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ch07oddslns - Angel Interactive Computer Graphics Fifth...

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Angel: Interactive Computer Graphics, Fifth Edition Chapter 7 Solutions 7.1 First, consider the problem in two dimensions. We are looking for an α and β such that both parametric equations yield the same point, that is x ( α ) = (1 - α ) x 1 + αx 2 = (1 - β ) x 3 + βx 4 , y ( α ) = (1 - α ) y 1 + αy 2 = (1 - β ) y 3 + βy 4 . These are two equations in the two unknowns α and β and, as long as the line segments are not parallel (a condition that will lead to a division by zero), we can solve for α β . If both these values are between 0 and 1, the segments intersect. If the equations are in 3D, we can solve two of them for the α and β where x and y meet. If when we use these values of the parameters in the two equations for z , the segments intersect if we get the same z from both equations. 7.3 If we clip a convex region against a convex region, we produce the intersection of the two regions, that is the set of all points in both regions, which is a convex set and describes a convex region. To see this, consider any two points in the intersection. The line segment connecting them must be in both sets and therefore the intersection is convex. 7.5 See Problem 6.22. Nonuniform scaling will not preserve the angle between the normal and other vectors. 7.7 Note that we could use OpenGL to, produce a hidden line removed image by using the z buffer and drawing polygons with edges and interiors the same color as the background. But of course, this method was not used in pre–raster systems.
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