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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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 Spring '14

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