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Cubic Surfaces

Cubic Surfaces. CGP&P Chapter 11. Modeling surfaces. Extension of parametric cubic curves called “parametric bicubic surfaces” Idea: infinite # of curves stacked together equations now have 2 parameters Q(s,t). P 1 (t). t=0.75. P 4 (t). t=0.25. t. s. Matrix representation.

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Cubic Surfaces

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  1. Cubic Surfaces • CGP&P Chapter 11

  2. Modeling surfaces • Extension of parametric cubic curves called “parametric bicubic surfaces” • Idea: infinite # of curves stacked together • equations now have 2 parameters • Q(s,t) P1(t) t=0.75 P4(t) t=0.25 t s

  3. Matrix representation • A single curve was expressed Q(t) = TMG • Now, the geometric information varies Q(s,t) = SMG each Gi(t) is itself a cubic curve

  4. Matrix representation (cont.) • Since each Gi is a cubic curve, it can be written: Gi(t) = TMGi

  5. Matrix representation (cont.) • Substituting, we obtain: Q(s,t) = SM = SMTM This formulation does not work in terms of matrix dimensions...

  6. Matrix representation (cont.) So, use the transpose rule: Gi(t) = GiT MT TT Q(s,t) = S M MT TT = S M MT TT

  7. Matrix representation (cont.) • Finally, remember that the large geometry matrix has 3 components (x, y, z) for each gij, so that we get three parametric equations: x(s,t) = S M Gx MT TT y(s,t) = S M Gy MT TT z(s,t) = S M Gz MT TT

  8. Hermite surfaces • extension of Hermite curves to parametric bicubic surfaces • four elements of the geometry matrix are now P1(t), P4(t), R1(t), R4(t) • can be thought of as interpolating the curves Q(s,0) and Q(s,1) or Q(0,t) and Q(1,t)

  9. Hermite surface matrices x(s,t) = S M GHx MT TT y(s,t) = S M GHy MT TT z(s,t) = S M GHz MT TT

  10. Hermite Surface Matrix • Upper left = x-coordinates of surface • Upper right = x derivatives in t at corners • Lower left = x derivatives in s at corners • Lower right = twist at corners

  11. Rendering surfaces • Can use iterative methods in s and t • Solve surface at points Q(s, t) and connect points with quadrilaterals • Expensive because iterating for small s and t results in many cubic surface evaluations • Forward Differencing • Our old friend… • Because we can differentiate cubic curves three times, we can increment all derivatives • x += Dx • D x += D2x • D2x += D3x

  12. Surface Rendering • Subdivision • As with cubic curves, Bezier cubic surface easily supports subdivision • Subdivision ceases when plane described by one quarter of the surface is nearly coplanar with the other three-fourths • Watch for abutting quadrilaterals that don’t match up • This happens when different levels of subdivision are applied to adjoining patches

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