Solution: Exercises 21 and 22 concern the way in which color

Problem 22E Exercises 21 and 22 concern the way in which color is specified for display in computer graphics. A color on a computer screen is encoded by three numbers (R, G, B) that list the amounts of energy an electron gun must transmit to red, green, and blue phosphor dots on the computer screen. (A fourth number specifies the luminance or intensity of the color.) [M] The signal broadcast by commercial television describes each color by a vector (Y, I, Q). If the screen is black and white, only the Y -coordinate is used. (This gives a better monochrome picture than using CIE data for colors.) The correspondence between YIQ and a “standard” RGB color is given by (A screen manufacturer would change the matrix entries to work for its RGB screens.) Find the equation that converts the YIQ data transmitted by the television station to the RGB data needed for the television screen.

Problem 3E In Exercises 3–8, find the 3 × 3 matrices that produce the described composite 2D transformations, using homogeneous coordinates. Translate by (2, 1), and then rotate about the origin.

Problem 2E Use matrix multiplication to find the image of the triangle with data matrix under the transformation that reflects points through the y-axis. Sketch both the original triangle and its image.

Problem 1E What 3 × 3 matrix will have the same effect on homogeneous coordinates for that the shear matrix A has in Example 2?

Problem 4E In Exercises 3–8, find the 3 × 3 matrices that produce the described composite 2D transformations, using homogeneous coordinates. Translate by (–1, 4) and then scale the x-coordinate by 1/2 and the y-coordinate by 3/2.

Problem 5E In Exercises 3–8, find the 3 × 3 matrices that produce the described composite 2D transformations, using homogeneous coordinates. Reflect points through the x-axis, and then rotate about the origin.

Problem 6E In Exercises 3–8, find the 3 × 3 matrices that produce the described composite 2D transformations, using homogeneous coordinates. Rotate points about the origin, then reflect through the x-axis.

Problem 7E In Exercises 3–8, find the 3 × 3 matrices that produce the described composite 2D transformations, using homogeneous coordinates. Rotate points through about the point (6, 8).

Problem 8E In Exercises 3–8, find the 3 × 3 matrices that produce the described composite 2D transformations, using homogeneous coordinates. Rotate points through about the point (3, 7).

Problem 9E A 2 × 100 data matrix D contains the coordinates of 100 points. Compute the number of multiplications required to transform these points using two arbitrary 2 × 2 matrices A and B. Consider the two possibilities A (BD) and (AB)D. Discuss the implications of your results for computer graphics calculations.

Problem 10E Consider the following geometric 2D transformations: D, a dilation (in which x-coordinates and y-coordinates are scaled by the same factor); R, a rotation; and T, a translation. Does D commute with R? That is, is Does D commute with T ? Does R commute with T ?

Problem 12E A rotation in usually requires four multiplications. Compute the product below, and show that the matrix for a rotation can be factored into three shear transformations (each of which requires only one multiplication).

Problem 11E A rotation on a computer screen is sometimes implemented as the product of two shear-and-scale transformations, which can speed up calculations that determine how a graphic image actually appears in terms of screen pixels. (The screen consists of rows and columns of small dots, called pixels.) The first transformation A1 shears vertically and then compresses each column of pixels; the second transformation A2 shears horizontally and then stretches each row of pixels. Let and show that the composition of the two transformations is a rotation in

Problem 13E The usual transformations on homogeneous coordinates for 2D computer graphics involve 3 × 3 matrices of the form Show that such a transformation amounts to a linear transformation on followed by a translation. [Hint: Find an appropriate matrix factorization involving partitioned matrices.]

Problem 14E Show that the transformation in Exercise 7 is equivalent to a rotation about the origin followed by a translation by p. Find p. Exercise 7: In Exercises 3–8, find the 3 × 3 matrices that produce the described composite 2D transformations, using homogeneous coordinates. Rotate points through about the point (6, 8).

Problem 15E What vector in has homogeneous coordinates

Problem 16E Are homogeneous coordinates for the same point in ? Why or why not?

Problem 17E Give the 4 × 4 matrix that rotates points in about the x-axis through an angle of 60°. (See the figure.)

Problem 18E Give the 4 × 4 matrix that rotates points in about the z-axis through an angle of , and then translates by

Problem 19E Give the 4 × 4 matrix that rotates points in about the z-axis through an angle of , and then translates by

Problem 20E Give the 4 × 4 matrix that rotates points in about the z-axis through an angle of , and then translates by

Problem 21E Exercises 21 and 22 concern the way in which color is specified for display in computer graphics. A color on a computer screen is encoded by three numbers (R, G, B) that list the amounts of energy an electron gun must transmit to red, green, and blue phosphor dots on the computer screen. (A fourth number specifies the luminance or intensity of the color.) [M] The actual color a viewer sees on a screen is influenced by the specific type and amount of phosphors on the screen. So each computer screen manufacturer must convert between the (R, G, B) data and an international CIE standard for color, which uses three primary colors, called X, Y , and Z. A typical conversion for short-persistence phosphors is A computer program will send a stream of color information to the screen, using standard CIE data (X, Y, Z) Find the equation that converts these data to the (R, G, B) data needed for the screen’s electron gun.