Involute
In mathematics, an involute (also known as an evolvent) is a particular type of curve that is dependent on another shape or curve. An involute of a curve is the locus of a point on a piece of taut string as the string is either unwrapped from or wrapped around the curve. It is a class of curves coming under the roulette family of curves. The evolute of an involute is the original curve. The notions of the involute and evolute of a curve were introduced by Christiaan Huygens in his work titled "Horologium oscillatorium sive de motu pendulorum  ad horologia aptato demonstrationes geometricae" (1673). Involute of a parameterized curve. Let formula_1 be a regular curve in the plane with its curvature nowhere 0 and formula_2, then the curve with the parametric representation formula_3 is an "involute" of the given curve. Adding an arbitrary but fixed number formula_4 to the integral formula_5 results in an involute corresponding to a string extended by formula_4 (like a ball of wool yarn having some length of thread already hanging before it is unwound). Hence, the involute can be varied by constant formula_7 and/or adding a number to the integral (see Involutes of  a semicubic parabola). If formula_8 one gets Properties of involutes. In order to derive properties of a regular curve it is advantageous to suppose the arc length formula_10 to be the parameter of the given curve, which lead to the following simplifications: formula_11 and formula_12, with formula_13 the curvature and formula_14 the unit normal. One gets for the involute: and the statement: and from formula_19 follows: Examples. Involutes of a circle. For a circle with parametric representation formula_25, one has formula_26. Hence formula_27, and the path length is formula_28. Evaluating the above given equation of the involute, one gets for  the parametric equation of the involute of the circle. The formula_7 term is optional; it serves to set the start location of the curve on the circle. The figure shows involutes for formula_31 (green), formula_32 (red), formula_33 (purple) and formula_34 (light blue). The involutes look like Archimedean spirals, but they are actually not. The arc length for formula_35 and formula_36 of the involute is Involutes of a semicubic parabola. The parametric equation formula_38 describes a semicubical parabola. From formula_39 one gets formula_40 and formula_41. Extending the string by formula_42 extensively simplifies further calculation, and one gets Eliminating yields formula_44 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 formula_45, the tangent vector is formula_46, and, as formula_47 its length is formula_48. Thus the arc length from the point is formula_49 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 formula_51 describes a cycloid. From formula_52, 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 (Parallel curves of a cycloid are not cycloids.) Involute and evolute. The evolute of a given curve formula_59 consists of the curvature centers of formula_59. Between involutes and evolutes the following statement holds: Application. The involute has some properties that makes it extremely important to the gear industry: If two intermeshed gears have teeth with the profile-shape of involutes (rather than, for example, a traditional triangular shape), they form an involute gear system. Their relative  rates of rotation are constant while the teeth are engaged. The gears also always make contact along a single steady line of force. 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 gear teeth bear the involute shape. 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.