Involute

An involute (also known as an evolvent) is a particular type of curve that is dependent on another shape or curve. It is the path taken by the end of an idealized string as it wraps (or unwraps) around a curve.Tractrix (red) as an involute of a catenaryThe evolute of a tractrix is a catenary An involute (also known as an evolvent) is a particular type of curve that is dependent on another shape or curve. It is the path taken by the end of an idealized string as it wraps (or unwraps) around a curve. Involute curves are described using the differential geometry of curves, and are obtained from another given curve by one of two methods. The evolute of an involute is the original curve, less portions of zero or undefined curvature. 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). Let be x → = c → ( t ) , t ∈ [ t 1 , t 2 ] {displaystyle {vec {x}}={vec {c}}(t),;tin } a regular curve in the plane with its curvature nowhere 0 and a ∈ ( t 1 , t 2 ) {displaystyle ain (t_{1},t_{2})} , then the curve with the parametric representation is an involute of the given curve.The integral describes the actual length of the free part of the string in the interval [ a , t ] {displaystyle } and the vector prior to that is the tangent unitvector. Adding an arbitrary but fixed number l 0 {displaystyle l_{0}} to the integral results in an involute corresponding to a string, which is extended by l 0 {displaystyle l_{0}} . Hence: the involute can be varied by parameter a {displaystyle a} and/or adding a number to the integral (see Involutes of a semicubic parabola). If x → = c → ( t ) = ( x ( t ) , y ( t ) ) T {displaystyle {vec {x}}={vec {c}}(t)=(x(t),y(t))^{T}} one gets In order to derive properties of a regular curve it is advantageous to suppose the arc length s {displaystyle s} to be the parameter of the given curve. Because of the simplifications in this case: | c → ′ ( s ) | = 1 {displaystyle ;|{vec {c}}'(s)|=1;} and c → ″ ( s ) = κ ( s ) n → ( s ) {displaystyle ;{vec {c}}''(s)=kappa (s){vec {n}}(s);} , with κ {displaystyle kappa } the curvature and n → {displaystyle {vec {n}}} the unit normal, one gets for the involute: