Suppose {p1, p2, p3} is an affinely independent set in Rn

Problem 24E In Exercises 21–24, a, b, and c are noncollinear points in R2 and p is any other point in R2. Let denote the closed triangular region determined by a, b, and c, and let be the region determined by p, b, and c. For convenience, assume that a, b, and c are arranged so that det is positive, where and are the standard homogeneous forms for the points. Take q on the line segment from b to c and consider the line through q and a, which may be written as for all real x. Show that, for each From this and earlier work, conclude that the parameter x is the first barycentric coordinate of p. However, by construction, the parameter x also determines the relative distance between p and q along the segment from q to a. When this fact is applied to Example 5, it shows that the colors at vertex a and the point q are smoothly interpolated as p moves along the line between a and q.

Problem 4E In Exercises 1–6, determine if the set of points is affinely dependent. (See Practice Problem 2.) If so, construct an affine dependence relation for the points. Reference Practice Problem 2:

Problem 5E In Exercises 1–6, determine if the set of points is affinely dependent. (See Practice Problem 2.) If so, construct an affine dependence relation for the points. Reference Practice Problem 2:

Problem 1E In Exercises 1–6, determine if the set of points is affinely dependent. (See Practice Problem 2.) If so, construct an affine dependence relation for the points. Reference Practice Problem 2:

Problem 2E In Exercises 1–6, determine if the set of points is affinely dependent. (See Practice Problem 2.) If so, construct an affine dependence relation for the points. Reference Practice Problem 2:

Problem 3E In Exercises 1–6, determine if the set of points is affinely dependent. (See Practice Problem 2.) If so, construct an affine dependence relation for the points. Reference Practice Problem 2:

Problem 6E In Exercises 1–6, determine if the set of points is affinely dependent. (See Practice Problem 2.) If so, construct an affine dependence relation for the points. Reference Practice Problem 2:

Problem 8E In Exercises 7 and 8, find the barycentric coordinates of p with respect to the affinely independent set of points that precedes it.

Problem 7E In Exercises 7 and 8, find the barycentric coordinates of p with respect to the affinely independent set of points that precedes it.

Problem 11E Explain why any set of five or more points in R3 must be affinely dependent.

Problem 10E In Exercises 9 and 10, mark each statement True or False. Justify each answer. a. is an affinely dependent set in Rn, then the set of homogeneous forms may be linearly independent. d. When color information is specified at each vertex v1, v2, v3 of a triangle in R3, then the color may be interpolated at a point p in aff using the barycentric coordinates of p. e. If T is a triangle in R2 and if a point p is on an edge of the triangle, then the barycentric coordinates of p (for this triangle) are not all positive.

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

Problem 12E Show that a set is affinely dependent when

Problem 13E Use only the definition of affine dependence to show that an indexed set is affinely dependent if and only if

Problem 14E 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 S1 and S2 in R2 such that S1 is affinely dependent and S2 is affinely independent. In each case, the set should contain either one, two, or three nonzero points.

Problem 18E Let T be a tetrahedron in “standard” position, with three edges along the three positive coordinate axes in R3, and suppose the vertices are 1, 2, 3, and 0, where [e1 e2 e3] = I3. Find formulas for the barycentric coordinates of an arbitrary point p in R3.

Problem 15E Let v1 = v2 = v3 = {v1, v2, v3}. a. Show that the set S is affinely independent. b. Find the barycentric coordinates of p1 = P2 = p3 = p4 = and p5 = with respect to S. c. Let T be the triangle with vertices v1, v2, and v3. When the sides of T are extended, the lines divide R2 into seven regions. See Fig. 8. Note the signs of the barycentric coordinates of the points in each region. For example, p5 is inside the triangle T and all its barycentric coordinates are positive. Point p1 has coordinates (–, +, +) Its third coordinate is positive because p1 is on the v3 side of the line through v1 and v2. Its first coordinate is negative because p1 is opposite the v1 side of the line through v2 and v3. Point p2 is on the v2v3 edge of T . Its coordinates are (0, +, +). Without calculating the actual values, determine the signs of the barycentric coordinates of points p6, p7, and p8 as shown in Fig. 8.

Problem 17E Prove Theorem 6 for an affinely independent set . [Hint: One method is to mimic the proof of Theorem 7 in Section 4.4.] Reference Theorem 6: Reference Theorem 7 in Section 4.4:

Problem 16E a. Show that the set S is affinely independent. b. Find the barycentric coordinates of p1, p2, and p3 with respect to S. c. On graph paper, sketch the triangle T with vertices v1, v2, and v3, extend the sides as in Fig. 5, and plot the points p4, p5, p6, and p7. Without calculating the actual values, determine the signs of the barycentric coordinates of points p4, p5, p6, and p7.

Problem 19E be an affinely dependent set of points in Rn and let be a linear transformation. Show that is affinely dependent in Rm.

Problem 20E Suppose {p1, p2, p3} is an affinely independent set in Rn and q is an arbitrary point in Rn. Show that the translated set {p1 + q, p2 + q, p3 + q} is also affinely independent.

Problem 22E In Exercises 21–24, a, b, and c are noncollinear points in R2 and p is any other point in R2. Let denote the closed triangular region determined by a, b, and c, and let be the region determined by p, b, and c. For convenience, assume that a, b, and c are arranged so that det is positive, where and are the standard homogeneous forms for the points. Let p be a point on the line through a and b. Show that

Problem 23E In Exercises 21–24, a, b, and c are noncollinear points in R2 and p is any other point in R2. Let denote the closed triangular region determined by a, b, and c, and let be the region determined by p, b, and c. For convenience, assume that a, b, and c are arranged so that det is positive, where and are the standard homogeneous forms for the points. Let p be any point in the interior of , with barycentric coordinates (r, s, t) so that Use Exercise 19 and a fact about determinants (Chapter 3) to show that

Problem 21E In Exercises 21–24, a, b, and c are noncollinear points in R2 and p is any other point in R2. Let denote the closed triangular region determined by a, b, and c, and let be the region determined by p, b, and c. For convenience, assume that a, b, and c are arranged so that det is positive, where and are the standard homogeneous forms for the points. Show that the area of  [Hint: Consult Sections 3.2 and 3.3, including the Exercises.]