In [url=https://www.geogebra.org/m/vF4PxQFz]https://www.geogebra.org/m/vF4PxQFz[/url] it was shown that the parallel curve to a parabola is not a parabola. Given the parabola with equation [math]y=\frac{1}{4a}{{x}^{2}}[/math] or, in parametric form, [br][math]\left\{ \begin{array}{*{35}{l}}[br] x=2at \\[br] y=a{{t}^{2}} \\[br]\end{array} \right.[/math][br]the parametric equations of the parallel curves were derived, with simple geometric arguments, as [br][math]\left\{ \begin{array}{*{35}{l}}[br] x=2at-d\frac{t}{\sqrt{1+{{t}^{2}}}} \\[br] y=a{{t}^{2}}+d\frac{1}{\sqrt{1+{{t}^{2}}}} \\[br]\end{array} \right.[/math][br][br]Now, there’s another way to trace the parallel curves to a parabola.[br]It’s not as simple as the first one but it’s rather interesting as it allows to understand the important concept, in the field of differential geometry, of [b]evolute[/b] and [b]involute[/b].[br]The [url=https://en.wikipedia.org/wiki/Evolute]evolute[/url] of a given curve can be defined as the locus of all the centers of curvature built from the points of the curve. Alternatively an evolute can be seen as the envelope of the normals drawn from the points of the starting curve. [br]In the case of a parabola its evolute is a [url=https://en.wikipedia.org/wiki/Semicubical_parabola]semi-cubical parabola[/url], an interesting curve that also has the property of being [i]isochrone[/i].The [url=https://en.wikipedia.org/wiki/Involute]involute[/url] of a curve is more difficult to understand and visualize.[br]One of its classical definitions says that:[br][i]“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.”[/i]I adopted this definition in the present worksheet.[br]Changing the point to which the string is attached and/or its length we can trace different involutes of a given curve.[br]In the present simulation we have the following equations:[br][math]\begin{align}[br] & \textbf{parabola: }y=\frac{1}{4a}{{x}^{2}}\ \ \ \ or\ \ \ \left\{ \begin{array}{*{35}{l}}[br] x=2at \\[br] y=a{{t}^{2}} \\[br]\end{array} \right. \\ [br] & \textbf{evolute: }\,\,\,{{\left( y-2a \right)}^{3}}=\frac{27}{4}a{{x}^{2}}\,\,\,\,or\,\,\,\,\,\left\{ \begin{array}{*{35}{l}}[br] x=2a{{t}^{3}} \\[br] y=3a{{t}^{2}}+2a \\[br]\end{array} \right. \\ [br]\end{align}[/math] [br]The involute is traced by attaching the string alternatively to the points [math]{{A}_{1}}[/math] and [math]{{A}_{2}}[[/math] with a lozenge shape. If we call [math]{{s}_{A}}[/math] the arc length from [math]V'[/math] to one of the lozenge shaped points, then the string length tracing the curve parallel to the parabola at distance [math]d[/math] will have length [math]{{s}_{A}}+2a-d[/math].[br]The arc length [math]{{s}_{A}}[[/math] can be calculated as[br][math]{{s}_{A}}=\int{ds}=\int_{0}^{{{t}_{A}}}{\sqrt{x'{{\left( t \right)}^{2}}+y'{{\left( t \right)}^{2}}}dt}=\int_{0}^{{{t}_{A}}}{6at\sqrt{1+{{t}^{2}}}d}t=2a\left( {{\left( 1+{{t}_{A}}^{2} \right)}^{3/2}}-1 \right) [[/math][br]Above definition of the involute doesn’t help understanding what happens when the involute presents self-intersecting points. [br]In this case, another way to construct the involute of a curve could be: [br][i]“...replace the taut string by a line segment that is tangent to the curve on one end, while the other end traces out the involute. The length of the line segment is changed by an amount equal to the arc length traversed by the tangent point as it moves along the curve.”[/i][br][br]The worksheet can also show the evolute of the evolute (or second evolute) of the parabola, that is the evolute of the semi-cubical parabola.[br]By checking “[i]show second svolute[/i]” and “[i]show trace[/i]” you’ll also see how an evolute can be thought of as the envelope of the normals drawn from the points of the starting curve.[br]