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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.
Definition. A curve that is obtained by attaching a string which is imaginary and then winding and unwinding it tautly on the curve given is called involute in differential geometry. Involute or evolvent is the locus of the free end of this string.
One of its classical definitions says that: “an involute is a curve obtained from another given curve by attaching an imaginary taut string to the given curve and tracing its free end as it is wound (or unwound) onto that given curve.”
Oct 28, 2024 · Extend the string so that it is tangent to the curve at the point of attachment. Then wind the string up, keeping it always taut. The locus of points traced out by the end of the string is called the involute of the original curve, and the original curve is called the evolute of its involute.
Aug 6, 2022 · Involute: Given a regular parametric curve $\gamma$, an involute for $\gamma$ is another parametric curve $\gamma^*$ defined on the same interval such that i) $\gamma^*(t)$ lie on $T(t)$ (tangent to $\gamma$ at $t$). ii) $T(t)$ and $T^*(t)$ (corresponding tangent vectors) are orthogonal.
All involutes of a given curve are parallel to each other. This property also makes it easy to see that evolute of a curve is the envelope of its normals. Two involutes of a ellipse. Involute and Evolute. If curve A is the evolute of curve B, then curve B is the involute of curve A.
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Involute : If C is a curve and C' is its evolute, then C is called an involute of C'. Any parallel curve to C is also an involute of C '. Hence a curve has a unique evolute but infinitely many involutes.
