Lecture 12 - Spline surfaces (slides)

the terrain map can be modelled with parametric

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Unformatted text preview: There are two gradients •  There are two gradients ! Graphics Lecture 12: Slide 14! Parametric patches " Parametric patches •  In practice we use the more general parametric patch formulation with two parameters µ and ν. ! In practice we use the more general parametric patch formulation with two parameters µ and ⌫ . •  The terrain map can be modelled with parametric patches. ! The terrain map can be modelled with parametric patches. •  need to match three values at each corner We We need to match three values at each corner ! P(µ, ⌫ ) Graphics Lecture 12: Slide 15! @ P(µ, ⌫ ) @µ @ P(µ, ⌫ ) @⌫ Corners " •  As usual we adopt the convention that the corners are at parameter values (0, 0), (0, 1), (1, 0) and (1, 1) ! •  We need to ensure that the patch joins its neighbours exactly at the edges. ! •  Hence we ensure that the edge contours are the same on adjacent patches ! Graphics Lecture 12: Slide 16! Edges Edges " •  We do this by designing the edge curves in an identical manner to the cubic spline curve patch. ! We do this by designing the edge curves in an identical manner to ! the cubic spline curve patch. Edge curve Points joined P(0, ⌫ ) Pi,j Pi,j +1 P(1, ⌫ ) Pi+1,j Pi+1,j +1 P(µ, 0) Pi,j Pi+1,j P(µ, 1) Pi,j +1 Pi+1,j +1 Graphics Lecture 12: Slide 17! A parametric spline patchpatch " A parametric spline As long the gradients are are the for the four patches that As long asas the gradients the samesame for the four patches that meet at point the will join will join seamlessly ! meet at a pointathe surface surface seamlessly ! Graphics Lecture 12: Slide 18! The Coons patch The Coons patch " To define the internal points we linearly interpolate the edge To define the internal points we linearly interpolate the curves: edge curves: ! ! P(µ, ⌫ ) = P(µ, 0)(1 ⌫ ) + P(µ, 1)⌫ + ! P(0, ⌫ )(1 µ) + P(1, ⌫ )µ ! P(0, 1)(1 µ)⌫ P(1, 0)µ(1 ⌫ ) ! P(0, 0)(1 µ)(1 ⌫ ) P(1, 1)µ⌫ ! ! Substituting val...
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