3rd practicals, 19. 10. 2022, (planar) curves & parametrization, part 1

1. Plot the parametric planar curve (Subscript[c, 1])={t,-(Sqrt[1-(t^2)])}t∈[0,1].

2. Do the points [1, 1] a [(1/4),- ((Sqrt[15])/4)] lie on the curve (Subscript[c, 1])? Find the point corresponding to t = (1/2).

3. Find the tangent and the unit tangent vector to the curve (Subscript[c, 1]) at the point t = (π/6).

4. Sample 10 points on the curve (Subscript[c, 1]) and plot tangents vectors at these points.

5. Compute the speed function of the curve (Subscript[c, 1]) and plot the function.

6. Compute the length of (Subscript[c, 1]).

7. Parametrize the curve (Subscript[c, 1]) by arc length and plot the speed function of the new curve.

8. Approximate (Subscript[c, 1]) by a line segments,start for instance with 5 points. Compute the length of the approximation. What happens, when you refine the division of the curve (sample more points)? Try it out!

9. Compute the speed function of a spatial curve  (Subscript[c, 2])={cos(t),sin(t),t}, t∈[0,6π] and plot it.

10. Compute the tangent vector to (Subscript[c, 2]) at the point t=2 and plot it together with the curve.

11. Parametrize (Subscript[c, 2]) by arc length.

12. Animate the change of tangent/normal/binormal vector of (Subscript[c, 2]) for t∈[0,6π]. Use Manipulate command.