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The parabola $y = x^2$ may be expressed as the parametric curve $\boldsymbol{\alpha}(u) = \langle u, u^2 \rangle$. Find a parametric equation $\boldsymbol{\beta}(u)$ of the involute that starts at the parabola's vertex.
The static and animated diagrams below correspond to the parameter range $0 \le u \le 1$. See what you can do to produce similar diagrams. Yours need not be exactly like mine.
![[involute-of-parabola.png]](involute-of-parabola.png)
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![[involute-of-parabola.gif]](involute-of-parabola.gif)
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A parabola is one of the rare curves whose involutes may be expressed symbolically in terms of elementary functions. The circle and the cycloid are another two such curves.
The diagrams below show a line segment of length $2\pi$ unwrapping off of a unit circle. Can you produce those diagrams? Yours need not be exactly like mine.
![[wrap-around-circle.png]](wrap-around-circle.png)
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![[wrap-around-circle.gif]](wrap-around-circle.gif)
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This demo's wrapping operation is applied (twice) to produce the animated torus on the course's main web page.