Let alpha (t) be a plane curve, and N(t) the unit normal vector as defined in the text. Now define a different curve. beta (t) = alpha(t) + 1 / K(t)N(t) as the evolute of alpha(t). In this exercise, we will figure out exactly what makes the evolute so special. Let alpha(t) = t, t2 . Compute the evolute of alpha (t). Sketch alpha (t) and its evolute. Show that |alpha(t) - beta(t)| is the radius of the osculating circle for the about curve. Show that the tangent to beta(t) passes through alpha(t) and is perpendicular to alpha(t) for the above curve. Show that the point beta(t) is the center of the osculating circle at the point alpha(t) for the above curve. Verify your solutions to the problem above by specifically computing the curvature of alpha(t) at t = 0, t = 1, t = -2.
Show transcribed image textLet alpha (t) be a plane curve, and N(t) the unit normal vector as defined in the text. Now define a different curve. beta (t) = alpha(t) + 1 / K(t)N(t) as the evolute of alpha(t). In this exercise, we will figure out exactly what makes the evolute so special. Let alpha(t) = t, t2 . Compute the evolute of alpha (t). Sketch alpha (t) and its evolute. Show that |alpha(t) - beta(t)| is the radius of the osculating circle for the about curve. Show that the tangent to beta(t) passes through alpha(t) and is perpendicular to alpha(t) for the above curve. Show that the point beta(t) is the center of the osculating circle at the point alpha(t) for the above curve. Verify your solutions to the problem above by specifically computing the curvature of alpha(t) at t = 0, t = 1, t = -2.