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Chapter 8: Q19E (page 437)

Question:19.In Exercises 17–20, prove the given statement about subsets\(A\)and\(B\)of\({\mathbb{R}^n}\). A proof for an exercise may use results of earlier exercises.

19. a. \(\left( {\left( {{\rm{conv}}\,A} \right) \cup \left( {{\rm{conv}}\,B} \right)} \right) \subset {\rm{conv}}\left( {A \cup B} \right)\),

b. Find an example in\({\mathbb{R}^2}\)to show that equality need not hold in part

(a)

Short Answer

Expert verified
  1. It is shown that \(\left( {{\rm{conv}}\,A\,\, \cup {\rm{conv}}\,B} \right) \subset {\rm{conv}}\,\left( {{\rm{A}} \cup \,B} \right)\).
  2. \(A\) be the two adjacent corners of the square, and B be the other two corners of the square.

Step by step solution

01

(a) Step 1:Use the result of Exercise 18

It is known that\({\rm{conv}}\,A \subset {\rm{conv}}\,B\). Moreover, the combination of \(A\) or \(B\)must contains all the combinations of A.

Similarly, convex combinations of \(A\) must contain every convex combination of \(A\) or \(B\); that is,\({\rm{conv}}\,A \subset {\rm{conv}}\,\left( {{\rm{A}} \cup \,B} \right)\).

02

Step 2:Draw a conclusion

If \({\rm{conv}}\,A \subset {\rm{conv}}\,\left( {{\rm{A}} \cup \,B} \right)\) is true, then \(\left( {\left( {conv\,A} \right)U\left( {conv\,B} \right)} \right) \subset conv\left( {AUB} \right)\) must also be true as a convex combination of points ofA orconvex combination of points of \(B\), must contain some or all points of convex combinations of \(A\).

Thus,\(\left( {\left( {conv\,A} \right)U\left( {conv\,B} \right)} \right) \subset conv\left( {AUB} \right)\).

03

(b) Step 3:  Assume an example in \({\mathbb{R}^2}\) as a requirement

Consider a square and assume \(A\) be the two adjacent corners of the square, whereas B be the other two corners of the square.

Then, \({\rm{conv}}\,A\,\, \cup {\rm{conv}}\,B\) is a set of convex of \(A\) or convex of \(B\), which represents the opposite sides of the square, whereas \({\rm{conv}}\,\left( {A\,\, \cup \,B} \right)\) is the convex combination of points of \(A\) or \(B\) which represents the opposite sides of the perimeter of the square.

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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, \({{\mathop{\rm r}\nolimits} _1} = \frac{1}{2}\left( {{{\mathop{\rm r}\nolimits} _2} + \frac{1}{2}\left( {{{\mathop{\rm p}\nolimits} _1} + {{\mathop{\rm p}\nolimits} _2}} \right)} \right)\).

Let\({v_1} = \left[ {\begin{array}{*{20}{c}}{ - 1}\\2\end{array}} \right]\),\({v_{\bf{2}}} = \left[ {\begin{array}{*{20}{c}}{\bf{0}}\\{\bf{4}}\end{array}} \right]\),\({v_{\bf{3}}} = \left[ {\begin{array}{*{20}{c}}{\bf{2}}\\{\bf{0}}\end{array}} \right]\), and let\(S = \left\{ {{v_1},{v_2},{v_3}} \right\}\).

  1. Show that the set is affinely independent.
  2. Find the barycentric coordinates of\({p_1} = \left[ {\begin{array}{*{20}{c}}2\\3\end{array}} \right]\),\({p_{\bf{2}}} = \left[ {\begin{array}{*{20}{c}}{\bf{1}}\\{\bf{2}}\end{array}} \right]\),\({p_{\bf{3}}} = \left[ {\begin{array}{*{20}{c}}{ - 2}\\{\bf{1}}\end{array}} \right]\),\({p_{\bf{4}}} = \left[ {\begin{array}{*{20}{c}}{\bf{1}}\\{ - {\bf{1}}}\end{array}} \right]\), and\({p_{\bf{5}}} = \left[ {\begin{array}{*{20}{c}}{\bf{1}}\\{\bf{1}}\end{array}} \right]\), with respect to S.
  3. Let\(T\)be the triangle with vertices\({v_1}\),\({v_{\bf{2}}}\), and\({v_{\bf{3}}}\). When the sides of\(T\)are extended, the lines divide\({\mathbb{R}^{\bf{2}}}\)into seven regions. See Figure 8. Note the signs of the barycentric coordinates of the points in each region. For example,\({{\bf{p}}_{\bf{5}}}\)is inside the triangle\(T\)and all its barycentric coordinates are positive. Point\({{\bf{p}}_{\bf{1}}}\)has coordinates\(\left( { - , + , + } \right)\). Its third coordinate is positive because\({{\bf{p}}_{\bf{1}}}\)is on the\({{\bf{v}}_{\bf{3}}}\)side of the line through\({{\bf{v}}_{\bf{1}}}\)and\({{\bf{v}}_{\bf{2}}}\). Its first coordinate is negative because\({{\bf{p}}_{\bf{1}}}\)is opposite the\({{\bf{v}}_{\bf{1}}}\)side of the line through\({{\bf{v}}_{\bf{2}}}\)and\({{\bf{v}}_{\bf{3}}}\). Point\({{\bf{p}}_{\bf{2}}}\)is on the\({{\bf{v}}_{\bf{2}}}{{\bf{v}}_{\bf{3}}}\)edge of\(T\). Its coordinates are\(\left( {0, + , + } \right)\). Without calculating the actual values, determine the signs of the barycentric coordinates of points\({{\bf{p}}_{\bf{6}}}\),\({{\bf{p}}_{\bf{7}}}\), and\({{\bf{p}}_{\bf{8}}}\)as shown in Figure 8.

The parametric vector form of a B-spline curve was defined in the Practice Problems as

\({\bf{x}}\left( t \right) = \frac{1}{6}\left[ \begin{array}{l}{\left( {1 - t} \right)^3}{{\bf{p}}_o} + \left( {3t{{\left( {1 - t} \right)}^2} - 3t + 4} \right){{\bf{p}}_1}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left( {3{t^2}\left( {1 - t} \right) + 3t + 1} \right){{\bf{p}}_2} + {t^3}{{\bf{p}}_3}\end{array} \right]\;\), for \(0 \le t \le 1\) where \({{\bf{p}}_o}\) , \({{\bf{p}}_1}\), \({{\bf{p}}_2}\) , and \({{\bf{p}}_3}\) are the control points.

a. Show that for \(0 \le t \le 1\), \({\bf{x}}\left( t \right)\) is in the convex hull of the control points.

b. Suppose that a B-spline curve \({\bf{x}}\left( t \right)\)is translated to \({\bf{x}}\left( t \right) + {\bf{b}}\) (as in Exercise 1). Show that this new curve is again a B-spline.

Question: 19. Let \(S\) be an affine subset of \({\mathbb{R}^n}\) , suppose \(f:{\mathbb{R}^n} \to {\mathbb{R}^m}\)is a linear transformation, and let \(f\left( S \right)\) denote the set of images \(\left\{ {f\left( {\rm{x}} \right):{\rm{x}} \in S} \right\}\). Prove that \(f\left( S \right)\)is an affine subset of \({\mathbb{R}^m}\).

Question: 17. Choose a set \(S\) of three points such that aff \(S\) is the plane in \({\mathbb{R}^3}\) whose equation is \({x_3} = 5\). Justify your work.
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