Evolute 842387 215540997 2008-05-28T17:35:45Z Giftlite 37986 /* Radial of a curve */ +. [[Image:Ellipse evolute.svg||right|thumb|240px|An [[ellipse]] (red) and its evolute (blue), the dots are the vertices of the curve, each vertex corresponds to a cusp on the evolute. the evolute of an ellipse is called an astroid.]] [[Image:Evolute1.gif||right|thumb|240px|How the above evolute is constructed.]] In the [[differential geometry of curves]], the '''evolute''' of a [[curve]] is the [[locus (mathematics)|locus]] of all its [[Osculating circle|centers of curvature]]. Equivalently, it is the [[envelope (mathematics)|envelope]] of the [[perpendicular|normals]] to a curve. The original curve is an [[involute]] of its evolute. (Compare [[Media:Evolute2.gif]] and [[Media:Involute.gif]]) ==History== [[Apollonius of Perga|Apollonius]] (c. 200 BC) discussed evolutes in Book V of his ''Conics''. However, [[Christiaan Huygens|Huygens]] is sometimes credited with being the first to study them (1673). ==Equations== Let <math>(x, y) = (x(t), y(t))</math> be a parametrically defined plane curve. Let <math>R = 1/\kappa</math> be the [[Osculating circle|radius of curvature]] and <math>\phi</math> be the [[tangential angle]]. Then the [[Osculating circle|center of curvature]] at <math>(x, y)</math> is given by <math>(x - R \sin \phi, y + R \cos \phi)</math> and we may take <math>(X, Y) = (x - R \sin \phi, y + R \cos \phi)</math> as parametric equations for the evolute. We have <math>(\cos \phi, \sin \phi) = \frac{(x', y')}{(x'^2+y'^2)^{1/2}}</math> and <math>R = 1/\kappa = \frac{(x'^2+y'^2)^{3/2}}{x'y''-x''y'}</math>, so we may eliminate <math>R</math> and <math>\phi</math> to obtain: <math>(X, Y)= (x-y'\frac{x'^2+y'^2}{x'y''-x''y'}, y+x'\frac{x'^2+y'^2}{x'y''-x''y'})</math> If the curve (x, y) is parametrized by [[arc length]] s (i.e. <math> (x, y) = (x(s), y(s))</math> where <math>|(x', y')|=1</math>; see [[Differential geometry of curves#Length and natural parametrization|natural parametrization]]) then this simplifies to: <math>(X, Y)= (x+\frac{x''}{x''^2+y''^2}, y+\frac{y''}{x''^2+y''^2}).</math> ==Properties== Differentiating <math>(X, Y) = (x - R \sin \phi, y + R \cos \phi)</math> with respect to <math>s</math> we obtain:<br><br> <math>\frac{d}{ds} (X, Y) = (\frac{dx}{ds} - R \cos \phi \frac{d\phi}{ds} - \frac{dR}{ds}\sin \phi, \frac{dy}{ds} - R \sin \phi \frac{d\phi}{ds} + \frac{dR}{ds}\cos \phi)</math>. <br><br> <math>\frac{dx}{ds} = \cos \phi</math>, <math>\frac{dy}{ds} = \sin \phi</math> and <math>\frac{d\phi}{ds} = \kappa = 1/R</math>, so this simplifies to <br><br> <math>\frac{d}{ds} (X, Y) = (-\frac{dR}{ds}\sin \phi, \frac{dR}{ds}\cos \phi) = \frac{dR}{ds}(-\sin \phi,\cos \phi).</math> <br><br> Which has magnitude <math>|\frac{dR}{ds}|</math> and direction <math>\phi \pm \pi/2</math>. This has the following implications: * The [[tangential angle]] of the evolute is <math>\phi \pm \pi/2</math>. (The sign of <math>\pm \pi/2</math> is determined by the sign of <math>\frac{dR}{ds}</math>.) * The tangent to the evolute is normal to the original curve. A curve is the envelope of its tangents so the evolute is also the envelope of the lines normal to the curve. * The arclength along the curve <math>(X, Y)</math> from <math>(X(s_1), Y(s_1))</math> to <math>(X(s_2), Y(s_2))</math> is given by <math>\int_{s_1}^{s_2}\frac{dR}{ds} ds = R(s_2)-R(s_1)</math>. * The original curve is an involute of the evolute. If <math>\phi</math> can be solved as a function of <math>R</math>, say <math>\phi = g(R)</math>, then the [[Whewell equation]] for the evolute is <math>\Phi = g(R) + \pi/2</math>, where <math>\Phi</math> is the tangential angle of the evolute and we take <math>R</math> as arclength along the evolute. From this we can derive the [[Cesàro equation]] as <math>\Kappa = g'(R)</math>, where <math>\Kappa</math> is the [[curvature]] of the evolute. ===Relationship between a curve and its evolute=== [[Image:Evolute and parallel.gif|right|thumb|240px|An ellipse (red), its evolute (blue) and some parallel curves. Note how the parallel curves have cusps when they touch the evolute]] By the above discussion, the derivative of <math>(X, Y)</math> vanishes when <math>\frac{dR}{ds} = 0</math>, so the evolute will have a [[cusp (singularity)|cusp]] when the curve has a [[Vertex (curve)|vertex]], that is when the curvature has a local maximum or minimum. At a point of inflection of the original curve the radius of curvature becomes infinite and so <math>(X, Y)</math> will become infinite, often this will result in the evolute having an [[asymptote]]. Similarly, when the original curve has a cusp where the radius of curvature is 0 then the evolute will touch the the original curve. This can be seen in the figure to the right, the blue curve is the evolute of all the other curves. The cusp in the blue curve corresponds to a vertex in the other curves. The cusps in the green curve are on the evolute. Curves with the same evolute are [[Parallel curve|parallel]]. ==Radial of a curve== A curve with a similar definition is the '''Radial''' of a given curve. For each point on the curve take the vector from the point to the center of curvature and translate it so that it begins at the origin. Then the locus of points at the end of such vectors is called the Radial of the curve. The equation for the radial is obtained by removing the x and y terms from the equation of the evolute. Ths produces <math>(X, Y)= (- R \sin \phi, R \cos \phi)</math> or <math>(X, Y)= (-y'\frac{x'^2+y'^2}{x'y''-x''y'}, x'\frac{x'^2+y'^2}{x'y''-x''y'}).</math> ==Examples== * The evolute of a [[parabola]] is a [[semicubical parabola]]. The cusp of the latter curve is the center of curvature of the parabola at its vertex. * The evolute of a [[Logarithmic spiral]] is a congruent spiral. * The evolute of a [[cycloid]] is a similar cycloid. ==References== [http://mathworld.wolfram.com/Evolute.html Weisstein, Eric W. "Evolute." From MathWorld--A Wolfram Web Resource.] Yates, R. C.: ''A Handbook on Curves and Their Properties'', J. W. Edwards (1952), "Evolutes." pp. 86ff [http://www.2dcurves.com/derived/curvature.html#evolute Evolute on 2d curves.] {{Geometry-stub}} {{Differential transforms of plane curves}} [[Category:differential geometry]] [[Category: Curves]] [[cs:Evoluta]] [[de:Evolute]] [[es:Evoluta]] [[fr:Développée]] [[it:Evoluta]] [[nl:Evolute]] [[pl:Ewoluta]] [[ru:Эволюта]] [[sk:Evolúta]] [[fi:Evoluutta]]