Homework 1 (until 27.3.2026)

Consider a planar Bézier curve c of degree 7, parameterized over the interval [0,1] and given by the following control points and 
{{-2, 0}, {-2, -1}, {5, 0}, {3, 3}, {-2, 4}, {0, -1}, {-2, 1}, {-2, 0}}.
Do not use the built-in functions for Bézier curves or Berstein basis!

1. Plot the curve c.

2. Compute and plot the speed function of c. Make sure that the axes of the graph intersect at {0,0}.

3. For values	a) k = 10,		b) k = 20,		c) k = 40,		d) k = 80.
sample k+1 points (Subscript[Q, 1]), (Subscript[Q, 2]), ..., (Subscript[Q, k+1]) on the curve c and use these points to approximate the curve by line segments. Plot both the approximation and the curve c into the same image.
(Create a separate image for each value of k.)

4. Compute the length of the curve using an integral of the first kind.  Create a table that for each k = 10, 20, 40, 80 lists the length of the approximation from 3. and the error (i. e. difference  from the length computed via integration).
Hint: You can create a table from any data stored in a list by using the command TableForm[list].

5. Compute the signed curvature of c as a function of t and plot its graph.

6. Determine the signed curvature at (Subscript[t, 0])=(3/10). Use the line segment approximations from 3. to find an approximate value of signed curvature at (Subscript[t, 0]). Create a table that for each k = 10, 20, 40, 80 lists the approximate curvature and the error. Plot the osculating circle at this point.

7. Find numerically all the local maxima and minima of the signed curvature of c. Plot the osculating circles at the corresponding points.
Hint: Use the command NSolve[...] but not more complex commands.

8. Verify that c is simple, closed and the tangent vectors at c(0) and c(1) coincide. Determine its orientation by computing the  integral of the first kind of its signed curvature. Then compute the area enclosed by curve c.

9. Translate and rotate the curve c, so that the point c((3/10)) lies at the origin and the tangent vectors at this point is a multiple of {1, 0}. Plot the new curve with the point and the tangent vector into one image.