Exercise set 2: Parametric Curves

1. Consider the parametric planar curve (Subscript[c, 1]) = {t, -(Sqrt[1-(t^2)])}, t ∈ [0, 1].
1.1 Plot the curve (Subscript[c, 1]).
1.2 Check whether the points [1,1] and [ (1/4), ((-(Sqrt[15]))/4)] lie on the curve.
1.3 Find the tangent and the unit tangent vector to the curve (Subscript[c, 1]) at the point t = (π/6).
1.4 Sample 10 points on the curve and plot the tangent vectors at these points.
1.5 Compute the speed function of the curve and plot the result.
1.6 Compute the length of the curve.
1.7 Parametrize the curve by arc length and plot the speed function of the new curve.
1.8 Approximate (Subscript[c, 1]) by line segments using 5 points and compute the length of the approximation. Refine your approximation by sampling more points and observe how the length changes.

2. Consider the spatial curve (Subscript[c, 2]) = {cos(t), sin(t), t} for t ∈ [0, 6π ].
2.1 Plot the curve (Subscript[c, 2]).
2.2 Compute the tangent vector and the unit tangent vector to the curve (Subscript[c, 2]) at the point t = 2. Plot the vector.
2.3 Compute the speed function of the curve.
2.4 Parametrize the curve by arc length.

3. Consider the planar curve (Subscript[c, 3]) = {sin((t^2)), t cos(t)}, t ∈ [-2,2].
3.1 Compute and plot the signed curvature of (Subscript[c, 3]).
3.2 Compute and plot osculating circles at the points t = 0, t = (5/4) and t = (7/4).
3.3 Approximate the curve by line segments and estimate its signed curvature at the points t = 0, t = (5/4) and t = (7/4). Compare the results for different numbers of points.

4. Consider the planar curve (Subscript[c, 4]) = {(t^2) + 3t, ((3t)/((t^2)+1))}, t ∈ [-3,0].
4.1 Check whether the points [-2, -(3/2)] and [-1, -(10/9)] lie on the curve.
4.2 Compute and plot the signed curvature of (Subscript[c, 4]). Check if there is any point of inflection.
4.3 Translate the curve so that its singular point (the point where the curve intersects itself) is at the origin and rotate the curve by angle (π/3). Plot the new curve.
4.4 Compute and plot the signed curvature of the new curve.
4.5 Plot the osculating circles of the curve (Subscript[c, 4]) at several points, that are close to the point of inflection.
4.6 Approximate the curve (Subscript[c, 4]) by line segments and estimate the signed curvature at the point of inflection.

5. Use the Manipulate[] function to animate the change of tangent, normal, and binormal vector of (Subscript[c, 2]) for t ∈ [0, 6π ].