The evolute of a curve (blue parabola) is the locus of all its centers of curvature (red).The evolute of a curve (in this case, an ellipse) is the envelope of its normals.
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.[1] 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 M be a smooth, regular submanifold in Rn. For each point p in M and each vector v, based at p and normal to M, we associate the point p + v. This defines a Lagrangian map, called the normal map. The caustic of the normal map is the evolute of M.[2]
Evolutes are closely connected to involutes: A curve is the evolute of any of its involutes.
^Weisstein, Eric W. "Circle Evolute". MathWorld.
^Arnold, V. I.; Varchenko, A. N.; Gusein-Zade, S. M. (1985). The Classification of Critical Points, Caustics and Wave Fronts: Singularities of Differentiable Maps, Vol 1. Birkhäuser. ISBN 0-8176-3187-9.
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...
used), cubocycloid, and paracycle. It is nearly identical in form to the evolute of an ellipse. If the radius of the fixed circle is a then the equation...
\varphi +\cos 3\varphi ,3\sin \varphi +\sin 3\varphi )} (see above). The evolute of a curve is the locus of centers of curvature. In detail: For a curve...
of the circle k {\displaystyle k} , one gets a limaçon of Pascal. The evolute of a curve is the locus of centers of curvature. In detail: For a curve...
as the string is either unwrapped from or wrapped around the curve. The evolute of an involute is the original curve. It is generalized by the roulette...
curvature, form another curve, called the evolute of C. Vertices of C correspond to singular points on its evolute. Within any arc of a curve C within which...
Mamikon's theorem. The envelope of the normals of the tractrix (that is, the evolute of the tractrix) is the catenary (or chain curve) given by y = a cosh x/a...
Anahoplites). Where it does not cover those preceding, the specimen is said to be evolute (e.g., Dactylioceras). A thin living tube called a siphuncle passed through...
the curve and the envelope of such normals is its evolute. Therefore, YR is tangent to the evolute and the point Y is the foot of the perpendicular from...
(red) and its evolute (blue), the locus of its centers of curvature. The four marked vertices of the ellipse correspond to the four cusps of the evolute....
amount of dispersion. As a mathematician, Huygens developed the theory of evolutes and wrote on games of chance and the problem of points in Van Rekeningh...
pendulum. These and other results led Huygens to develop his theory of evolutes and provided the incentive to write a much larger work, which became the...
An ellipse (red), its evolute (blue), and its medial axis (green). The symmetry set, a super-set of the medial axis, is the green and yellow curves. One...
Ptychitidae and, Isculitidae. Nannites is very small, subglobose, generally evolute and smooth, with a rounded venter, and simple goniatitic sutures. Treatise...
between the origin and point (4,8), one gets the arc length 9.073. The evolute of the parabola ( t 2 , t ) {\displaystyle (t^{2},t)} is a semicubical...
varied genus that makes up the Jurassic Cenoceras complex. Cenoceras is evolute to involute, and globular to lentincular; with a suture that generally...
A. (1991–2001) El Car Atom (2022–present) Aurus Motors (2018–present) Evolute (2022–present) GAZ (1932–present) Lada (1966–present) Moskvich (1930–present)...
contribution from Mikhail Gaichenkov). As the Archimedean spiral grows, its evolute asymptotically approaches a circle with radius |v|/ω. Sometimes the term...
The shell, which grew to be rather large, is evolute, strongly and sharply ribbed. Ribbing is both simple and biplicate, with primary ribs bifurcating...
one cusp is a cardioid, two cusps is a nephroid. An epicycloid and its evolute are similar. We assume that the position of p {\displaystyle p} is what...
arbitrary dimension Dioptre, a measurement of curvature used in optics Evolute, the locus of the centers of curvature of a given curve Fundamental theorem...
an epicycloid with k cusps moves snugly inside one with k+1 cusps. The evolute of a hypocycloid is an enlarged version of the hypocycloid itself, while...
Global Information
