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Chapter 8: Q10E (page 437)

Repeat Exercise 9 for the points \({{\bf{q}}_{\bf{1}}}\),鈥�.\({{\bf{q}}_{\bf{5}}}\) whose barycentric coordinates with respect to S are given by \(\left( {\frac{{\bf{1}}}{{\bf{8}}},\frac{{\bf{1}}}{{\bf{4}}},\frac{{\bf{1}}}{{\bf{8}}},\frac{{\bf{1}}}{{\bf{2}}}} \right)\), \(\left( {\frac{{\bf{3}}}{{\bf{4}}}, - \frac{{\bf{1}}}{{\bf{4}}},{\bf{0}},\frac{{\bf{1}}}{{\bf{2}}}} \right)\),\(\left( {{\bf{0}},\frac{{\bf{3}}}{{\bf{4}}},\frac{{\bf{1}}}{{\bf{4}}},{\bf{0}}} \right)\),\(\left( {{\bf{0}}, - {\bf{2}},{\bf{0}},{\bf{3}}} \right)\), and \(\left( {\frac{{\bf{1}}}{{\bf{3}}},\frac{{\bf{1}}}{{\bf{3}}},\frac{{\bf{1}}}{{\bf{3}}},{\bf{0}}} \right)\), respectively.

Short Answer

Expert verified

\({{\bf{q}}_1}\) is inside the tetrahedron \({\rm{conv}}\,S\), \({{\bf{q}}_2}\) is outside \({\rm{conv}}\,S\), \({{\bf{q}}_3}\) is the edge between \({{\bf{v}}_2}\) and \({{\bf{v}}_3}\). \({{\bf{q}}_4}\) is outside the tetrahedron and \({{\bf{q}}_5}\) is on the face with vertices \({{\bf{v}}_1}\), \({{\bf{v}}_2}\), and \({{\bf{v}}_3}\).

Step by step solution

01

Check for the first coordinate

The barycentric coordinateof the point \({{\bf{q}}_1}\) are all positive, so the point \({{\bf{q}}_1}\) is inside the tetrahedron convex S.

02

Check for the second coordinate

The barycentric coordinate of the point \({{\bf{q}}_2}\)is not positive, so the point \({{\bf{q}}_2}\) is outside the tetrahedron convexS.

03

Check for the third coordinate

The first and fourth barycentric coordinate of the point \({{\bf{q}}_3}\)is zero, representing the edge between \({{\bf{v}}_2}\) and \({{\bf{v}}_3}\).

04

Check for the fourth coordinate

The barycentric coordinate of the point \({{\bf{q}}_4}\)is not positive, so the point \({{\bf{q}}_4}\) is outside the tetrahedron convex S.

05

Check for the fifth coordinate

The point \({{\bf{q}}_5}\) is a convex combination of vectors \({{\bf{v}}_1}\), \({{\bf{v}}_2}\), and \({{\bf{v}}_3}\), hence \({{\bf{q}}_5}\) lies on face containing the vertices \({{\bf{v}}_1}\), \({{\bf{v}}_2}\), and \({{\bf{v}}_3}\).

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Most popular questions from this chapter

In Exercises 21-26, prove the given statement about subsets A and B of \({\mathbb{R}^n}\), or provide the required example in \({\mathbb{R}^2}\). A proof for an exercise may use results from earlier exercises (as well as theorems already available in the text).

21. If \(A \subset B\), then B is affine, then \({\mathop{\rm aff}\nolimits} A \subset B\).

The conditions for affine dependence are stronger than those for linear dependence, so an affinely dependent set is automatically linearly dependent. Also, a linearly independent set cannot be affinely dependent and therefore must be affinely independent. Construct two linearly dependent indexed sets\({S_{\bf{1}}}\)and\({S_{\bf{2}}}\)in\({\mathbb{R}^2}\)such that\({S_{\bf{1}}}\)is affinely dependent and\({S_{\bf{2}}}\)is affinely independent. In each case, the set should contain either one, two, or three nonzero points.

Question: Suppose that the solutions of an equation \(A{\bf{x}} = {\bf{b}}\) are all of the form \({\bf{x}} = {x_{\bf{3}}}{\bf{u}} + {\bf{p}}\), where \({\bf{u}} = \left( {\begin{array}{*{20}{c}}{\bf{5}}\\{\bf{1}}\\{ - {\bf{2}}}\end{array}} \right)\) and \({\bf{p}} = \left( {\begin{array}{*{20}{c}}{\bf{1}}\\{ - {\bf{3}}}\\{\bf{4}}\end{array}} \right)\). Find points \({{\bf{v}}_{\bf{1}}}\) and \({{\bf{v}}_{\bf{2}}}\) such that the solution set of \(A{\bf{x}} = {\bf{b}}\) is \({\bf{aff}}\left\{ {{{\bf{v}}_{\bf{1}}},\,{{\bf{v}}_{\bf{2}}}} \right\}\).

Let\({v_1} = \left[ {\begin{array}{*{20}{c}}1\\3\\{ - 6}\end{array}} \right]\),\({v_{\bf{2}}} = \left[ {\begin{array}{*{20}{c}}{\bf{7}}\\3\\{ - {\bf{5}}}\end{array}} \right]\), \({v_{\bf{3}}} = \left[ {\begin{array}{*{20}{c}}{\bf{3}}\\{\bf{9}}\\{ - {\bf{2}}}\end{array}} \right]\), \({\bf{a}} = \left[ {\begin{array}{*{20}{c}}{\bf{0}}\\{\bf{0}}\\{\bf{9}}\end{array}} \right]\), \({\bf{b}} = \left[ {\begin{array}{*{20}{c}}{1.4}\\{{\bf{1}}.{\bf{5}}}\\{ - {\bf{3}}.{\bf{1}}}\end{array}} \right]\), and \({\bf{x}}\left( t \right) = {\bf{a}} + t{\bf{b}}\)for \(t \ge {\bf{0}}\).Find the point where the ray\({\bf{x}}\left( t \right)\)intersects the plane that contains the triangle with vertices\({v_1}\),\({v_{\bf{2}}}\), and\({v_{\bf{3}}}\). Is this point inside the triangle?

In Exercises 21鈥�24, a, b, and c are noncollinear points in\({\mathbb{R}^{\bf{2}}}\)and p is any other point in\({\mathbb{R}^{\bf{2}}}\). Let\(\Delta {\bf{abc}}\)denote the closed triangular region determined by a, b, and c, and let\(\Delta {\bf{pbc}}\)be the region determined by p, b, and c. For convenience, assume that a, b, and c are arranged so that\(\left[ {\begin{array}{*{20}{c}}{\overrightarrow {\bf{a}} }&{\overrightarrow {\bf{b}} }&{\overrightarrow {\bf{c}} }\end{array}} \right]\)is positive, where\(\overrightarrow {\bf{a}} \),\(\overrightarrow {\bf{b}} \)and\(\overrightarrow {\bf{c}} \)are the standard homogeneous forms for the points.

23. Let p be any point in the interior of\(\Delta {\bf{abc}}\), with barycentric coordinates\(\left( {r,s,t} \right)\), so that

\(\left[ {\begin{array}{*{20}{c}}{\overrightarrow {\bf{a}} }&{\overrightarrow {\bf{b}} }&{\overrightarrow {\bf{c}} }\end{array}} \right]\left[ {\begin{array}{*{20}{c}}r\\s\\t\end{array}} \right] = \widetilde {\bf{p}}\)

Use Exercise 21 and a fact about determinants (Chapter 3) to show that

\(r = \left( {area of \Delta pbc} \right)/\left( {area of \Delta abc} \right)\)

\(s = \left( {area of \Delta apc} \right)/\left( {area of \Delta abc} \right)\)

\(t = \left( {area of \Delta abp} \right)/\left( {area of \Delta abc} \right)\)
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