Evolute

In the differential geometry of curves, the evolute of a curve is the locus of all its centers of curvature. That is to say that when the center of curvature of each point on a curve is drawn, the resultant shape will be the evolute of that curve. The evolute of a circle is therefore a single point at its center. Equivalently, an evolute is the envelope of the normals to a curve. The evolute of a curve, a surface, or more generally a submanifold, is the caustic of the normal map. Let be a smooth, regular submanifold in . For each point in and each vector , based at and normal to , we associate the point . This defines a Lagrangian map, called the normal map. The caustic of the normal map is the evolute of . Evolutes are closely connected to involutes: A curve is the evolute of any of its involutes.

In the differential geometry of curves, the evolute of a curve is the locus of all its centers of curvature. That is to say that when the center of curvature of each point on a curve is drawn, the resultant shape will be the evolute of that curve. The evolute of a circle is therefore a single point at its center. Equivalently, an evolute is the envelope of the normals to a curve. The evolute of a curve, a surface, or more generally a submanifold, is the caustic of the normal map. Let be a smooth, regular submanifold in . For each point in and each vector , based at and normal to , we associate the point . This defines a Lagrangian map, called the normal map. The caustic of the normal map is the evolute of . Evolutes are closely connected to involutes: A curve is the evolute of any of its involutes. If \vec x= \vec c(t),\; t\in [t_1, t_2] is the parametric representation of a regular curve in the plane with its curvature nowhere 0 and \rho(t) its curvature radius and \vec n(t) the unit normal pointing to the curvature center, then \vec E(t) = \vec c(t) + \rho (t) \vec n(t) describes the evolute of the given curve. For \vec c(t)=(x(t),y(t))^\mathsf{T} and \vec E=(X,Y)^\mathsf{T} one gets X(t) = x(t) - \frac{y'(t) \Big(x'(t)^2+y'(t)^2\Big)}{x'(t) y(t) - x(t) y'(t)} and Y(t) = y(t) + \frac{x'(t) \Big(x'(t)^2+y'(t)^2\Big)}{x'(t) y(t) - x(t) y'(t)}. For a curve \Gamma expressed implicitly in the form f(x, y) = 0 , the Evolute can be determined through a system of equations that relates the original curve to its normal lines. At each point on the curve \Gamma , the direction of the normal vector is given by: \left( -\frac{f_y}{f_x}, \, \frac{x f_y - y f_x}{f_x} \right) where f_x = \frac{\partial f}{\partial x} and f_y = \frac{\partial f}{\partial y} . These components describe the direction of the normal vector at each point on the curve. The Evolute can be obtained by solving the following system (see for details): \begin{aligned} 0 &= f(x, y), \\ u &= -\frac{f_y}{f_x}, \\ v &= \frac{x f_y - y f_x}{f_x}, \end{aligned} where u and v represent the components of the normal vector. Eliminating the variables x and y from this system yields a single algebraic equation in terms of u and v , which defines the evolute of the curve. In order to derive properties of a regular curve it is advantageous to use the arc length s of the given curve as its parameter, because of \left|\vec c'\right| = 1 and \vec n' = -\vec c'/\rho (see Frenet–Serret formulas). Hence the tangent vector of the evolute \vec E=\vec c +\rho \vec n is: \vec E' = \vec c' +\rho'\vec n + \rho\vec n' = \rho'\vec n\ . From this equation one gets the following properties of the evolute: At points with \rho' = 0 the evolute is not regular. That means: at points with maximal or minimal curvature (vertices of the given curve) the evolute has cusps. (See the diagrams of the evolutes of the parabola, the ellipse, the cycloid and the nephroid.) For any arc of the evolute that does not include a cusp, the length of the arc equals the difference between the radii of curvature at its endpoints. This fact leads to an easy proof of the Tait–Kneser theorem on nesting of osculating circles. The normals of the given curve at points of nonzero curvature are tangents to the evolute, and the normals of the curve at points of zero curvature are asymptotes to the evolute. Hence: the evolute is the envelope of the normals of the given curve. At sections of the curve with \rho' > 0 or \rho' the curve is an involute of its evolute. (In the diagram: The blue parabola is an involute of the red semicubic parabola, which is actually the evolute of the blue parabola.) Proof of the last property: Let be \rho' > 0 at the section of consideration. An involute of the evolute can be described as follows: \vec C_0=\vec E -\frac{\vec E'}{\left|\vec E'\right|} \left(\int_0^s\left|\vec E'(w)\right| \mathrm dw + l_0 \right) , where l_0 is a fixed string extension (see Involute of a parameterized curve ). With \vec E=\vec c +\rho\vec n\; ,\; \vec E'=\rho'\vec n and \rho'>0 one gets \vec C_0 = \vec c +\rho\vec n-\vec n \left(\int_0^s \rho'(w) \; \mathrm dw \;+l_0\right)= \vec c + (\rho(0) - l_0)\; \vec n\, . That means: For the string extension l_0=\rho(0) the given curve is reproduced. Parallel curves have the same evolute. Proof: A parallel curve with distance d off the given curve has the parametric representation \vec c_d = \vec c + d\vec n and the radius of curvature \rho_d=\rho -d (see parallel curve). Hence the evolute of the parallel curve is \vec E_d = \vec c_d +\rho_d \vec n =\vec c +d\vec n +(\rho -d)\vec n=\vec c +\rho \vec n = \vec E\; . X=\cdots=-4t^3 Y=\cdots=\frac{1}{2} + 3t^2 \, , which describes a semicubic parabola X= \cdots = \frac{a^2-b^2}{a}\cos ^3t Y= \cdots = \frac{b^2-a^2}{b}\sin ^3t \; . These are the equations of a non symmetric astroid. Eliminating parameter t leads to the implicit representation (aX)^{\tfrac{2}{3}} +(bY)^{\tfrac{2}{3}} = (a^2-b^2)^{\tfrac{2}{3}}\ . === Evolute of a cycloid=== For the cycloid with the parametric representation (r(t - \sin t), r(1 - \cos t)) the evolute will be: X=\cdots=r(t + \sin t) Y=\cdots=r(\cos t - 1) which describes a transposed replica of itself. The evolute of a log-aesthetic curve is another log-aesthetic curve. One instance of this relation is that the evolute of an Euler spiral is a spiral with Cesàro equation \kappa(s) = -s^{-3} . (X, Y)= \left(-y'\frac{{x'}^2+{y'}^2}{x'y-xy'}\; ,\; x'\frac{{x'}^2+{y'}^2}{x'y-xy'}\right) .

Text derived from https://en.wikipedia.org/w/index.php?oldid=1369257467 under CC BY-SA 4.0; modifications have been made.