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3D Computer Graphics : A Mathematical Introduction with by Samuel R. Buss

By Samuel R. Buss

This creation to 3D special effects emphasizes basics and the maths underlying special effects, whereas additionally protecting programming innovations utilizing OpenGL, a platform-independent pics programming atmosphere. The minimum must haves make it compatible for self-study or to be used as a sophisticated undergraduate or introductory graduate textual content because the writer leads step by step from the fundamentals of differences to complex subject matters resembling animations and kinematics. Accompanying software program, together with resource code for a ray tracing software program package deal, is out there freely from the book's website.

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Extra info for 3D Computer Graphics : A Mathematical Introduction with OpenGL

Example text

By convention, the axis of rotation always passes through the origin, and thus the rotation fixes the origin. 14 on page 37 illustrates the action of Rθ,u on a point v. Clearly, Rθ,u is a linear transformation and is rigid and orientation-preserving. 4 below shows that every rigid, orientation-preserving, linear transformation in 3-space is a rotation.

3 is correct. 4 Every rigid, orientation-preserving, affine transformation is either a translation or a generalized rotation. Obviously, the converse of this theorem holds too. Proof Let A be a rigid, orientation-preserving, affine transformation. Let u = A(0). 2 implies that A is a rotation. So suppose u = 0. It will suffice to prove that either A is a translation or there is some point v ∈ R2 that is a fixed point of A, that is, such that A(v) = v. 2 shows that A is a generalized rotation around v.

If A(u) = u + u, then A must be the translation Tu . This follows because, again by the rigidity of A, every point x ∈ L must map to x + u and, by the rigidity and orientation-preserving properties, the same holds for every point not on L. On the other hand, if A(u) = 0, then rigidity implies that v = 12 u is a fixed point of A, and thus A is a generalized rotation around v. Second, suppose that the line L is mapped to a different line L . 8. Since L = L, θ is nonzero and is not a multiple of 180◦ .

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