Bezier curves

1. Plot Bernstein bases of degrees 6 and 7 (on the interval [0,1]). Sum the values of the bases functions at the point (π/4). 
Verify, that some of the relations from the lecture hold: symmetry, recurrence, derivative... (verify it just by a computation or a figure)

2. Consider the following points: {-3,-1},{-2,-(3/2)},{-1,-(4/3)},{0,-(1/2)},{1,(1/2)},{(1/2),(7/6)},{-(1/2),2} and plot a Bezier curve c with these points as the control points. Together with the curve plot also its control polygon.

3.  Slightly change one of the points and observe how does the curve change.

4. Rotate the curve c by (π/3) around the point [1,(1/2)].

5. Compute and plot the derivative of the curve c.

6. Compute the signed curvature of c. Plot an osculating circle at some point c(t). Are there any points of inflection?

7. Write a function that for a given Bezier curve and a parameter t finds the corresponding point on the curve (De Casteljau algorithm).

8. Write a function that for a given Bezier curve c of degree n and a parameter t  divides the curve c at the point c(t) into two curves of the same degree (Subdivision).

9. Write a function that for a given Bezier curve of degree n returns a curve with the same image but of degree n+1 (Degree Elevation).