1. Given three control points on the xy-plane (-1,0), (0,1) and (2,0), do the following:
  2. In the variation diminishing property, what if you have a line or a plane that passes through a control point or contains a line segment of the control polyline? Suggest a proper counting of intersection points and verify your claim with examples.
  3. A Bézier curve of degree 2 defined by three control points P0, P1 and P2 is a portion of a conic section. What type of this conic section is it? Is it a portion of a parabola, a hyperbola or an ellipse? You can assume the given control points are in the xy-coordinate plane.
  4. Suppose Bézier curve C(u) (resp., D(u)) of degree n is defined by control points P0, P1, ..., Pn (resp., O0, Q1, ..., Qn). If the curves are identical (i.e., C(u) = D(u) for every u in [0,1]), then the corresponding control points are also identical (i.e., Pi = Qi for all 0 <= i <= n).
    Hint: First show that if (1-u)A+uB is a zero vector for every u in [0,1], then A and B are both zero vectors. Then, work the de Casteljau's algorithm backward to show that Pi - Qi is a zero vector for all 0 <= i <= n.
  5. Suppose Bézier curve C(u) of degree n is defined by control points P0, P1, ..., Pn.
    1. Prove the following:

    2. Show that curve C(u) can be rewritten to the following matrix form:

      where entry mij is defined as follows:

    Therefore, a Bézier curve can be rewritten using the traditional polynomial form in u0 = 1, u1, u2, ...., un. This is the so-called monomial form and the basis functions are u0 = 1, u1, u2, ...., un. However, the use of this monomial form is computationally unstable.
  6. Show that the maximum of Bn,i(u) occurs at u = i/n and that the maximum value is

  7. Verify the following results with your calculus knowledge:
  8. The discussion of joining two Bézier curves with C1-continuity assumes the domain of the curves is [0,1]. Suppose the domain of the first curve is [0,s] and the domain of the second curve is [s,1]. Redo the calculation. What is your conclusion? Is there any modification required?
  9. Prove the following:

    where Dik's are the k-th difference points and C(k,j) is the binomial coefficient defined as follows:

    With this formula, we can express a higher derivative using the original control points rather than using finite difference points.

  10. After subdividing a Bézier curve of degree p at s, we have two Bézier curves of degree p, one on interval [0,s] while the other on [s,1]. Show that these two curves are of C1 continuous at the joining point.
    Hint: Suppose the last two control points of the curve on [0,s] are Pp-1 and Pp, and the first two control points of the curve on [s,1] are Q0 and Q1. Then, we have Pp-1, Pp = Q0 and Q1 are on the same line, and the ratio of the distance from Pp-1 to Pp = Q0 and the distance from Pp = Q0 to Q1 is equal to s due to subdivision. Now, change the variables of both curves so that they have domain on [0,1]. A simple calculation will lead to the desired conclusion.