By David Salomon

Computer images is critical in lots of components together with engineering layout, structure, schooling, and computing device paintings and animation. This ebook examines a wide range of present tools utilized in growing real-looking gadgets within the laptop, one of many major goals of machine graphics.

Key features:

- stable foundational mathematical creation to curves and surfaces; no complex math required

- issues prepared by means of varied interpolation/approximation concepts, each one procedure offering necessary information regarding curves and surfaces

- Exposition influenced by means of a variety of examples and workouts sprinkled all through, helping the reader

- features a gallery of colour photographs, Mathematica code listings, and sections on curves and surfaces by means of refinement and on sweep surfaces

- website maintained and up to date by means of the writer, delivering readers with errata and auxiliary material

This enticing textual content is geared to a vast and normal readership of laptop science/architecture engineers utilizing special effects to layout items, programmers for machine gamemakers, utilized mathematicians, and scholars majoring in special effects and its purposes. it can be utilized in a school room environment or as a basic reference.

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Additional resources for Curves and Surfaces for Computer Graphics

Sample text

Pnm gm (w) ... 28) Notice that it uses a matrix whose elements are nonscalar quantities (triplets). 9 The Cartesian Product 39 coordinate axes used if ij hij (u, w) = 1. If the two original curves Q(u) and R(w) are isotropic, then it’s easy to see that the surface is also isotropic because hij (u, w) = fi gj = ij i j gj fi j = 1. i The following two examples illustrate the importance of the Cartesian product. 3) from that of a straight segment. 5)]. 8). The second example starts with the parametric cubic polynomial that passes through four given points.

2. Denote by s the arc length from point h to point i. The torsion of the curve at point i is defined as the limit of the ratio θ/s when h approaches i. 15 shows how the rectifying plane rotates about the tangent as we move on the curve from h to i. The torsion can be expressed by means of the derivatives of the curve and by means of the curvature τ (t) = |Pt (t) Ptt (t) Pttt (t)| |Pt (t) Ptt (t) Pttt (t)| = ρ(t)2 . 6 Curvature and Torsion 33 (The numerator is a determinant and the denominator is an absolute value.

A C k continuity is more restrictive than Gk , so a curve that has C k continuity at a join point also has Gk continuity at the point, but there is an exception. Imagine two segments connecting at a point, where both have tangent vectors of (0, 0, 0) at the point. The vectors are identical, so the curve has C 1 continuity at the point. 3 (page 146) shows that the two segments may move in different directions at the point, in which case the curve will not have G1 continuity. Parameter Substitution Instead of naming the parameter t, we can give it a different name.

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