Let x: I → R2 be a path of class C2 that is not a straight line and such that x’(t) 0. Let
This is the path traced by the center of the osculating circle of the path x . The quantity ρ = 1/κ is the radius of the osculating circle and is called the radius of curvature of the path x. The path e is called the evolute of the path x. Exercises 20–25 involve evolutes of paths.
Use a computer algebra system to calculate the formula for the evolute of the cardioid x(t) = (2a cos t(1 + a cos t), 2a sin t(1 + a cos t)).
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