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Lecture Notes
Dr. Tong Lai Yu, 2010
    1. Introduction
    2. OpenGL Shading Language ( GLSL ) I
    3. GLSL II
    4. Curve and Surface Design
    5. Modeling Shapes with Polygonal Meshes
    6. Texture Mapping
    7. Casting Shadows
    8. Tools for Raster Display
    9. Parsing External Objects

    Curves and Surfaces

    Hermite Interpolation

    1. Consider both derivatives at data points as well as data points themselves; closely related to "Newton divided difference" method.

    2. May consider Cubic polynomials; curve passing through 5 points P0, P1,P2,P3,P4, ( a set of data points can be broken down to groups of 5 points )

    3. Pk = ( xk, yk, zk ), we can consider each component separately

    4. Subinterval (xk, xk+1) can be normalized to ( 0, 1 ) via t = ( x - xk ) / ( xk+1 - xk )

    5. The k-th cubic segment of the interpolated curve is given by so that xk(0) = xk,   xk(1) = xk+1

    6. So,

    7. The derivatives are

    8. Let x'k = x'k(0),   x'k+1 = x'k(1)

    9. From f) and h), we can solve the coefficients:

    How to find the derivatives ( slopes ) ?

    Consider second derivatives:

    We require the second derivatives to be continuous, i.e. x"k-1(1) = x"k(0), So

    Substituting ak, and bk of i) into (2), we have

    which can be simplified to

    This gives us 4 sets of equations:

    Now we have 3 equations but 5 unknowns. We need 2 more; so we choose,

    which gives ( from 1 ),

    Substituting these into i), we have

    So we have 5 equations, 4a-4e, and can solve for the 5 unknowns x'i, i = 0, 1, 2, 3, 4