Homework 1

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

2. Compute and plot the speed function of c. Find its extrema.

3. Compute (at least numerically) the length of c.

4. Sample 10 points (Subscript[P, 1, ])(Subscript[P, 2, ])..., (Subscript[P, 10, ]) on the curve c and approximate the curve by line segments using these points. Compute the length of this polygonal chain. How many points do we need for the estimation error to be less then (10^(-5))? (Only experimentally, try using more points).

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

6. Translate and rotate the curve c so that the point c(-(1/2)) lies in the origin and the tangent vector at this point is a multiple of the vector (-1,1). Compute signed curvature of the new curve.

7. Compute and plot the function of signed curvature of c.

8. Compute the points of inflection of c (only numerically and only the real ones :)).

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

10. Using a polygonal chain (connected line segments) estimate the value of the signed curvature at the extremal points.