6. Problems

Problem E-1. Prove the half-way-point property of Bézier curves.

Problem E-2. (a) Find the coordinate functions of the Bézier curve with control points (0,0), (1,0), (1,1), (0,2). (b) Sketch this curve. (Instead of plotting many points, it is usually better to plot a few points and find the tangent vectors there to tell the direction of the curve.)

Problem E-3. Here is a graphical construction of a cubic Bézier curve: For each $t$ value you want, first mark the point corresponding to $t$ on each segment $\overline {P_0 P_1}$, $\overline {P_1 P_2}$, and $\overline {P_2 P_3}$. Call the resulting points $Q_0$, $Q_1$, $Q_2$. (For example, if $t = {\frac 1 2}$, each $Q_i$ is the midpoint of its segment.) Then mark the point corresponding to $t$ on $\overline {Q_0 Q_1}$ and $\overline {Q_1 Q_2}$. Call the resulting points $R_0$, $R_1$. Finally mark the point corresponding to $t$ on $\overline {R_0 R_1}$. Call the resulting point $S_0$. Then $P(t) = S_0$.

(a) Carry out this construction for the control points in Problem E-, for the cases $t = .25$, $t = .5$, $t = .75$.

(b) Prove algebraically that the construction works. (Express the $Q_i$, $R_i$ and $S_0$ in terms of $t$ and the $P_i$, and simplify.)

(c) Prove the Half-way Point property by reasoning graphically--why does the point $P({\frac 1 2})$ come out three-quarters of the way from the midpoint of $\overline {P_0 P_3}$ to the midpoint of $\overline {P_1 P_2}$?

Problem E-4. Which of the properties listed for the cubic Bernstein polynomials are fairly evident from the graphs shown, and which are not as evident?

Problem E-5. Prove the unit sum property of the cubic Bernstein polynomials.

Problem E-6. Prove the values given for the first and second derivatives of the cubic Bernstein polynomials at $t = 0$ and $t = 1$.

Problem E-7. Prove the maximum property for the cubic Bernstein polynomials. (Do derivatives help for all $k$?)

Problem E-8. Prove the linear sum property for the cubic Bernstein polynomials. (One way: First consider just the case where $x_k = {{\frac{\displaystyle k}{\displaystyle 3}}}$ for $k = 0, 1, 2, 3$. The general case follows from this case and the unit sum property.)

Problem E-9. Prove the basis property for the cubic Bernstein polynomials.

(Since the dimension of the vector space is known to be four, it is enough to show either that the cubic Bernstein polynomials span the vector space or that they are linearly independent. For the first way: Show how to express each of $1,t,t^2, t^3$ as a linear combination of cubic Bernstein polynomials. For the second way: Given a linear combination that is the zero function, i.e., is zero for all values of $t$, look at the values of the linear combination at $t = 0$ and at $t = 1$ and also the values of the first derivative at the same places.)

Problem E-10. Give an example of a ``cubic'' Bézier curve (using the $J_{3,k}$) that actually has (a) degree 1; (b) degree 0; (c) degree 2. (Recall that the degree of a polynomial parametric curve is the maximum of the degrees of its coordinate polynomials.)

Problem E-11. For cubic Bézier curves, verify (a) the formulas for the first derivatives at $t = 0$ and $t = 1$; (b) the formulas for the second derivatives at $t = 0$ and $t = 1$; (c) the tangency property (by using (a)). (You may quote any relevant properties of cubic Bernstein polynomials.)

Problem E-12. Prove this version of Taylor's Theorem for cubic Bézier curves: $P(t) = P_0 + 3t(P_1 - P_0) + {\frac{\displaystyle t^2}{\displaystyle 2}} 6(P_0... ...2) + {\frac{\displaystyle t^3}{\displaystyle 6}} 6(-P_0 + 3 P_1 - 3 P_2 + P_3)$. (One method: algebraic expansion. Another method: Verify that both sides have the same values and first, second, and third derivatives at $t = 0$, because then by the regular Taylor's Theorem they must have the same cubic polynomials for each coordinate.)

Problem E-13. Prove the convex hull property of cubic Bézier curves. (You may quote any needed properties of convex combinations and of cubic Bernstein polynomials.)

Problem E-14. If the four control points for a cubic Bézier curve are all equal, then $P(t)$ is constant. Prove this fact two ways: (a) Using the convex hull property of cubic Bézier curves; and (b) using the unit sum property of Bernstein polynomials.

Problem E-15. Consider the affine compatibility property of cubic Bézier curves. On what property of affine transformations and linear combinations does it depend?

Problem E-16. (a) In pictures of examples like those of Section 1, why is it acceptable not to indicate the $x$ and $y$ axes and a scale? (b) Why is it acceptable not to indicate which end of the curve has $t = 0$? (You may quote relevant properties of Bézier curves, if you explain how they apply.)

Problem E-17. Prove the property of Bézier curves with evenly spaced control points. (You may quote any relevant properties of cubic Bernstein polynomials.)

Problem E-18. Find the coordinate functions of the Bézier curve $P(t)$ for the control points $P_0 = (123,123)$, $P_1 = (124,124)$, $P_2 = (125,125)$, and $P_3 = (126,126)$. (Do not attempt a long algebraic method; instead, quote any relevant properties of Bézier curves.)

Problem E-19. In the upper left diagram in Figure , indicate the point on the curve where each of the two non-end control points has its maximum influence. (Trace the curve and control polygon, regard the vertices as the vertices of the unit square, calculate the two points of maximum influence, and indicate them.)

Problem E-20. For an unspecified positive constant $h$, consider the loop made by taking a Bézier curve with control points $(0,0)$, $(h,0)$, $(0,h)$, $(0,0)$. (a) Find $P({\frac 1 2})$. (b) Find the value of $t$ at which the $x$ coordinate is largest, and the corresponding point on the curve. (c) For what value of $h$ would the loop just fit inside a unit square?

Problem E-21. (a) Find control points for a cubic Bézier curve that has exactly the same shape as the arc in Example 2 of Section 5, but goes from $(1,0)$ to $(2,1)$. (b) Find points that give an arrowhead so that the whole curved arrow has exactly the same shape as the one in Example 3 of Section 5.

Problem E-22. Prove the formulas given for finding Bézier control points from Hermite data.

Problem E-23. Sketch a cubic parametric curve $Q(t)$ such that $Q(0)$ $=$ $(0,1)$, $Q(1)$ $=$ $(0,-1)$, $Q'(0)$ $=$ $(6,0)$, and $Q'(1)$ $=$ $(6,0)$.

Problem E-24. Prove the algebraic generality property of cubic Bézier curves.

(Method: Write $P(t) = (x(t),y(t))$. You may quote the ``basis'' property of Bernstein polynomials. A suitable linear combination of them gives the $x(t)$, and similarly for $y(t)$. How should you choose the control points?)

Problem E-25. Prove the geometric generality property of cubic Bézier curves.

(Method: The difference from the algebraic generality property is that you must consider any segment of a cubic parametric curve, not just the segment $0 \leq t \leq 1$. Accordingly, suppose $C(t)$ is a cubic parametric curve and consider a segment given by $a \leq t \leq b$. Say how to make a re-parameterization $Q(t)$ that gives the same points but for which the same segment is given by $0 \leq t \leq 1$. Then use the algebraic generality property.)

Problem E-26. (a) Prove this derivative property of Bernstein polynomials: $J _ {n,k} ^ \prime (t) = n \left (J _ {n-1,k-1}(t) - J _ {n-1,k}(t) \right )$, except that a term is omitted if its second subscript is out of bounds; specifically, $J _ {n-1,-1}(t)$ and $J _ {n-1,n}(t)$ are omitted.

(Method: You will need the fact that $\binom nk = {\frac{\displaystyle n!}{\displaystyle k! (n-k)!}}$. Discuss separately the end cases with omitted terms.)

(b) Prove the ``derivative property at any point'' for Bézier curves of any degree, as given in Section .

(c) Use the derivative property at any point to prove the derivative properties listed in Section for cubic Bézier curves.

Problem E-27. (a) Starting from the fact that $(s+t)^2 = s^2 + 2 st + t^2$, invent the quadratic Bernstein polynomials and define quadratic Bézier curves (using three control points). (b) Sketch the quadratic Bézier curve with control points $(1,0)$, $(0,0)$, $(0,1)$ for $0 \leq t \leq 1$. (c) Extend your sketch to $-2 \leq t \leq 3$.

Problem E-28. Show that in    R$^2$ any quadratic Bézier curve with noncollinear control points can be mapped onto any other quadratic Bézier curve with non-collinear control points by a suitable affine transformation, so that the control polygon of the first is mapped onto the control polygon of the second.

(``Noncollinear'' means ``not in a straight line''. You may assume any needed properties of quadratic Bézier curves that are similar to properties of cubic Bézier curves. These curves are parabolas (unless they are lines), so this problem says that arbitrary parabolas can be taken to arbitrary parabolas by an affine transformation, but it says more: This can be done so that the $t$ values match up, with $P(t)$ on one curve going to $Q(t)$ on the other for every $t$.)

Problem E-29. Find the coordinate functions $x(t)$, $y(t)$, $z(t)$ of a Bézier curve in    R$^3$ with control points $(0,0,0)$, $(1,0,0)$, $(0,1,0)$, $(0,0,1)$.

Problem E-30. Show that in    R$^3$, any nonplanar cubic Bézier curve can be mapped by an affine transformation to any other nonplanar Bézier curve. (You may quote any relevant properties of Bézier curves.)

Problem E-31. Show that as a parametric function of $t$, the first derivative of the cubic Bézier curve with control points $P_0,\dots, P_3$ is the same as the quadratic Bézier curve with control points $3(P_1 - P_0)$, $3(P_2 - P_1)$, and $3(P_3 - P_2)$. (One method: Expand both sides. Another method: Verify that both sides have the same values and first, second, and third derivatives at $t = 0$, because then by Taylor's Theorem they must have the same cubic polynomials for each coordinate.)

Problem E-32. A certain machine tool is programmed to follow a cubic Bézier curve in    R$^2$. However, it is not necessarily at point $P(t)$ at time $t$. Instead, the $x$ and $y$ coordinates are driven by separate identical motors and there is a maximum rate at which each motor can go. Therefore the machine is programmed so that at any moment either the $x$ motor or the $y$ motor is going at its maximum speed, whether forwards or backwards. For example, it might be that for the points with $0 \leq t \leq .21$ on the curve, the $x$ motor is at its maximum rate, then for $.21 \leq t \leq .53$, the $y$ motor is at its maximum rate, and then for $.53 \leq t \leq 1.0$ the $x$ motor is at its maximum rate again. In this example, there were three ranges of $t$. Find the maximum number of ranges there could be for a cubic Bézier curve in general.

(Method: Imagine the path in    R$^2$ of the velocity vector $P'(t)$. By Problem E-, this curve is a quadratic Bézier curve itself. By an earlier problem, the graph is a parabola. In which regions of    R$^2$ is the $x$-motor limiting, and in which regions is the $y$-motor limiting? Into how many pieces could the curve be broken by these regions?)

Problem E-33. For given Hermite data, is there necessarily a quadratic Bézier curve that satisfies the data? (Either give a method or give a counterexample.)

Problem E-34. (a) Invent a ``half-way derivative'' property for cubic Bézier curves, by expressing $P'({\frac 1 2})$ in terms of the control points and looking for a geometrical interpretation. (It will involve a vector between midpoints of two sides of the control polygon, times a factor.)

(b) Show that at $t = {\frac 1 2}$ the curve is parallel to the line segment joining the midpoints of the first and last legs of the control polygon.

Problem E-35. When we say a ``cubic'' Bézier curve we really mean a Bézier curve with four control points and of degree at most 3. Invent a criterion for a cubic Bézier curve to have degree at most two. Express your criterion in terms of the control points. If possible, given a pictorial interpretation.

(Method: A curve is quadratic when its third derivative is always zero. Analogously to the ``Second derivative at ends'' property, derive a ``Third derivative at ends'' property. Actually, since the third derivative of a cubic function is constant, the third derivative will be the same at both ends and everywhere else. Now set the third derivative = 0.)

Problem E-36. For each graph in Figure , do the following, separately for each: Choose $x,y$-axes so that the lower-left vertex is at the origin and so that all control points have integer coordinates, with all their coordinates together having greatest common divisor 1. Then calculate $P({\frac 1 2})$ in decimals, and see if it does seem to match the graph.

Problem E-37. Plot the following functions together on one graph, for $0 \leq t \leq 1$:
$J_{3,0} (t)$,
$J _ {3,0}(t) + J _ {3,1}(t)$,
$J _ {3,0}(t) + J _ {3,1}(t) + J _ {3,2}(t)$,
$J _ {3,0}(t) + J _ {3,1}(t) + J _ {3,2}(t) + J _ {3,3}(t)$.

Problem E-38. On a single graph, plot the Bézier curve with control points $(1,0)$, $(1,{\frac 1 2})$, $({\frac 1 2},1)$, $(0,1)$ and also the three other Bézier curves whose control points are the same as these times $R _ {90 ^\circ }$, $R _ {180 ^\circ }$, and $R _ {270 ^\circ }$, respectively. This gives a pretty good approximation to a circle.

Problem E-39. For a Bézier curve of degree 4, with control points $P _ 0,\dots, P _ 4$, invent a Half-way Point Property in terms of $P_2$ and the midpoints of $\overline {P _ 0 P _ 4}$ and $\overline {P _ 1 P _ 3}$.

Problem E-40. Sketch a parametric cubic curve $P(t)$ for which $P(0)=(-1,0)$, $P(1)= (1,0)$, $P'(0)=(-3,3)$, $P'(1)=(3,-3)$.

Problem E-41. Show that for a cubic Bézier curve with control points $P_0$, $P_1$, $P_2$, $P_3$, the center of mass of the ``control triangle'' for $P'(t)$ is the same as the ``missing leg'' $P _ 3 - P _ 0$ (as a vector) in the original control polygon. (You may quote the result of Problem E-.)

Problem E-42. Find the set of $t$ with $0 \leq t \leq 1$ for which $J_{3,0} (t)$ has the largest value among the four Bernstein polynomials of degree 3. This is the same as the set of $t$ for which $P_0$ has the largest influence among the four control points. Correspondingly, where does each of $P_1$, $P_2$, and $P_3$ have the largest influence, in terms of sets of $t$?