Involutes of a semicubic parabola The parametric equation describes a semicubical parabola. From one gets and . Extending the string by extensively simplifies further calculation, and one gets Eliminating yields showing that this involute is a parabola. The other involutes are thus parallel curves of a parabola, and are not parabolas, as they are curves of degree six (See ). Involutes of a catenary For the catenary , the tangent vector is , and, as its length is . Thus the arc length from the point is Hence the involute starting from is parametrized by and is thus a tractrix. The other involutes are not tractrices, as they are parallel curves of a tractrix. Involutes of a cycloid The parametric representation describes a cycloid. From , one gets (after having used some trigonometric formulas) and Hence the equations of the corresponding involute are which describe the shifted red cycloid of the diagram. Hence The involutes of the cycloid are parallel curves of the cycloid (Parallel curves of a cycloid are not cycloids.) Involute and evolute The evolute of a given curve consists of the curvature centers of . Between involutes and evolutes the following statement holds: A curve is the evolute of any of its involutes. Application The most common profiles of modern gear teeth are involutes of a circle. In an involute gear system the teeth of two meshing gears contact at a single instantaneous point that follows along a single straight line of action. The forces exerted the contacting teeth exert on each other also follow this line, and are normal to the teeth. The involute gear system maintaining these conditions follows the fundamental law of gearing: the ratio of angular velocities between the two gears must remain constant throughout. With teeth of other shapes, the relative speeds and forces rise and fall as successive teeth engage, resulting in vibration, noise, and excessive wear. For this reason, nearly all modern planar gear systems are either involute or the related cycloidal gear system. The involute of a circle is also an important shape in gas compressing, as a scroll compressor can be built based on this shape. Scroll compressors make less sound than conventional compressors and have proven to be quite efficient. The High Flux Isotope Reactor uses involute-shaped fuel elements, since these allow a constant-width channel between them for coolant. See also Envelope (mathematics) Evolute Goat grazing problem