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Chapter 8: Q11E (page 437)

Question: In Exercises 11 and 12, mark each statement True or False. Justify each answer.

11.a. The cubic Bezier curve is based on four control points.

b. Given a quadratic Bezier curve \({\mathop{\rm x}\nolimits} \left( t \right)\) with control points \({{\mathop{\rm p}\nolimits} _0},{{\mathop{\rm p}\nolimits} _1},\) and \({{\mathop{\rm p}\nolimits} _2}\), the directed line segment \({{\mathop{\rm p}\nolimits} _1} - {{\mathop{\rm p}\nolimits} _0}\) (from \({{\mathop{\rm p}\nolimits} _0}\) to \({{\mathop{\rm p}\nolimits} _1}\)) is the tangent vector to the curve at \({{\mathop{\rm p}\nolimits} _0}\).

c. When two quadratic Bezier curves with control points \(\left\{ {{{\mathop{\rm p}\nolimits} _0},{{\mathop{\rm p}\nolimits} _1},{{\mathop{\rm p}\nolimits} _2}} \right\}\) and \(\left\{ {{{\mathop{\rm p}\nolimits} _2},{{\mathop{\rm p}\nolimits} _3},{{\mathop{\rm p}\nolimits} _4}} \right\}\) are joined at \({{\mathop{\rm p}\nolimits} _2}\), the combined Bezier curve will have \({C^1}\) continuity at \({{\mathop{\rm p}\nolimits} _2}\)if\({{\mathop{\rm p}\nolimits} _2}\) is the midpoint of the line segment between \({{\mathop{\rm p}\nolimits} _1}\) and \({{\mathop{\rm p}\nolimits} _3}\).

Short Answer

Expert verified
  1. The statement is True.
  2. The statement is False.
  3. The statement is True.

Step by step solution

01

Determine whether the given statement is True or False

a)

Thecubic Bezier curveis determined by four control points. The standard form is shown below:

\({\bf{x}}\left( t \right) = {\left( {1 - t} \right)^3}{{\bf{p}}_0} + 3t{\left( {1 - t} \right)^2}{{\bf{p}}_1} + 3{t^2}\left( {1 - t} \right){{\bf{p}}_2} + {t^3}{{\bf{p}}_3}\)

Thus, the given statement (a) is True.

02

Determine whether the given statement is True or False

b)

Recall that thetangent vectorat \({{\mathop{\rm p}\nolimits} _0}\), for instance, from \({{\mathop{\rm p}\nolimits} _0}\) to \({{\mathop{\rm p}\nolimits} _1}\), and it is twice as long as the segment from \({{\mathop{\rm p}\nolimits} _0}\) to \({{\mathop{\rm p}\nolimits} _1}\).

Thus, the given statement (b) is False.

03

Determine whether the given statement is True or False

c)

It is true that when two quadratic Bezier curve with control points \(\left\{ {{{\mathop{\rm p}\nolimits} _0},{{\mathop{\rm p}\nolimits} _1},{{\mathop{\rm p}\nolimits} _2}} \right\}\)and \(\left\{ {{{\mathop{\rm p}\nolimits} _2},{{\mathop{\rm p}\nolimits} _3},{{\mathop{\rm p}\nolimits} _4}} \right\}\) then \({{\mathop{\rm p}\nolimits} _2}\) is the midpoint of the line segment from \({{\mathop{\rm p}\nolimits} _1}\) to \({{\mathop{\rm p}\nolimits} _3}\).

The Bezier curve has \({C^1}\) continuity at \({{\mathop{\rm p}\nolimits} _2}\) when joined at \({{\mathop{\rm p}\nolimits} _2}\), then \({{\mathop{\rm p}\nolimits} _2}\) is the midpoint of the line segment from \({{\mathop{\rm p}\nolimits} _1}\) to \({{\mathop{\rm p}\nolimits} _3}\).

Thus, the given statement (c) is True.

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Most popular questions from this chapter

In Exercises 13-15 concern the subdivision of a Bezier curve shown in Figure 7. Let \({\mathop{\rm x}\nolimits} \left( t \right)\) be the Bezier curve, with control points \({{\mathop{\rm p}\nolimits} _0},...,{{\mathop{\rm p}\nolimits} _3}\), and let \({\mathop{\rm y}\nolimits} \left( t \right)\) and \({\mathop{\rm z}\nolimits} \left( t \right)\) be the subdividing Bezier curves as in the text, with control points \({{\mathop{\rm q}\nolimits} _0},...,{{\mathop{\rm q}\nolimits} _3}\) and \({{\mathop{\rm r}\nolimits} _0},...,{{\mathop{\rm r}\nolimits} _3}\), respectively.

14.a. Justify each equal sign:

\(3\left( {{{\mathop{\rm r}\nolimits} _3} - {{\mathop{\rm r}\nolimits} _2}} \right) = z'\left( 1 \right) = .5x'\left( 1 \right) = \frac{3}{2}\left( {{{\mathop{\rm p}\nolimits} _3} - {{\mathop{\rm p}\nolimits} _2}} \right)\)

b. Show that \({{\mathop{\rm r}\nolimits} _2}\) is the midpoint of the segment from \({{\mathop{\rm p}\nolimits} _2}\) to \({{\mathop{\rm p}\nolimits} _3}\).

c. Justify each equal sign: \(3\left( {{{\mathop{\rm r}\nolimits} _1} - {{\mathop{\rm r}\nolimits} _0}} \right) = z'\left( 0 \right) = .5x'\left( {.5} \right)\).

d. Use part (c) to show that \(8{{\mathop{\rm r}\nolimits} _1} = - {{\mathop{\rm p}\nolimits} _0} - {{\mathop{\rm p}\nolimits} _1} + {{\mathop{\rm p}\nolimits} _2} + {{\mathop{\rm p}\nolimits} _3} + 8{{\mathop{\rm r}\nolimits} _0}\).

e. Use part (d) equation (8), and part (a) to show that \({{\mathop{\rm r}\nolimits} _1}\) is the midpoint of the segment from \({{\mathop{\rm r}\nolimits} _2}\) to the midpoint of the segment from \({{\mathop{\rm p}\nolimits} _1}\) to \({{\mathop{\rm p}\nolimits} _2}\). That is, \({{\mathop{\rm r}\nolimits} _1} = \frac{1}{2}\left( {{{\mathop{\rm r}\nolimits} _2} + \frac{1}{2}\left( {{{\mathop{\rm p}\nolimits} _1} + {{\mathop{\rm p}\nolimits} _2}} \right)} \right)\).

In Exercises 9 and 10, mark each statement True or False. Justify each answer.

9.

a. If \({{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p}\) are in \({\mathbb{R}^n}\) and if the set \(\left\{ {{{\mathop{\rm v}\nolimits} _1} - {{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3} - {{\mathop{\rm v}\nolimits} _2},...,{{\mathop{\rm v}\nolimits} _p} - {{\mathop{\rm v}\nolimits} _2}} \right\}\) is linearly dependent, then \(\left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p}} \right\}\) is affinely dependent. (Read this carefully.)

b. If \({{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p}\) are in \({\mathbb{R}^n}\) and if the set of homogeneous forms \(\left\{ {{{\overline {\mathop{\rm v}\nolimits} }_1},...,{{\overline {\mathop{\rm v}\nolimits} }_p}} \right\}\) in \({\mathbb{R}^{n + 1}}\) is linearly independent, then \(\left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p}} \right\}\) is affinely dependent.

c. A finite set of points \(\left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _k}} \right\}\) is affinely dependent if there exist real numbers \({c_1},...,{c_k}\) , not all zero, such that \({c_1} + ... + {c_k} = 1\) and \({c_1}{{\mathop{\rm v}\nolimits} _1} + ... + {c_k}{{\mathop{\rm v}\nolimits} _k} = 0\).

d. If \(S = \left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p}} \right\}\) is an affinely independent set in \({\mathbb{R}^n}\) and if p in \({\mathbb{R}^n}\) has a negative barycentric coordinate determined by S, then p is not in \({\mathop{\rm aff}\nolimits} S\).

e.

If \({{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3},a,\) and \(b\) are in \({\mathbb{R}^3}\) and if ray \({\mathop{\rm a}\nolimits} + t{\mathop{\rm b}\nolimits} \) for \(t \ge 0\) intersects the triangle with vertices \({{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},\) and \({{\mathop{\rm v}\nolimits} _3}\) then the barycentric coordinates of the intersection points are all nonnegative.

Let \({\bf{x}}\left( t \right)\) be a B-spline in Exercise 2, with control points \({{\bf{p}}_o}\), \({{\bf{p}}_1}\) , \({{\bf{p}}_2}\) , and \({{\bf{p}}_3}\).

a. Compute the tangent vector \({\bf{x}}'\left( t \right)\) and determine how the derivatives \({\bf{x}}'\left( 0 \right)\) and \({\bf{x}}'\left( 1 \right)\) are related to the control points. Give geometric descriptions of the directions of these tangent vectors. Explore what happens when both \({\bf{x}}'\left( 0 \right)\)and \({\bf{x}}'\left( 1 \right)\)equal 0. Justify your assertions.

b. Compute the second derivative and determine how and are related to the control points. Draw a figure based on Figure 10, and construct a line segment that points in the direction of . [Hint: Use \({{\bf{p}}_2}\) as the origin of the coordinate system.]

Question: In Exercises 15-20, write a formula for a linear functional f and specify a number d so that \(\left( {f:d} \right)\) the hyperplane H described in the exercise.

Let A be the \({\bf{1}} \times {\bf{4}}\) matrix \(\left( {\begin{array}{*{20}{c}}{\bf{1}}&{ - {\bf{3}}}&{\bf{4}}&{ - {\bf{2}}}\end{array}} \right)\) and let \(b = {\bf{5}}\). Let \(H = \left\{ {{\bf{x}}\,\,{\rm{in}}\,{\mathbb{R}^{\bf{4}}}:A{\bf{x}} = {\bf{b}}} \right\}\).

Question: In Exercise 9, let Hbe the hyperplane through the listed points. (a) Find a vector n that is normal to the hyperplane. (b) Find a linear functional f and a real number d such that \(H = \left( {f:d} \right)\).

9. \(\left( {\begin{array}{*{20}{c}}1\\0\\1\\0\end{array}} \right),\left( {\begin{array}{*{20}{c}}2\\3\\1\\0\end{array}} \right),\left( {\begin{array}{*{20}{c}}1\\2\\2\\0\end{array}} \right),\left( {\begin{array}{*{20}{c}}1\\1\\1\\1\end{array}} \right)\)
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