By Tom McReynolds, David Blythe

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**Additional info for Advanced Graphics Programming Techniques Using OpenGL**

**Sample text**

A CSG Tree in Normal Form 7. 8. 9. X , Y Z ! X Z , Y X Y , Z ! X , Z Y , Z X Y Z ! X Z Y Z X, Y, and Z here match either primitives or subtrees. (isUnionOperation(t) || (isPrimitive(t->right) && ! isUnionOperation(T->left)))); normalize(t->right); } Normalization may increase the size of the tree and add primitives which do not contribute to the final image. The bounding volume of each CSG subtree can be used to prune the tree as it is normalized. Bounding volumes for the tree may be calculated using the following algorithm: 18 Programming with OpenGL: Advanced Rendering findBounds(tree *t) { if (isPrimitive(t)) return; findBounds(t->left); findBounds(t->right); switch (t->operation){ case union: t->bounds = unionOfBounds(t->left->bounds, t->right->bounds); case intersection: t->bounds = intersectionOfBounds(t->left->bounds, t->right->bounds); case subtraction: t->bounds = t->left->bounds; } } CSG subtrees rooted by the intersection or subtraction operators may be pruned at each step in the normalization process using the following two rules: 1.

No matter how far back the far clip plane is, at least half of the available depth range is present in the first “unit” of distance. In other words, if the distance from the eye to the near clip plane is one unit, at least half of the z range is used up in the first “unit” from the near clip plane towards the far clip plane. Figure 18 plots the z range for the first unit distance for various ranges. 5 at one unit of distance. As long as the data is mostly drawn close to the near plane, the z precision is good.

The algorithm presented here assumes that the CSG tree is in “normal” form. A tree is in normal form when all intersection and subtraction operators have a left subtree which contains no union operators and a right subtree which is simply a primitive (a set of polygons representing a single solid object). All union operators are pushed towards the root, and all intersection and subtraction operators are pushed towards the leaves. For example, A B , C D E G , F H is in normal form; Figure 13 illustrates the structure of that tree and the characteristics of a tree in normal form.

### Advanced Graphics Programming Techniques Using OpenGL by Tom McReynolds, David Blythe

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