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Chapter 13

# A line l in the scene passes through the points 230

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Unformatted text preview: re therefore called clipping planes. 1. Suppose the screen is represented by a rectangle in the yz-plane with vertices 0, 400, 0 and 0, 400, 600 , and the camera is placed at 1000, 0, 0 . A line L in the scene passes through the points 230, 285, 102 and 860, 105, 264 . At what points should L be clipped by the clipping planes? 2. If the clipped line segment is projected on the screen window, identify the resulting line segment. 3. Use parametric equations to plot the edges of the screen window, the clipped line segment, and its projection on the screen window. Then add sight lines connecting the viewpoint to each end of the clipped segments to verify that the projection is correct. 4. A rectangle with vertices 621, 147, 206 , 563, 31, 242 , 657, 111, 86 , and 599, 67, 122 is added to the scene. The line L intersects this rectangle. To make the rectangle appear opaque, a programmer can use hidden line rendering which removes portions of objects that are behind other objects. Identify the portion of L that should be removed. |||| 13.6 Cylinders and Quadric Surfaces We have already looked at two special types of surfaces—planes (in Section 13.5) and spheres (in Section 13.1). Here we investigate two other types of surfaces—cylinders and quadric surfaces. In order to sketch the graph of a surface, it is useful to determine the curves of intersection of the surface with planes parallel to the coordinate planes. These curves are called traces (or cross-sections) of the surface. Cylinders A cylinder is a surface that consists of all lines (called rulings) that are parallel to a given line and pass through a given plane curve. EXAMPLE 1 Sketch the graph of the surface z x 2. x 2, doesn’t involve y. This means that any vertical plane with equation y k (parallel to the x z-plane) intersects the graph in a curve with equation z x 2. So these vertical traces are parabolas. Figure 1 shows how the graph is formed by taking the parabola z x 2 in the x z-plane and moving it in the SOLUTION Notice that the equation of the graph, z 5E-13(pp 868-877) 1/18/06 11:34 AM Page 869 S ECTION 1...
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