, the line of sight should be directed to its center. On a plan drawing, the line of sight should correspond to the bisector of the angle containing the extreme projections of the object(s) to the station point. The area on a plan or on an elevation or section, between the pair of lines that correspond to the extremes of the view and converge at the station point is called the drawing´s visual angle and its breadth depends on the distance from the observer to the object(s). The line of sight should always bisect the observer´s
Perspective projection distortion
13690260
length of this projection is given by: Now, thus Since "Q" is a point on the line, formula_27, and so, It is possible to produce another expression to find the shortest distance of a point to a line. This derivation also requires that the line is not vertical or horizontal. The point P is given with coordinates (formula_29). The equation of a line is given by formula_30. The equation of the normal of that line which passes through the point P is given formula_31. The point at which these two lines intersect is the closest point on the original line
Distance from a point to a line
13690259
formula_19, and formula_20in terms of the coordinates of P and the coefficients of the equation of the line to get the indicated formula. Let "P" be the point with coordinates ("x", "y") and let the given line have equation "ax" + "by" + "c" = 0. Also, let "Q" = ("x", "y") be any point on this line and n the vector ("a", "b") starting at point "Q". The vector n is perpendicular to the line, and the distance "d" from point "P" to the line is equal to the length of the orthogonal projection of formula_21 on n. The
Distance from a point to a line
651313
unit of length and a direction along the line to call positive. The second point is then uniquely determined as the point on the line that is at a distance of one positive unit from the origin. The distance between any two points on the real line is the absolute value of the numerical difference of their coordinates. It is common to identify the name of a point with its Cartesian coordinate. Thus if "p" and "q" are two points on the real line, then the distance between them is given by: In one dimension, there is a single homogeneous,
Euclidean distance
13690262
equation of a line can be given in vector form: Here is the position of a point on the line, and is a unit vector in the direction of the line. Then as scalar "t" varies, gives the locus of the line. The distance of an arbitrary point to this line is given by This formula can be derived as follows: formula_39 is a vector from to the point on the line. Then formula_40 is the projected length onto the line and so is a vector that is the projection of formula_39 onto the line. Thus is the component of
Distance from a point to a line
13690261
to the point P. Hence: We can solve this equation for "x", The y coordinate of the point of intersection can be found by substituting this value of "x" into the equation of the original line, Using the equation for finding the distance between 2 points, formula_35, we can deduce that the formula to find the shortest distance between a line and a point is the following: Recalling that "m" = -"a"/"b" and "k" = - "c"/"b" for the line with equation "ax" + "by" + c = 0, a little algebraic simplification reduces this to the standard expression. The
Distance from a point to a line
651312
direction from the origin to the second point is defined as the "positive direction". This line segment may be translated along the line to build longer segments whose lengths correspond to multiples of the unit distance. In this manner real numbers can be associated to points on the line (as the distance from the origin to the point) and these are the Cartesian coordinates of the points on what may now be called the real line. As an alternate way to establish the metric, instead of choosing two points on the line, choose one point to be the origin, a
Euclidean distance
5148008
compute the 3D volume of a mesh (discrete volumetric integral)<br> Surface Area - compute the surface area of a mesh (discrete surface integral)<br> Collision Detection - determine if two complex meshes in motion have collided<br> Fitting - construct a parametric surface (NURBS, bicubic spline) by fitting it to a given mesh<br> Point-Surface Distance - compute distance from a point to the mesh<br> Line-Surface Distance - compute distance from a line to the mesh<br> Line-Surface Intersection - compute intersection of line and the mesh<br> Cross Section - compute the curves created by a cross-section of a plane through a mesh<br> Centroid
Polygonal modeling
858903
problems in geometry to be translated into problems about numbers and "vice versa"; this is the basis of analytic geometry. The simplest example of a coordinate system is the identification of points on a line with real numbers using the "number line". In this system, an arbitrary point "O" (the "origin") is chosen on a given line. The coordinate of a point "P" is defined as the signed distance from "O" to "P", where the signed distance is the distance taken as positive or negative depending on which side of the line "P" lies. Each point is given a unique
Coordinate system
13690254
the same point and the line is |"ax" + "c"| / |"a"|, as measured along a horizontal line segment. If the line passes through two points "P"=("x","y") and "P"=("x","y") then the distance of (x,y) from the line is: The denominator of this expression is the distance between "P" and "P". The numerator is twice the area of the triangle with its vertices at the three points, (x,y), "P" and "P". See: . The expression is equivalent to formula_4, which can be obtained by rearranging the standard formula for the area of a triangle: formula_5, where "b" is the length of
Distance from a point to a line
2343045
vanishing point are drawn (in grey). These lines give the basic, one point "railroad tracks" perspective. The closer it is the horizon line, the farther away it is from the viewer, and the smaller it will appear. The farther away from the viewer it is, the closer it is to being perpendicular to the picture plane. A new point ("the eye") is now chosen, on the horizon line, either to the left or right of the vanishing point. The distance from this point to the vanishing point represents the distance of the viewer from the drawing. If this point is
Perspective (graphical)
2831773
horizon line, while orthogonals are created using a compass set on a line made at a 90-degree angle through each of the four vanishing points. Distances a and c between the viewer and the wall are greater than the b distance, so adopting the principle that when an object is a greater distance from the observer, it becomes smaller, the wall is reduced and thus appears distorted at the edges. If a point has the 3D Cartesian coordinates ("x","y","z"): Denoting distance from the point to the origin by "d" = , then the transformation of the point to a curvilinear
Curvilinear perspective
13690258
are vertical lines). Corresponding sides of these triangles are in the same ratio, so: If point "S" has coordinates ("x","m") then |"PS"| = |"y" - "m"| and the distance from "P" to the line is: Since "S" is on the line, we can find the value of m, and finally obtain: A variation of this proof is to place V at P and compute the area of the triangle ∆"UVT" two ways to obtain that formula_17 where D is the altitude of ∆"UVT" drawn to the hypoteneuse of ∆"UVT" from "P". The distance formula can then used to express formula_18,
Distance from a point to a line
11741704
the circle that passes through the three points P, Q and T. If the line "m" is "not" parallel to the given line "l", then it intersects "l" at a point G. By the power of a point theorem, the distance from G to a tangent point T must equal the geometric mean Two points on the given line "L" are located at a distance formula_12 from the point G, which may be denoted as T and T. The two solution circles are the circles that pass through the three points (P, Q, T) and (P, Q, T), respectively. The
Special cases of Apollonius' problem
5168559
are measured along the "x" axis of an EPR spectrum, from a line's center to a chosen reference point of the line. These defined widths are called halfwidths and possess some advantages: for asymmetric lines, values of left and right halfwidth can be given. The halfwidth formula_61 is the distance measured from the line's center to the point in which absorption value has half of maximal absorption value in the center of resonance line. First inclination width formula_62 is a distance from center of the line to the point of maximal absorption curve inclination. In practice, a full definition of
Electron paramagnetic resonance
8647576
diagram a line has been drawn from the origin to the point with minimum sink. The slope of the line from the origin gives the glide angle, because it is the ratio of the distance along the airspeed axis to the distance along the sink rate axis. A whole series of lines could be drawn from the origin to each of the data points, each line showing the glide angle for that speed. However the best glide angle is the line with the least slope. In the second diagram, the line has been drawn from the origin to the point
Polar curve (aerodynamics)
17204436
one distance to the trunk calculate the area. The fourth method, the Azimuth Method, requires only azimuths and distance measurements from point to point around the crown perimeter. The measurer walks the perimeter of the crown following the drip line fairly closely. Points are identified on the ground that represent the outline of the crown and marked in a way so that the next point is always visible from the prior one. For the first point, the distance to the center of the tree's trunk is measured along with the vertical angle to the point. Then the distance to the
Tree crown measurement
5520084
in 1987, a year after the NCAA. The NCAA used the FIBA three-point line (see below) in the National Invitation Tournament (NIT) For three seasons beginning in 1994–95, the NBA attempted to address decreased scoring by shortening the distance of the line from ( at the corners) to a uniform around the basket. From the 1997–98 season on, the NBA reverted the line to its original distance of ( at the corners, with a 3-inch differential). Ray Allen is currently the NBA all-time leader in career made three-pointers with 2,973. In 2008, FIBA announced that the distance would be increased
Three-point field goal
3185129
scoring by shortening the overall distance of the line to a uniform around the basket. It was moved back to its original distance after the 1996–97 season. FIBA and the NCAA both adopted the three-point line in 1985. In most high school associations in the United States, the distance is 19.75 feet. This was formerly the distance for college basketball as well. On May 26, 2007, the NCAA playing rules committee agreed to move the three-point line back one foot to 20.75 feet for the men. This rule went into effect for the 2008–2009 season. The three-point line for women
Basketball court
9662017
"RS" and "RW" are in proportion formula_1. In this figure, segments "PU" and "VT" are of equal length, and "RV" is perpendicular to "TU". These properties can be used as part of an equivalent alternative definition for the Philo line for a point "P" and angle edges "d" and "e": it is a line segment connecting "d" to "e" through "P" such that the distance along the segment from "P" to "d" is equal to the distance along the segment from "V" to "e", where "V" is the closest point on the segment to the corner point of the angle.
Philo line
3152423
do not intersect. Equivalently, they are parallel if and only if the distance from a point "P" on line "m" to the nearest point in plane "q" is independent of the location of "P" on line "m". Similar to the fact that parallel lines must be located in the same plane, parallel planes must be situated in the same three-dimensional space and contain no point in common. Two distinct planes "q" and "r" are parallel if and only if the distance from a point "P" in plane "q" to the nearest point in plane "r" is independent of the location
Parallel (geometry)
1537554
racing patterns call for a precise distance between the start line and the first barrel, from the first to the second barrel, and from the second to the third barrel. The pattern from every point of the cloverleaf will have a precisely measured distance from one point to the next. Usually the established distances are as follows: "Note:" In a standard WPRA pattern, the score line begins at the plane of arena, meaning from fence to fence regardless of the position of the electric eye or timer. In larger arenas, there is a maximum allowable distance of 105 feet between
Barrel racing
8059764
the inexact differential is very simple conceptually. There are many everyday examples that are much more relevant to inexact differentials in the actual context in which it is used. The easiest example is the difference between net distance and total distance. For example, in walking from Point "A" to Point "B" along a straight line, one covers a net distance "B" − "A" that is equal to the total distance. If one then returns to Point "A", however, net distance is now 0 while total distance covered is 2*("B" − "A"). This example captures the essential idea behind the inexact
Inexact differential
17301592
south crossing a small portion of Corio Bay parallel with and at a distance of 2 miles from the eastern boundary of the said township to a point bearing east from the south-east corner of the said Township of Geelong thence by a line drawn westerly at a distance of 2 miles from the southern boundary of the said township crossing the Barwon River and Waurn Chain of Ponds to a point 2 miles west of the south-west corner of the said township thence by a line bearing north parallel with and at a distance of 2 miles from the
Electoral district of Geelong (Victorian Legislative Council)
5228291
indifferent between the actual cities. In analogy with Newton's law of gravitation, the point of indifference is the point at which the "attractiveness" of the two retail centres (postulated to be proportional to their size and inversely proportional to the square of the distance to them) is equal: Where formula_2 is the distance of the point of indifference from "A", formula_3 is its distance from "B", and formula_4 is the relative size of the two centres. If the customer is on the line connecting "A" and "B", then if "D" is the distance between the centres, the point of indifference
Reilly's law of retail gravitation
3407448
Shear mapping In plane geometry, a shear mapping is a linear map that displaces each point in fixed direction, by an amount proportional to its signed distance from a line that is parallel to that direction. This type of mapping is also called shear transformation, transvection, or just shearing. An example is the mapping that takes any point with coordinates formula_1 to the point formula_2. In this case, the displacement is horizontal, the fixed line is the formula_3-axis, and the signed distance is the formula_4 coordinate. Note that points on opposite sides of the reference line are displaced in opposite
Shear mapping
13690253
has coordinates: Horizontal and vertical lines In the general equation of a line, "ax" + "by" + "c" = 0, "a" and "b" cannot both be zero unless "c" is also zero, in which case the equation does not define a line. If "a" = 0 and "b" 0, the line is horizontal and has equation "y" = -"c"/"b". The distance from ("x", "y") to this line is measured along a vertical line segment of length |"y" - (-"c"/"b")| = |"by" + "c"| / |"b"| in accordance with the formula. Similarly, for vertical lines ("b" = 0) the distance between
Distance from a point to a line
3112185
dynamics and in geomagnetics, the terms "isocline" and "isoclinic line" have specific meanings which are described below. A curve of equidistant points is a set of points all at the same distance from a given point, line, or polyline. In this case the function whose value is being held constant along a contour line is a distance function. In geography, the word "isopleth" (from or "plethos", meaning 'quantity') is used for contour lines that depict a variable which cannot be measured at a point, but which instead must be calculated from data collected over an area. An example is population
Contour line
1787928
pole" of the given line. The perpendiculars on the other side also intersect at a point, which is different from the other absolute pole only in spherical geometry, for in elliptic geometry the poles on either side are the same. There are no antipodal points in elliptic geometry. Every point corresponds to an "absolute polar line" of which it is the absolute pole. Any point on this polar line forms an "absolute conjugate pair" with the pole. Such a pair of points is "orthogonal", and the distance between them is a "quadrant". The distance between a pair of points is
Elliptic geometry
12788352
Hesse normal form The Hesse normal form named after Otto Hesse, is an equation used in analytic geometry, and describes a line in formula_1 or a plane in Euclidean space formula_2 or a hyperplane in higher dimensions. It is primarily used for calculating distances (see point-plane distance and point-line distance). It is written in vector notation as The dot formula_4 indicates the scalar product or dot product. The vector formula_5 represents the unit normal vector of "E" or "g", that points from the origin of the coordinate system to the plane (or line, in 2D). The distance formula_6 is the
Hesse normal form
19816202
Orthodontics. He developed the holdaway lip analysis. The lip analysis, which was called H Line, was known to run from the cephalometric point Soft Tissue Pogonion to upper lip. "H" angle of Holdaway was between the Nasion-B Point (NB) line and the H line. For a normal ANB value of 2-3 degree, H angle is around 7-8 degrees. He was also known for developing the Holdaway ratio to compare the prominence of the lower incisors compare to that of the bony chin. According to him, the distance from labial surface of mandibular incisor to the N-B line and distance from
Reed Holdaway
9249981
Note that the distance from a point, "x" to the line formula_60 is given by And so the squared distance from a point, "x", to a line is The sum of squared distances to many lines is the cost function: This can be rearranged: To find the minimum, we differentiate with respect to "x" and set the result equal to the zero vector: so and so While formula_55 is not well-defined in more than two dimensions, this can be generalized to any number of dimensions by noting that formula_69 is simply the (symmetric) matrix with all eigenvalues unity except for
Line–line intersection
5520088
then becomes parallel to the sidelines from the plane of the basket center to the baseline ( in college, in high schools). The distance of the three-point line to the center of the hoop varies by level: A player's feet must be completely behind the three-point line at the time of the shot or jump in order to make a three-point attempt; if the player's feet are on or in front of the line, it is a two-point attempt. A player is allowed to jump from outside the line and land inside the line to make a three-point attempt, as
Three-point field goal
12971305
and logical arguments were either ignored or slurred over. In Mac Lane's system there are four primitive notions (undefined terms): "point", "distance", "line" and "angle measure". There are also 14 axioms, four giving the properties of the distance function, four describing properties of lines, four discussing angles (which are directed angles in this treatment), a similarity axiom (essentially the same as Birkhoff's) and a continuity axiom which can be used to derive the Crossbar theorem and its converse. The increased number of axioms has the pedagogical advantage of making early proofs in the development easier to follow and the use
Foundations of geometry
3185128
line would give the player three free throws if the shot does not go in, and one if it does. The distance to the three-point line from the center of the basket varies depending on the level or league, and has changed several times. These are the current distances, with the league or level using each distance: The NBA adopted the three-point line at the start of the 1979–80 season. This is of variable distance, ranging from in the corners to behind the top of the key. During the 1994–95, 1995–96 and 1996–97 seasons, the NBA attempted to address decreased
Basketball court
9249982
a zero eigenvalue in the direction along the line providing a seminorm on the distance between formula_54 and another point giving the distance to the line. In any number of dimensions, if formula_71 is a unit vector "along" the "i"-th line, then where "I" is the identity matrix, and so Line–line intersection In Euclidean geometry, the intersection of a line and a line can be the empty set, a point, or a line. Distinguishing these cases and finding the intersection point have use, for example, in computer graphics, motion planning, and collision detection. In three-dimensional Euclidean geometry, if two lines
Line–line intersection
3158806
standard court size in U.S. college and WNBA play is 94 feet long by 50 feet wide. The FIBA standard court is slightly smaller at 28 metres long by 15 metres wide (91 ft 10.4 in by 49 ft 2.6 in). For most of its distance, the three-point line is from the middle of the basket under both FIBA and WNBA rules. Near the sidelines, the three-point line runs parallel to the.sideline, at a distance of exactly 3 feet in the WNBA and 0.9 m in FIBA play. Under NCAA rules, the three-point distance is for most of the width
Women's basketball
12392545
would not change because the distance from each point to the line grows on one side of the line, while the distance to each point on the opposite side of the line diminishes by exactly the same amount. Thus the sum of absolute errors remains the same. Also, since one can tilt the line in infinitely small increments, this also shows that if there is more than one solution, there are infinitely many solutions. The least absolute deviation problem may be extended to include multiple explanators, constraints and regularization, e.g., a linear model with linear constraints: where formula_5 is a
Least absolute deviations
390742
is often quite different from a value of instantaneous speed. If the average speed and the time of travel are known, the distance travelled can be calculated by rearranging the definition to Using this equation for an average speed of 80 kilometres per hour on a 4-hour trip, the distance covered is found to be 320 kilometres. Expressed in graphical language, the slope of a tangent line at any point of a distance-time graph is the instantaneous speed at this point, while the slope of a chord line of the same graph is the average speed during the time interval
Speed
4605283
for a point where they meet, a fix is generated. However, in the case of fast-moving receivers, the position of the signal moves as signals are received from several satellites. In addition, the radio signals slow slightly as they pass through the ionosphere, and this slowing varies with the receiver's angle to the satellite, because that changes the distance through the ionosphere. The basic computation thus attempts to find the shortest directed line tangent to four oblate spherical shells centred on four satellites. Satellite navigation receivers reduce errors by using combinations of signals from multiple satellites and multiple correlators, and
Satellite navigation
20615238
The through line (VzG line number 6151) begins at Glasower Damm Ost junction. However, running distance on the line (chainage) does not start at zero, but follows that of the Berlin outer ring. This in turn continues the chainage of the Michendorf–Großbeeren railway, opened in 1926, which formed part of the Brandenburg Bypass Railway. The starting point of the chainage is therefore at kilometre 28.220 km, the distance from Michendorf. The outer ring to the east and the line to the new airport have parallel chainages measured from that point. The end of line 6151 is at Bohnsdorf Süd junction
Glasower Damm Ost–Bohnsdorf Süd railway
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points are granted for the critical point length. Based on the hills length, distance points are calculated, which is added for each meter beyond the critical point and subtracted for each point shorter than the critical point. A meter has more distance points in smaller hills. Construction point The construction point (), also known as the critical point, calculation point, calculation line, K-point and K-spot, is a line across a ski jumping hill which indicates the hill's steepest point in meters. It was formerly used to classify the size of a ski jumping hill, and to calculate the number of
Construction point
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Cross-ratio In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points "A", "B", "C" and "D" on a line, their cross ratio is defined as where an orientation of the line determines the sign of each distance and the distance is measured as projected into Euclidean space. (If one of the four points is the line's point at infinity, then the two distances involving that point are dropped from the formula.) The point "D" is the harmonic conjugate
Cross-ratio
5520077
Three-point field goal A three-point field goal (also 3-pointer or informally, trey) is a field goal in a basketball game made from beyond the three-point line, a designated arc surrounding the basket. A successful attempt is worth three points, in contrast to the two points awarded for field goals made within the three-point line and the one point for each made free throw. The distance from the basket to the three-point line varies by competition level: in the National Basketball Association (NBA) the arc is from center of the basket; in FIBA and the WNBA the arc is ; and
Three-point field goal
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a side, and "h" is the perpendicular height from the opposite vertex. This proof is only valid if the line is neither vertical nor horizontal, that is, we assume that neither "a" nor "b" in the equation of the line is zero. The line with equation "ax" + "by" + "c" = 0 has slope -"a"/"b", so any line perpendicular to it will have slope "b"/"a" (the negative reciprocal). Let ("m", "n") be the point of intersection of the line "ax" + "by" + "c" = 0 and the line perpendicular to it which passes through the point ("x", "y").
Distance from a point to a line
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of the numbers 0.9, 0.99, 0.999, etc. Because there is ultimately no room between 1 and these numbers, the point 1 must be this least point, and so . If one places 0.9, 0.99, 0.999, etc. on the number line, one sees immediately that all these points are to the left of 1, and that they get closer and closer to 1. More precisely, the distance from 0.9 to 1 is , the distance from 0.99 to 1 is , and so on. The distance to 1 from the th point (the one with 9s after the decimal point) is
0.999...
6196705
Shin-Aomori Station Shin-Aomori Station is the northern terminus of the Ōu Main Line from via (a distance of ), although most trains continue on to . It also forms the northern terminus of the high-speed Tōhoku Shinkansen line from (a distance of ), operated by JR East, and the starting point of the Hokkaido Shinkansen to (a distance of ), operated by JR Hokkaido. The conventional Ōu Main Line has a single island platform, serving two tracks. In addition to regular Ōu Main Line trains, the station serves two round-trips per day of Aoimori Railway trains, as well two round-trips
Shin-Aomori Station
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cross product of α and β. The special perpendicular is employed to compute the volume of a pyramid (p 35), an equation on skew lines that reduces to zero when they are coplanar, a property of a spherical triangle, and the coincidence of the perpendiculars in a tetrahedron. Chapter four describes equations of geometric figures: line, plane, circle, sphere. The definition of a conic section is taken from Kelland and Tait: "the locus of a point which move so that its distance from a fixed point bears a constant ratio to its distance from a fixed straight line." Ellipse, hyperbola,
Hüseyin Tevfik Pasha
516635
given time. For instance, the distance from the center of gravity of the Earth "A" and the center of gravity of the Moon "B" (which does not strictly imply motion from "A" to "B") falls into this category. A directed distance along a curved line is not a vector and is represented by a segment of that curved line defined by endpoints "A" and "B", with some specific information indicating the sense (or direction) of an ideal or real motion from one endpoint of the segment to the other (see figure). For instance, just labelling the two endpoints as "A"
Distance
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from this point, passing a water tank and then connecting to a small spur with a maintenance shed. Shortly north of the spur, the line turns almost due west to follow the northern shore of Osbourne's Lake, then turns roughly north again after a short distance. The line continues north to a point just south of Peninsula Lake, where it turns to the west for a distance. This is a spur that forms a switchback; the train travels to the west onto this line, switches, and then travels back eastward into North Portage Station. Here there were two parallel spurs
Huntsville and Lake of Bays Transportation Company
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is, A particular tangent line distinguishes the hyperbola from the other conic sections. Let "f" be the distance from the vertex "V" (on both the hyperbola and its axis through the two foci) to the nearer focus. Then the distance, along a line perpendicular to that axis, from that focus to a point P on the hyperbola is greater than 2"f". The tangent to the hyperbola at P intersects that axis at point Q at an angle ∠PQV of greater than 45°. In the case formula_63 the hyperbola is called "rectangular" (or "equilateral"), because its asymptotes intersect rectangularly (i.e., are
Hyperbola
15860224
QBR is calculated as the following: formula_1, where g() is a function that scales from 0-100, where 50 is average. Total QBR is the raw QBR adjusted for the strength of the opponent. EPA is calculated based on the down, distance, and the yard line at snap, with each combination having its own point value. The point values are the average net point advantage the team on offense can expect given the particular down, distance, and field position. For example, a 1st and goal chance on the opponent's’ 1 yard line heavily favors the offense, yielding a positive point value.
Total quarterback rating
480293
parts of the surface are more than 9 kilometres away in straight-line distance. A suitable analogy would be the distance from the top centre of a bun to the bottom centre as compared to the distance from the top centre to a point on the bun's circumference: top-to-bottom is a longer distance than top-to-periphery when measured along the surface but shorter than it in direct straight-line terms. Compression from the same impact is believed to have created the thrust fault Hinks Dorsum. Data from the Near Earth Asteroid Rendezvous spacecraft collected on Eros in December 1998 suggests that it could
433 Eros
4628405
shape analysis, the topological skeleton or medial axis of a shape is a thin version of that shape that is equidistant from its boundaries. In Euclidean geometry, parallel lines (lines that never intersect) are equidistant in the sense that the distance of any point on one line from the nearest point on the other line is the same for all points. In hyperbolic geometry the set of points that are equidistant from and on one side of a given line form an hypercycle (which is a curve not a line). Equidistant A point is said to be equidistant from a
Equidistant
2015928
the reflection of a point drop a perpendicular from the point to the line (plane) used for reflection, and extend it the same distance on the other side. To find the reflection of a figure, reflect each point in the figure. To reflect point through the line using compass and straightedge, proceed as follows (see figure): Point is then the reflection of point through line . The matrix for a reflection is orthogonal with determinant −1 and eigenvalues −1, 1, 1, ..., 1. The product of two such matrices is a special orthogonal matrix that represents a rotation. Every rotation
Reflection (mathematics)
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Distance line A distance line, penetration line, cave line or guide line is an item of diving equipment used by scuba divers as a means of returning to a safe starting point in conditions of low visibility, water currents or where pilotage is difficult. They are often used in cave diving and wreck diving where the diver must return to open water after a penetration when it may be difficult to discern the return route. Guidelines are also useful in the event of silt out. Distance lines are wound on to a spool or a reel. The length of the
Distance line
1348998
the perpendicular to a side from the point of intersection of the diagonals always bisects the opposite side. Algebraically, the perpendicular bisector of a line segment with endpoints formula_1 and formula_2 is given by the equation which states that the squared distance of a point on the bisector to one endpoint equals the squared distance from that point to the other endpoint. An angle bisector divides the angle into two angles with equal measures. An angle only has one bisector. Each point of an angle bisector is equidistant from the sides of the angle. The "interior" or "internal bisector" of
Bisection
715370
lines (or circles), and a point; and then draw a line which passes through the given point and intersects three lines, and such that the distance between the points of intersection equals the given segment. This the Greeks called "neusis" ("inclination", "tendency" or "verging"), because the new line "tends" to the point. In this expanded scheme, we can trisect an arbitrary angle (see Archimedes' trisection) or extract an arbitrary cube root (due to Nicomedes). Hence, any distance whose ratio to an existing distance is the solution of a cubic or a quartic equation is constructible. Using a markable ruler, regular
Straightedge and compass construction
11659505
places at nearly the same longitude by measuring a base line and a chain of triangles. (Suitable stations for the end points are rarely at the same longitude). The distance Δ along the meridian from one end point to a point at the same latitude as the second end point is then calculated by trigonometry. The surface distance Δ is reduced to Δ', the corresponding distance at mean sea level. The intermediate distances to points on the meridian at the same latitudes as other stations of the survey may also be calculated. The geographic latitudes of both end points, φ
Earth ellipsoid
4685429
the point is not on the line, the pair is called an "anti-flag". There is no natural concept of distance (a metric) in an incidence structure. However, a combinatorial metric does exist in the corresponding incidence graph (Levi graph), namely the length of the shortest path between two vertices in this bipartite graph. The distance between two objects of an incidence structure – two points, two lines or a point and a line – can be defined to be the distance between the corresponding vertices in the incidence graph of the incidence structure. Another way to define a distance again
Incidence geometry
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the original curve. The starting curve is an ordered set of points or lines and the distance dimension "ε" > 0. The algorithm recursively divides the line. Initially it is given all the points between the first and last point. It automatically marks the first and last point to be kept. It then finds the point that is furthest from the line segment with the first and last points as end points; this point is obviously furthest on the curve from the approximating line segment between the end points. If the point is closer than "ε" to the line segment,
Ramer–Douglas–Peucker algorithm
312998
angle PVF is complementary to angle VPF, therefore angle PVF is . Since the length of is , the distance of F from the vertex of the parabola is . It is shown above that this distance equals the focal length of the parabola, which is the distance from the vertex to the focus. The focus and the point F are therefore equally distant from the vertex, along the same line, which implies that they are the same point. Therefore, the point F, defined above, is the focus of the parabola. An alternative proof can be done using Dandelin spheres.
Parabola
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points on the line "OK" have polar angle formula_4, and the points at distance "d" from "K" on this line are distance formula_5 from the origin. Therefore, the equation of the strophoid is given by Let "C" be given parametrically by ("x"("t"), "y"("t")). Let "A" be the point (a, b) and let "O" be the point ("p", "q"). Then, by a straightforward application of the polar formula, the strophoid is given parametrically by: where The complex nature of the formulas given above limits their usefulness in specific cases. There is an alternative form which is sometimes simpler to apply. This
Strophoid
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diagram above. Note that formula_30 by Pythagorean theorem. Substituting with appropriate functions - Rearranging gives: Draw a horizontal line (the "x"-axis); mark an origin O. Draw a line from O at an angle formula_34 above the horizontal line and a second line at an angle formula_35 above that; the angle between the second line and the "x"-axis is formula_36. Place P on the line defined by formula_36 at a unit distance from the origin. Let PQ be a line perpendicular to line defined by angle formula_34, drawn from point Q on this line to point P. formula_39 OQP is a
Proofs of trigonometric identities
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Equipollence (geometry) In Euclidean geometry, equipollence is a binary relation between directed line segments. A line segment "AB" from point "A" to point "B" has the opposite direction to line segment "BA". Two directed line segments are equipollent when they have the same length and direction. The concept of equipollent line segments was advanced by Giusto Bellavitis in 1835. Subsequently the term vector was adopted for a class of equipollent line segments. Bellavitis's use of the idea of a relation to compare different but similar objects has become a common mathematical technique, particularly in the use of equivalence relations. Bellavitis
Equipollence (geometry)
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distances and azimuths from that point to any other point will be correct. That point, ("φ", "λ"), will project to the center of a circular projection, with "φ" referring to latitude and "λ" referring to longitude. All points along a given azimuth will project along a straight line from the center, and the angle "θ" that the line subtends from the vertical is the azimuth angle. The distance from the center point to another projected point is given as "ρ". By this description, then, the point on the plane specified by ("θ","ρ") will be projected to Cartesian coordinates: The relationship
Azimuthal equidistant projection
12707831
did not materialise but a line running the short distance from Hendon to Allora was constructed and opened on 21 April 1897. Up to four trains daily connected via Kates with the main line services at Hendon. After development of land to the east of Allora, moves were afoot to extend the line about 20 kilometres to Goomburra. Approved in 1910, construction of the extension commenced in 1911 and opened on 8 July 1912. Stops were located at Kital, Berat and Kunda. The Goomburra extension was constructed from a point a short distance down from Allora station, turning Allora station
Goomburra railway line
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sight. The difference between the reflected line doppler shift velocity and the direct line velocity gives the actual (i.e. not projected) distance of the shell from the central star, again assuming expansion at constant velocity since the Great Eruption. Observations of the spectrum of the Homunculus at a particular angular distance from the central star has shown the actual linear distance of that point from the central star, which defines the distance. Values obtained using this method are around 2.3 kpc with a margin of error around 100pc. The same calculations also return the inclination of the axis of the
Homunculus Nebula
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colors and formula_4 symmetry. With a slight extension, Wythoff's symbol can be applied to all uniform polyhedra. However, the construction methods do not lead to all uniform tilings in Euclidean or hyperbolic space. In three dimensions, Wythoff's construction begins by choosing a "generator point" on the triangle. If the distance of this point from each of the sides is non-zero, the point must be chosen to be an equal distance from each edge. A perpendicular line is then dropped between the generator point and every face that it does not lie on. The three numbers in Wythoff's symbol, formula_5, formula_6
Wythoff symbol
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the distance between a given point and the origin and θ is the angle between a line of reference (the "x"-axis) and the line through the origin and the point. The angular coordinate is the same as for polar coordinates, while the radial coordinate is transformed according to the rule where formula_2 is the distance to the origin. The formulas for transformation from Cartesian coordinates to log-polar coordinates are given by and the formulas for transformation from log-polar to Cartesian coordinates are By using complex numbers ("x", "y") = "x" + "iy", the latter transformation can be written as i.e.
Log-polar coordinates
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Limiting point (geometry) In geometry, the limiting points of two disjoint circles "A" and "B" in the Euclidean plane are points "p" that may be defined by any of the following equivalent properties: The midpoint of the two limiting points is the point where the radical axis of "A" and "B" crosses the line through their centers. This intersection point has equal power distance to all the circles in the pencil containing "A" and "B". The limiting points themselves can be found at this distance on either side of the intersection point, on the line through the two circle centers.
Limiting point (geometry)
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shot, is located within the three-point arc at 15 feet from the basket. A foul shot is worth 1 point, but if a shot is made from the foul line while in play it is still worth 2 points. † The FIBA three-point line is 2.95 ft (0.90 m) from the sideline in a zone starting at the baseline and ending when it crosses the 22.1 ft (6.75 m) arc. The 21.65 ft (6.60 m) distance exists only at the points on the three-point line that are directly to the left and right of the basket center. The only two
Basketball court
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the object, runs into obstacles or runs out of line, air or time. The amount of distance line increment for each sweep should allow some overlap of sweeps to avoid the risk of missing the target between sweeps. If a buddy is involved the most efficient place is alongside the controlling diver on the line, and the extension of distance line for each sweep can be roughly doubled. Depending on the circumstances, control of the pattern may be from the surface, from a diver at the central point, or by the diver at the end of the search line, who
Underwater searches
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now worth 1, 2, or 3 points depending upon distance or game situation. Basically, all non-power play goals scored from inside the yellow line were worth 2 points while non-powerplay goals from outside the yellow line (50 feet from the goal line) were worth 3 points. Any power play goal was worth 1 point, as was any goal scored during a penalty shootout. Before the 1994–1995 season, the three-point line was changed to a 45-foot arc. Eventually, power play goals were worth either two or three points, but penalty shootouts were still kept at one point. National Professional Soccer League
National Professional Soccer League (1984–2001)
1374797
it was to the latter by the Nivelles–Namur paved high road over a distance of about , offered great facility for co-operation and mutual support upon whichever point the greater part of the French Army of the North might be directed. Should it prove tenable, then, considered in conjunction with the advance of the Russians from the Rhine, the whole line of the river Meuse below Namur, and the communications with Aachen and the Prussian States, were secured. If, on the other hand, either position should be forced by the French, then Mont-Saint-Jean and Wavre, upon parallel lines of retreat
Battle of Ligny
4628404
all four vertices, and it is equidistant from two opposite sides and also equidistant from the other two opposite sides. A point on the axis of symmetry of a kite is equidistant between two sides. The center of a circle is equidistant from every point on the circle. Likewise the center of a sphere is equidistant from every point on the sphere. A parabola is the set of points in a plane equidistant from a fixed point (the focus) and a fixed line (the directrix), where distance from the directrix is measured along a line perpendicular to the directrix. In
Equidistant
2727888
there is a safety or operational requirement to return to that point. Other dives where it is necessary or highly desirable to return to a starting point: This may include dives where a shot line or anchor line is to be used for ascent, and other dives where a specific exit point is chosen for safety or convenience. Working divers may use a guide line to allow confident and efficient movement to, from and around the workplace. This may also be referred to as a jackstay. In these applications the guideline is generally more for efficiency than for safety, but
Distance line
13337161
790.2-3. The line is fully electrified and the western section from Stuttgart to Schorndorf is part of the Stuttgart S-Bahn. The line’s chainage is measured from Stuttgart Bad Cannstatt station. A short distance from the station the four tracks of the Rems line separate from the Fils Valley Railway at a flying junction. At about the same point the tracks of the Rems line are rearranged to be paired by direction of operation, with S-Bahn trains running on the outer tracks and regional and long distance trains running on the inner tracks. The line climbs about 80 metres on its
Stuttgart-Bad Cannstatt–Aalen railway
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The line through these two points is perpendicular to the original line, so Thus, formula_7 and by squaring this equation we obtain: Now consider, using the above squared equation. But we also have, since ("m", "n") is on "ax" + "by" + "c" = 0. Thus, and we obtain the length of the line segment determined by these two points, This proof is valid only if the line is not horizontal or vertical. Drop a perpendicular from the point "P" with coordinates ("x", "y") to the line with equation "Ax" + "By" + "C" = 0. Label the foot of
Distance from a point to a line
19795114
the midpoint of the line segment "NL" connecting the Nagel point "N" and the Longchamps point "L". The radius of the Bevan circle is 2"R", that is twice the radius of the circumcircle. Bevan point The Bevan point, named after Benjamin Bevan, is a triangle center. It is defined as center of the Bevan circle, that is the circle through the centers of the three excircles of a triangle. The Bevan point M of triangle "ABC" has the same distance from its Euler line "e" as its incenter "I" and the circumcenter "O" is the midpoint of the line segment
Bevan point
3498366
Displacement (vector) A displacement is a vector whose length is the shortest distance from the initial to the final position of a point P. It quantifies both the distance and direction of an imaginary motion along a straight line from the initial position to the final position of the point. A displacement may be identified with the translation that maps the initial position to the final position. A displacement may be also described as a 'relative position': the final position of a point (x) relative to its initial position (x), and a displacement vector can be mathematically defined as the
Displacement (vector)
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is the distance from the origin to this perpendicular, and "d" is the length of the segment between reference and the given line. The motions of the line geometry are described with linear fractional transformations on the appropriate complex planes. Line coordinates In geometry, line coordinates are used to specify the position of a line just as point coordinates (or simply coordinates) are used to specify the position of a point. There are several possible ways to specify the position of a line in the plane. A simple way is by the pair where the equation of the line is
Line coordinates
12559596
point b, let's say, on the fovea. Likewise, endpoint A is imaged at point a. The optical (physical) angle between those chief rays is the visual angle "θ" which can be calculated: The retinal images at b and a are separated by the distance "R", given by the equation in which "n" is the eye's nodal distance that averages about 17 mm. That is, a viewed object's retinal image size is approximately given by . The line from point O outward through object point B specifies the optical direction, "d", of the object's base from the eye, let's say toward
Perceived visual angle
19315498
end of the "x"-axis. From point P draw the line "p" asymptotic to the "x"-axis to the right ideal point formula_26. "P" is the intersection of line "p" and horocycle "h". The coordinate "x" is the distance from P to "P" – positive if P is between "P" and formula_26, negative if "P" is between P and formula_26. The coordinate "y" is the arclength along horocycle "h" from the origin to "P". The distance between two points given in these coordinates is The corresponding metric tensor is: formula_31 The straight lines are described by equations of the form "y" =
Coordinate systems for the hyperbolic plane