Problem 4
Assume a polygon mesh contains 250,000 vertices. If a single matrix multiplication requires 28 floating-point operations, how fast a GPU is needed (floating-point operations per second) to produce real-time graphics at the rate of 30 frames per second?
Problem 8
Again assume you are working in two, rather than three, dimensions. Determine the four entries of the \(2 \times 2\) reflection matrix that takes a vertex point at position \((x, y)\) and reflects it around the \(y\)-axis. That is, assume the mirror line in Figure \(16.6(c)\) is the \(y\)-axis. This reflection operation is shown here:
Problem 10
Would a flight simulator package used to teach pilots to fly an airplane be a real-time graphical environment? Explain your answer.
Problem 11
The diagram on the next page shows a single triangular face in the wireframe representation of an object. The three vertices of the triangle are labeled \(v_{1}, v_{2}\), and \(v_{3}\), and each has been assigned a color, either red, blue, or green. The vertex color is stored as a three-tuple, with each entry an integer in the range 0 to 255 , representing the contribution of the components red, green, and blue, respectively. (Note: This is identical to the RGB color model introduced in Chapter 4, page 171.) So, for example, the color red is represented by the three-tuple \((255,0,0)\). Purple, an equal mix of red and blue, would be represented as \((128,0,128)\). During the rendering phase, a computer must shade in the entire triangular face, according to the colors assigned to each of the three vertices. Describe an algorithm that would do color shading and blending of the triangular face in a visually attractive manner.
Problem 12
You are given the three-dimensional coordinates of a point \(P 1\left(x_{1}, y_{1}, z_{1}\right)\) and a point \(P 2\left(x_{2}, y_{2}, z_{2}\right)\). You are also given the coordinates of the location point of a viewer \(\left(x_{v}, y_{v}, z_{1}\right)\). You may assume that \(\mathrm{P} 1\) and \(\mathrm{P} 2\) are located on the same side of the viewer. Describe informally (you do not need to write out an algorithm) exactly how to determine if, from the point of view of the viewer, it is possible to see both points P1 and \(\mathrm{P} 2\), or if one of these points is obstructed and not visible. In the latter case, describe how you can determine which is the occluded point.
