ch13oddslns

# ch13oddslns - Angel Interactive Computer Graphics Fifth...

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Angel: Interactive Computer Graphics, Fifth Edition Chapter 13 Solutions 13.1 Let’s do the problem in two dimensions. The solution in three dimensions is essentially the same. Assume that the vertices are used in a consistent clockwise or counterclockwise manner. Starting at some vertex, that vertex and the next determine a line of the form ax + by + c = 0 . If we evaluate ax + by + c for a given point, the result will be positive or negative depending on which side of the line the point lies. If we are following the vertices in a clockwise manner, the point is inside the polygon if and only if it is to the right of each of these lines. 13.3 Consider two identical circles of radius r centered at ( a, 0) and ( - a, 0). We can describe them through the single implicit equation (( x - a ) 2 + y 2 - r 2 )(( x + a ) 2 + y 2 - r 2 ) , by simply multiplying together their individual implicit equations. We can form the torus by rotation these circles about the y axis which is equivalent to replacing x 2 by x 2 + z 2 .

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