In the animation in Figure 3.1.3(a), the ellipse $C_1$ is drawn in red; the curve $C_2$, in black; and the line normal to $C_1$, in green. Note that each such normal line is actually tangent to $C_2$.

The curve $C_2$ is called the evolute of $C_1$, and $C_1$ is called the involute of $C_2$.

The evolute of a plane curve can be characterized as the envelope of the family of lines normal to the involute.

The evolute is also the locus of the centers of curvature of the involute. See Section 3.5 for a discussion of curvature and circles of curvature.

Write the equations $u\left(\mathrm{\θ}\right)\=\dots$ and $v\left(\mathrm{\θ}\right)\=\dots$ 

Context Panel: Assign Function

Write the equations $U\left(\mathrm{\θ}\right)\=\dots$ and $V\left(\mathrm{\θ}\right)\=\dots$ 

Context Panel: Assign Function

Write the negative reciprocal of the slope along $C_1$.

Context Panel: Evaluate and Display Inline

Write the expression for the slope along $C_2$.

Context Panel: Evaluate and Display Inline

Write the list $\left[u\left(\mathrm{\θ}\right)\,v\left(\mathrm{\θ}\right)\right]$.

Context Panel: Plot Builder≻2-D plot (parametric)
Global Options≻scaling≻constrained

Write the list $\left[U\left(\mathrm{\θ}\right)\,V\left(\mathrm{\θ}\right)\right]$.

Context Panel: Plot Builder≻2-D plot (parametric)
Global Options≻scaling≻constrained