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Chapter 8: Q1E (page 437)

Suppose a Bézier curve is translated to \({\bf{x}}\left( t \right) + {\bf{b}}\) . That is, for\(0 \le t \le 1\), the new curve is

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

Show that this new curve is again a Bézier curve.

Short Answer

Expert verified

It is shown that the new curve is also a Bézier curve.

Step by step solution

01

Find the new control points

If the curve is shifted to\({\rm{x}}\left( t \right) + {\rm{b}}\),then the new control points become\({{\rm{p}}_o} + {\rm{b}},\,\,{{\rm{p}}_1} + {\rm{b}},\,\,{{\rm{p}}_2} + {\rm{b}},\,\,{{\rm{p}}_3} + {\rm{b}}\).

02

Find the new Bezier curve

The new Bezier curve is represented as shown below:

\(\begin{array}{c}{\bf{y}}\left( t \right) = {\left( {1 - t} \right)^3}\left( {{{\bf{p}}_o} + {\bf{b}}} \right) + 3t\left( {1 - {t^2}} \right)\left( {{{\bf{p}}_1} + {\bf{b}}} \right) + 3{t^2}\left( {1 - t} \right)\left( {{{\bf{p}}_2} + {\bf{b}}} \right) + {t^3}\left( {{{\bf{p}}_3} + {\bf{b}}} \right)\\ = {\left( {1 - t} \right)^3}{{\bf{p}}_o} + 3t\left( {1 - {t^2}} \right){{\bf{p}}_1} + 3{t^2}\left( {1 - t} \right){{\bf{p}}_2} + {t^3}{{\bf{p}}_3} + {\left( {1 - t} \right)^3}{\bf{b}} + 3t\left( {1 - {t^2}} \right){\bf{b}}\\ + 3{t^2}\left( {1 - t} \right){\bf{b}} + {t^3}{\bf{b}}\end{array}\)

03

Verify the sum of weight is equal to 1 or not

The weighted sum of linear combination must be equal to 1.

\(\begin{array}{c}{\left( {1 - t} \right)^3} + 3t\left( {1 - {t^2}} \right) + 3{t^2}\left( {1 - t} \right) + {t^3} = 1 - {t^3} - 3t + 3{t^2} + 3t - 3{t^2} + {t^3}\\ = 1\end{array}\)

04

 Draw a conclusion

As \({\left( {1 - t} \right)^3} + 3t\left( {1 - {t^2}} \right) + 3{t^2}\left( {1 - t} \right) + {t^3} = 1\), for all \(t\). This implies that \({\rm{y}}\left( t \right) = {\rm{x}}\left( t \right) + {\rm{b}}\) is also true for all \(t\).

Thus, translation by \(b\)maps a Bézier curve into a new Bézier curve.

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Most popular questions from this chapter

Question 2: Given points \({{\mathop{\rm p}\nolimits} _1} = \left( {\begin{array}{*{20}{c}}0\\{ - 1}\end{array}} \right),{\rm{ }}{{\mathop{\rm p}\nolimits} _2} = \left( {\begin{array}{*{20}{c}}2\\1\end{array}} \right),\) and \({{\mathop{\rm p}\nolimits} _3} = \left( {\begin{array}{*{20}{c}}1\\2\end{array}} \right)\) in \({\mathbb{R}^{\bf{2}}}\), let \(S = {\mathop{\rm conv}\nolimits} \left\{ {{{\mathop{\rm p}\nolimits} _1},{{\mathop{\rm p}\nolimits} _2},{{\mathop{\rm p}\nolimits} _3}} \right\}\). For each linear functional \(f\), find the maximum value \(m\) of \(f\), find the maximum value \(m\) of \(f\) on the set \(S\), and find all points x in \(S\) at which \(f\left( {\mathop{\rm x}\nolimits} \right) = m\).

a. \(f\left( {{x_1},{x_2}} \right) = {x_1} + {x_2}\)

b. \(f\left( {{x_1},{x_2}} \right) = {x_1} - {x_2}\)

c. \(f\left( {{x_1},{x_2}} \right) = - 2{x_1} + {x_2}\)

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 H be the column space of the matrix \(B = \left( {\begin{array}{*{20}{c}}{\bf{1}}&{\bf{0}}\\{ - {\bf{4}}}&{\bf{2}}\\{\bf{7}}&{ - {\bf{6}}}\end{array}} \right)\). That is, \(H = {\bf{Col}}\,B\).(Hint: How is \({\bf{Col}}\,B\) related to Nul \({B^T}\)? See section 6.1)

A quartic Bézier curve is determined by five control points,

\({{\bf{p}}_{\bf{o}}}{\bf{,}}\,{\rm{ }}{{\bf{p}}_{\bf{1}}}\,{\bf{,}}{\rm{ }}{{\bf{p}}_{\bf{2}}}\,{\bf{,}}{\rm{ }}{{\bf{p}}_{\bf{3}}}\)and \({{\bf{p}}_4}\):

\({\bf{x}}\left( t \right) = {\left( {1 - t} \right)^4}{{\bf{p}}_0} + 4t{\left( {1 - t} \right)^3}{{\bf{p}}_1} + 6{t^2}{\left( {1 - t} \right)^2}{{\bf{p}}_2} + 4{t^3}\left( {1 - t} \right){{\bf{p}}_3} + {t^4}{{\bf{p}}_4}\)for \(0 \le t \le 1\)

Construct the quartic basis matrix \({M_B}\) for \({\bf{x}}\left( t \right)\).

Question: Let \({\bf{p}} = \left( {\begin{array}{*{20}{c}}{\bf{1}}\\{ - {\bf{3}}}\\{\bf{1}}\\{\bf{2}}\end{array}} \right)\), \({\bf{n}} = \left( {\begin{array}{*{20}{c}}{\bf{2}}\\{\bf{1}}\\{\bf{5}}\\{ - {\bf{1}}}\end{array}} \right)\), \({{\bf{v}}_{\bf{1}}} = \left( {\begin{array}{*{20}{c}}{\bf{0}}\\{\bf{1}}\\{\bf{1}}\\{\bf{1}}\end{array}} \right)\), \({{\bf{v}}_{\bf{2}}} = \left( {\begin{array}{*{20}{c}}{ - {\bf{2}}}\\{\bf{0}}\\{\bf{1}}\\{\bf{3}}\end{array}} \right)\), and \({{\bf{v}}_{\bf{3}}} = \left( {\begin{array}{*{20}{c}}{\bf{1}}\\{\bf{4}}\\{\bf{0}}\\{\bf{4}}\end{array}} \right)\), and let H be the hyperplane in\({\mathbb{R}^{\bf{4}}}\) with normal n and passing through p. Which of the points \({{\bf{v}}_{\bf{1}}}\), \({{\bf{v}}_{\bf{2}}}\), and \({{\bf{v}}_{\bf{3}}}\) are on the same side of H as the origin, and which are not?

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.
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