3D Graphics for Game Pgmg. (rev.) by J. Han

By J. Han

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View-frustum culling can save a fair amount of GPU computing cost with a little CPU overhead, especially when a polygon mesh comprises a large number of vertices. In Fig. 14, only the teapot would survive the view-frustum culling. However, its handle intersects the far plane of the view frustum. If a polygon intersects the boundary of the view frustum, it is clipped with respect to the boundary, and only the portion inside the view frustum is processed for display. See Fig. 15. The clipped polygons with black edges are further processed whereas those with red edges are discarded.

The second is a valid basis for R2 , but is neither standard nor orthonormal. The third is not the standard but an orthonormal basis. The vectors v1 , v2 , . . , vn form a basis for the vector space V if and only if (1) v1 , v2 , . . , vn are linearly independent, and (2) v1 , v2 , . . , vn span V. Fig. 12 shows three examples of the basis for the 2D Euclidean vector space R2 . The standard basis is denoted by {e1 ,e2 }, where e1 = (1, 0) and e2 = (0, 1). There are many other bases that can be chosen for R2 .

Compute a 2D affine transform matrix that rotates a 2D point by θ about a point (a, b), not about the origin. 2. In linear algebra, it is proven that the inverse of a rotation is simply its transpose. Given two non-standard orthonormal bases {a,b,c} and {d,e,f }, compute a 3×3 matrix that converts the vector defined in terms of {a,b,c} into the vector of {d,e,f }. 3. Let us define a matrix for scaling along 3 orthonormal vectors, a, b, and c, which are not identical to the standard basis vectors e1 , e2 , and e3 .

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