Download Curves and Surfaces for Computer Graphics by David Salomon PDF

By David Salomon

Computer images is critical in lots of parts together with engineering layout, structure, schooling, and computing device artwork and animation. This e-book examines a wide range of present equipment utilized in developing real-looking gadgets within the computing device, one of many major goals of laptop graphics.

Key features:

- reliable foundational mathematical advent to curves and surfaces; no complicated math required

- issues prepared by way of assorted interpolation/approximation innovations, each one method supplying invaluable information regarding curves and surfaces

- Exposition prompted through quite a few examples and workouts sprinkled all through, supporting the reader

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

- site maintained and up to date by way of the writer, offering readers with errata and auxiliary material

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

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Extra resources for Curves and Surfaces for Computer Graphics

Example text

However, their calculated tangent vectors have different magnitudes because dQ(t2 ) dQ(t) dP(t) = 2t = 2t . dt dt dt This is why we separate the direction and the magnitude of the tangent vectors when considering curve continuities. If the directions of the tangent vectors are equal, they produce a smooth join and we call this case G1 continuity (which is often all that is required in practice). Example: Consider the two straight segments P(t) = (8t, 6t) and Q(t) = (4(t + 2), 3(t + 2)). The first goes from (0, 0) to (8, 6) and the second goes from (8, 6) to (12, 9).

Such a curve has a tangent vector dP/dt, but not a slope. 10) (where α is any real number) is a straight line, even though it is a polynomial of degree 2 in t. Note that the curve goes from point P to point Q. 16 1. 1 Uniform and Nonuniform Parametric Curves So far, we have assumed that the parameter t of a parametric curve P(t) = (x(t), y(t)) varies in the interval [0, 1]. It is also possible to vary t in other ranges, and such curves may be useful in special applications. This idea arises naturally when we try to fit a curve to a given set of data points.

31) (Where matrix elements Mij are derived from the 16 points Pij and from the elements of matrix N. ) For a fixed w, the surface P(u, w) reduces to a PC curve in the u direction Pw (u) = Au3 + Bu2 + Cu + D. Each of the four coefficients is a cubic polynomial in w as follows: A(w) = M00 w3 + M01 w2 + M02 w + M03 , B(w) = M10 w3 + M11 w2 + M12 w + M13 , C(w) = M20 w3 + M21 w2 + M22 w + M23 , D(w) = M30 w3 + M31 w2 + M32 w + M33 . 1, we can compute the n points Pw (0), Pw (∆), Pw (2∆),. . , Pw ([n − 1]∆) [where (n − 1)∆ = 1] with three additions and three assignments for each point.

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