# Advanced Graphics Programming Techniques Using OpenGL P2

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## Advanced Graphics Programming Techniques Using OpenGL P2

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A fairly simple method of converting a model into triangle strips is sometimes known as greedy tristripping.One of the early greedy algorithms was developed for IRIS GL which allowed swapping of vertices to create direction changes to the facet with the least neighbors.

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## Nội dung Text: Advanced Graphics Programming Techniques Using OpenGL P2

1. 11 7 9 12 5 8 3 6 10 1 4 2 0 Figure 11. “Greedy” Triangle Strip Generation 3.4.1 Greedy Tri-stripping A fairly simple method of converting a model into triangle strips is sometimes known as greedy tri- stripping. One of the early greedy algorithms was developed for IRIS GL which allowed swapping of vertices to create direction changes to the facet with the least neighbors. However, with OpenGL the only way to get the equivalent behavior of swapping vertices is to repeat a vertex and create a degenerate triangle, which is much more expensive than the original vertex swap operation. For OpenGL a better algorithm is to choose a polygon, convert it to triangles, then continue onto the neighboring polygon from the last edge of the previous polygon. For a given starting polygon beginning at a given edge, there are no choices as to which polygon is the best to choose next since there is only one choice. The strip is continued until the triangle strip runs off the edge of the model or runs into a polygon that is already a part of another strip (see Figure 11). For best results, pick a polygon and go both directions as far as possible, then start the triangle strip from one end. A triangle strip should not cross a hard edge, unless the vertices on that edge are repeated redun- dantly, since you’ll want different normals for the two triangles on either side of that edge. Once one strip is complete, the best polygon to choose for the next strip is often a neighbor to the polygon at one end or the other of the previous strip. More advanced triangulation methods don’t try to keep all triangles of a polygon together. For more information on such a method refer to [17]. 3.5 Capping Clipped Solids with the Stencil Buffer When dealing with solid objects it is often useful to clip the object against a plane and observe the cross section. OpenGL’s user-deﬁned clipping planes allow an application to clip the scene by a plane. The stencil buffer provides an easy method for adding a “cap” to objects that are intersected by the clipping plane. A capping polygon is embedded in the clipping plane and the stencil buffer is used to trim the polygon to the interior of the solid. 15 Programming with OpenGL: Advanced Rendering
2. For more information on the techniques using the stencil buffer, see Section 14. If some care is taken when constructing the object, solids that have a depth complexity greater than 2 (concave or shelled objects) and less than the maximum value of the stencil buffer can be rendered. Object surface polygons must have their vertices ordered so that they face away from the interior for face culling purposes. The stencil buffer, color buffer, and depth buffer are cleared, and color buffer writes are disabled. The capping polygon is rendered into the depth buffer, then depth buffer writes are disabled. The stencil operation is set to increment the stencil value where the depth test passes, and the model is drawn with glCullFace(GL BACK). The stencil operation is then set to decrement the stencil value where the depth test passes, and the model is drawn with glCullFace(GL FRONT). At this point, the stencil buffer is 1 wherever the clipping plane is enclosed by the frontfacing and backfacing surfaces of the object. The depth buffer is cleared, color buffer writes are enabled, and the polygon representing the clipping plane is now drawn using whatever material properties are desired, with the stencil function set to GL EQUAL and the reference value set to 1. This draws the color and depth values of the cap into the framebuffer only where the stencil values equal 1. Finally, stenciling is disabled, the OpenGL clipping plane is applied, and the clipped object is drawn with color and depth enabled. 3.6 Constructive Solid Geometry with the Stencil Buffer Before continuing, the it may help for the reader to be familiar with the concepts of stencil buffer usage presented in Section 14. Constructive solid geometry (CSG) models are constructed through the intersection ( ), union ( ), and subtraction (,) of solid objects, some of which may be CSG objects themselves[23]. The tree formed by the binary CSG operators and their operands is known as the CSG tree. Figure 12 shows an example of a CSG tree and the resulting model. The representation used in CSG for solid objects varies, but we will consider a solid to be a collection of polygons forming a closed volume. “Solid”, “primitive”, and “object” are used here to mean the same thing. CSG objects have traditionally been rendered through the use of ray-casting, which is slow, or through the construction of a boundary representation (B-rep). B-reps vary in construction, but are generally deﬁned as a set of polygons that form the surface of the result of the CSG tree. One method of generating a B-rep is to take the polygons forming the surface of each primitive and trim away the polygons (or portions thereof) that don’t satisfy the CSG oper- ations. B-rep models are typically generated once and then manipulated as a static model because they are slow to generate. Drawing a CSG model using stencil usually means drawing more polygons than a B-rep would con- tain for the same model. Enabling stencil also may reduce performance. Nonetheless, some portions 16 Programming with OpenGL: Advanced Rendering
3. Resulting solid CGS tree Figure 12. An Example Of Constructive Solid Geometry of a CSG tree may be interactively manipulated using stencil if the remainder of the tree is cached as a B-rep. The algorithm presented here is from a paper by Tim F. Wiegand describing a GL-independent method for using stencil in a CSG modeling system for fast interactive updates. The technique can also process concave solids, the complexity of which is limited by the number of stencil planes avail- able. A reprint of Wiegand’s paper is included in the Appendix. 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. A CSG tree can be converted to normal form by repeatedly applying the following set of production rules to the tree and then its subtrees: 1. X , Y Z  ! X , Y  , Z 2. X Y Z  ! X Y  X Z  3. X , Y Z  ! X , Y  X , Z  4. X Y Z  ! X Y  Z 5. X , Y , Z  ! X , Y  X Z  6. X Y , Z  ! X Y  , Z 17 Programming with OpenGL: Advanced Rendering
4. Union at top of tree Left child of intersection Key or subtraction is never union H Union intersection C Subtraction F A B A Primitive G Right child of intersection D E or subtraction always a primitive ((((A B) - C) (((D E) G) - F)) H) Figure 13. A CSG Tree in Normal Form 7. X , Y  Z ! X Z  , Y 8. X Y  , Z ! X , Z  Y , Z  9. X Y  Z ! X Z  Y Z  X, Y, and Z here match either primitives or subtrees. Here’s the algorithm used to apply the produc- tion rules to the CSG tree: normalize(tree *t) { if (isPrimitive(t)) return; do { while (matchesRule(t)) /* Using rules given above */ applyFirstMatchingRule(t); normalize(t->left); } while (!(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
5. 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. if T is an intersection and not intersects(T->left->bounds, T->right->bounds), delete T. 2. if T is a subtraction and not intersects(T->left->bounds, T->right->bounds), re- place T with T->left. The normalized CSG tree is a binary tree, but it’s important to think of the tree rather as a “sum of products” to understand the stencil CSG procedure. Consider all the unions as sums. Next, consider all the intersections and subtractions as products. (Subtraction is equivalent to intersection with the complement of the term to the right. For example,  A , B = A B .) Imagine all the unions ﬂattened out into a single union with multiple children; that union is the “sum”. The resulting subtrees of that union are all composed of subtractions and intersections, the right branch of those operations is always a single primitive, and the left branch is another operation or a single primitive. You should read each child subtree of the imaginary multiple union as a single expression containing all the intersection and subtraction operations concatenated from the bottom up. These expressions are the “products”. For example, you should think of A B  , C  G D , E  F  H as meaning A B , C  G D , E F  H . Figure 14 illustrates this process. At this time redundant terms can be removed from each product. Where a term subtracts itself (A , A), the entire product can be deleted. Where a term intersects itself (A A), that intersec- tion operation can be replaced with the term itself. 19 Programming with OpenGL: Advanced Rendering
6. H H C G -F F E A B D G C B- A D E ((((A B) - C) (((D E) G) - F)) H) (A B - C) (D E G - F) H Figure 14. Thinking of a CSG Tree as a Sum of Products All unions can be rendered simply by ﬁnding the visible surfaces of the left and right subtrees and letting the depth test determine the visible surface. All products can be rendered by drawing the visible surfaces of each primitive in the product and trimming those surfaces with the volumes of the other primitives in the product. For example, to render A , B , the visible surfaces of A are trimmed by the complement of the volume of B, and the visible surfaces of B are trimmed by the volume of A. The visible surfaces of a product are the front facing surfaces of the operands of intersections and the back facing surfaces of the right operands of subtraction. For example, in A , B C , the visible surfaces are the front facing surfaces of A and C, and the back facing surfaces of B. Concave solids are processed as sets of front or back facing surfaces. The “convexity” of a solid is deﬁned as the maximum number of pairs of front and back surfaces that can be drawn from the viewing direction. Figure 15 shows some examples of the convexity of objects. The nth front sur- face of a k-convex primitive is denoted Anf , and the nth back surface is Anb . Because a solid may vary in convexity when viewed from different directions, accurately representing the convexity of a primitive may be difﬁcult and may also involve reevaluating the CSG tree at each new view. In- stead, the algorithm must be given the maximum possible convexity of a primitive, and draws the nth visible surface by using a counter in the stencil planes. The CSG tree must be further reduced to a “sum of partial products” by converting each product to a union of products, each consisting of the product of the visible surfaces of the target primitive with the remaining terms in the product. 20 Programming with OpenGL: Advanced Rendering
7. 1 1 2 1 2 3 2 3 4 4 5 6 1-Convex 2-Convex 3-Convex Figure 15. Examples of n-convex Solids For example, if A, B, and D are 1-convex and C is 2-convex: A , B C D ! A0f , B C D B0b A C D C0f A , B D C1f A , B D D0f A B C  Because the target term in each product has been reduced to a single front or back facing surface, the bounding volumes of that term will be a subset of the bounding volume of the original complete primitive. Once the tree is converted to partial products, the pruning process may be applied again with these subset volumes. In each resulting child subtree representing a partial product, the leftmost term is called the “target” surface, and the remaining terms on the right branches are called “trimming” primitives. The resulting sum of partial products reduces the rendering problem to rendering each partial prod- uct correctly before drawing the union of the results. Each partial product is rendered by drawing the target surface of the partial product and then “classifying” the pixels generated by that surface with the depth values generated by each of the trimming primitives in the partial product. If pixels drawn by the trimming primitives pass the depth test an even number of times, that pixel in the target primitive is “out”, and discarded. If the count is odd, the target primitive pixel is “in”’, and kept. Because the algorithm saves depth buffer contents between each object, we optimize for depth saves and restores by drawing as many of target and trimming primitives for each pass as we can ﬁt in the stencil buffer. 21 Programming with OpenGL: Advanced Rendering
8. The algorithm uses one stencil bit (Sp ) as a toggle for trimming primitive depth test passes (parity), n stencil bits for counting to the nth surface (Scount ), where n is the smallest number for which 2n is larger than the maximum convexity of a current object, and as many bits are available (Sa ) to accumulate whether target pixels have to be discarded. Because Scount will require the GL INCR operation, it must be stored contiguously in the least-signiﬁcant bits of the stencil buffer. Sp and Scount are used in two separate steps, and so may share stencil bits. For example, drawing 2 5-convex primitives would require 1 Sp bit, 3 Scount bits, and 2 Sa bits. Because Sp and Scount are independent, the total number of stencil bits required would be 5. Once the tree has been converted to a sum of partial products, the individual products are rendered. Products are grouped together so that as many partial products can be rendered between depth buffer saves and restores as the stencil buffer has capacity. For each group, writes to the color buffer are disabled, the contents of the depth buffer are saved, and the depth buffer is cleared. Then, every target in the group is classiﬁed against its trimming primitives. The depth buffer is then restored, and every target in the group is rendered against the trimming mask. The depth buffer save/restore can be optimized by saving and restoring only the region containing the screen-projected bounding volumes of the target surfaces. for each group glReadPixels(...); glStencilMask(0); /* so DrawPixels won’t affect Stencil */ glDrawPixels(...); Classiﬁcation consists of drawing each target primitive’s depth value and then clearing those depth values where the target primitive is determined to be outside the trimming primitives. glClearDepth(far); glClear(GL_DEPTH_BUFFER_BIT); a = 0; for (each target surface in the group) for (each partial product targeting that surface) for (each trimming primitive in that partial product) a++; The depth values for the surface are rendered by drawing the primitive containing the the target sur- face with color and stencil writes disabled. ( Scount ) is used to mask out all but the target surface. In practice, most CSG primitives are convex, so the algorithm is optimized for that case. if (the target surface is front facing) glCullFace(GL_BACK); 22 Programming with OpenGL: Advanced Rendering