Homework 1 (until 7.11.2025)

Consider the planar curve c = { ((-1+(t^3))/(1+(t^2))), (t/2)-2(t^2)}, t ∈ [-1, 1].

1. Compute and plot the speed function of c. Compute (at least numerically) the length of c.

2. Sample 10 points (Subscript[P, 1]), (Subscript[P, 2]), ..., (Subscript[P, 10]) on the curve c and use them to approximate the curve by line segments. Compute the length of this approximation.

3. Sample enough points, so that the length of approximation differs from the length of c by less then (10^(-5)). (Determine experimentally, how many points is “enough”.)

4. Compute and plot the unit tangent and normal vectors at the sampled points (Subscript[P, 1]), (Subscript[P, 2]), ..., (Subscript[P, 10]).

5. Translate and rotate the curve c, so that the point corresponding to t = -(1/2) lies at the origin and the tangent vector at this point is a multiple of (-1,1).

6. Compute the inflection points of c numerically (only the real ones).

7. Find the extrema of signed curvature of c. Plot the osculating circles at these points.

8. Pick one of the extremal points from 7. Use line segments to estimate the signed curvature at this point.