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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 Spring '14
 Surface, Parametric equation, Parametric surface

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