Lecture 12 - Spline surfaces (slides)

The boundaries are all second thusothe contours that

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Unformatted text preview: d⌫ 2 + f a + 2(b + e)⌫ + 2c + d⌫ + f 2 aµ2 + 2cµ + f aµ + 2 c µ + f 2 aµ2 + 2(b + c)µ + d + 2e + f aµ + 2(b + c)µ + d + 2e + f 9 / 35 9 / 35 Getting the edges from the surface equation The P(µ, ⌫ ) = aµ2 + d⌫ 2 " 2bµ⌫ + 2cµ + 2e⌫ + f resulting surface+ µ and ⌫ are in the range [0, 1]. The boundaries are all second Thusothe contours that bound nice rder curves and so will be the patchsmooth ! found by and can be substituting 0 or 1 for one of µ or ! ⌫ in the patch equation. There isP(0, ⌫a lot= flexibility2in⌫ + f quite ) of d⌫ 2 + e this formulation, but it is still only suitable for simple surfaces. ! 2 ! P(1, ⌫ ) = a + 2(b + e)⌫ + 2c + d⌫ + f P(µ, 0) = aµ2 + 2cµ + f P(µ, 1) = aµ2 + 2(b + c)µ + d + 2e + f Graphics Lecture 12: Slide 10! 9 / 35 We can use higher orders We can use higher orders " E.g. using the tensor product! E.g. using the tensor product ! 0 1 0 31 abcd ⌫ ! B g C B⌫ 2 C 3 µ2 µ 1 Bb e f CB C ! P(µ, ⌫ ) = µ @c f h j A @ ⌫ A ! dgjk 1 ! Using higher orders givesgives varietyvariety inand better control Using higher orders more more in shape shape and better control ! But the method is hard to apply and generalise, and so is not ! usually done But the method is hard to apply and generalise, and so is not usually done ! ! Graphics Lecture 12: Slide 11! ! 11 / 35 Cubic Spline Patches " •  The patch method is generally effective in creating more complex surfaces. ! •  The idea is, as in the case of the curves, to create a surface by joining a lot of simple surfaces continuously. ! Graphics Lecture 12: Slide 12! Cartesian surface patches - terrain map Cartesian surface patches - terrain map " Graphics Lecture 12: Slide 13! 13 / 35 Points and Gradients Points and Gradients " At each corner of the patch we need to interpolate the points and • At gradients to of the patch we need to set   the each cornermatch the adjacent patch. interpolate the points and set the gradients to match the adjacent patch.!...
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