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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. [1] The evolute of an involute is the original curve. It is generalized by the roulette family of curves. That is, the involutes of a curve are the roulettes of the curve generated by a straight line.
Aug 6, 2022 · Definition 1 1. The evolute of a given curve γ γ is another curve to which all the normals of γ γ are tangent. Definition 2 2. Given a γ γ, another curve is called an evolute of γ γ if it is an involute of the second. With the second definition, and arc-length parametrization, an internet source shows that its evolute as.
The involutes of a plane curve are the curves traced by the end of a wire tightened along and winding itself along . In other words, they are the traces on the plane of a point on a line pivoting without slipping on (therefore, they are special cases of roulettes). Thus, they also are the orthogonal trajectories of the family of the tangents to ...
is called the center of curvature. Here, nis the unit normal of . (b) The evolute (caustic) of the curve is the curve traced by the centers of curvature. Thus, a parametrization of the evolutive is e: I! R2; e(s) = (s) + 1 (s) n(s): (c) The involute of a plane curve is a curve whose evolute is the initial curve . Remark. Properties of the evolute.
Nov 14, 2024 · An evolute is the locus of centers of curvature (the envelope) of a plane curve's normals. The original curve is then said to be the involute of its evolute. Given a plane curve represented parametrically by , the equation of the evolute is given by. where are the coordinates of the running point, is the radius of curvature.
In the study of plane curves, once we defined the evolute of a curve, we considered the inverse problem of finding all curves that have a given curves as evolute. We may consider the analogous problem for space curves. If a curve . Y is the evolute of a curve X, then X is said to be an "involute" of the curve Y. As in the planar case, we may ...
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