4th practicals, 21. 10. 2021, Planar curves, signed curvature, osculating circle, signed curvature estimate

Consider the planar curves  (Subscript[c, 1])(t) = {sin ((t^2)), t cos (t)}, t ∈ [-2, 2] and (Subscript[c, 2])(t) = {(t^2)+ 3 t, ((3t)/((t^2)+1))}, t ∈ [-3, 0].

1. Compute and plot signed curvature of (Subscript[c, 1]).

2.Compute and plot osculating circles at points (Subscript[c, 1])(t) for t ∈ {0, (5/4), (7/4)}.

3. Approximate the curve (Subscript[c, 1]) by line segments and estimate its signed curvature at the points (Subscript[c, 1])(t) for t ∈ {0, (5/4), (7/4)}. Does the estimate improve with finer division of [-2,2]?

4. Compute and plot signed curvature of (Subscript[c, 2]). Is there a point of inflection?

5.Translate (Subscript[c, 2]) so that its singular point (the point where the curve intersects itself) is at the origin and rotate it by (π/3). Compute and plot signed curvature of the new curve.

6. Plot osculating circles of the curve (Subscript[c, 2]) at few points that are close to the point of inflection.

7. Approximate the curve (Subscript[c, 2]) by line segments and estimate the signed curvature at the point of inflection (or as close as possible to it).