Unformatted text preview: d⌫ 2 + f
a + 2(b + e)⌫ + 2c + d⌫ + f
aµ2 + 2cµ + f
aµ + 2 c µ + f
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
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
1 0 31
g C B⌫ 2 C
3 µ2 µ 1 Bb e f
P(µ, ⌫ ) = µ
@c f h j A @ ⌫ A
Using higher orders givesgives varietyvariety inand better control
Using higher orders more more in shape shape and better
But the method is hard to apply and generalise, and so is not
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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- Spring '14
- Surface, Parametric equation, Parametric surface