Exercise set 2: 2D Curves

1. 2D curves
Consider a parametric curve c(t) = {Cos[2t]+Sin[(t/2)], (t^3)-2t}, t ∈ [-1, 1].
1.1 Plot the curve c. Make sure that both axes have the same length unit (are not scaled).
1.2 Compute the speed function (derivative length) and plot its graph. Make sure that the axes intersect at {0,0}.
1.3 Compute numerically the length of c.
1.4 Compute the signed curvature as a function of t and plot its graph.
1.5 Approximate the curve c by line segments and use these segments to find an approximate value of signed curvature at (Subscript[t, 0])=0. What is the difference between this approximation and an actual value?
1.6 Compute the center and radius of osculating circle at the point (Subscript[t, 0]) = 0. Plot the osculating circle at this point into one picture with the curve c.
1.7 Construct matrix A and vector p, such that x ↦ Ax + p is a rotation about the point {1,0} by the angle (π/3).
1.8 Use the rotation from 1.7 to transform the curve c. Plot the new curve. Plot the osculating circle of the new curve at (Subscript[t, 0]) = 0 into the same picture.

2. Speed and curvature
A curve is given uniquely (up to a direct isometry) by its speed function s and its signed curvature κ.
2.1 Find a curve d on the interval t ∈ (0, 1), whose speed function is s(t)=(1/(Sqrt[1+t]))and whose signed curvature is κ(t)=(1/(Sqrt[1-t])). Plot the result.
2.2 Modify the curve d from 2.1 so that d(0) = {0,0} and d´(0) = {1,1}.