Unformatted text preview: e⌫ + f
P(µ, ⌫ ) = aµ2 + d⌫ 2 + 2bµ⌫ + 2cµ + 2e⌫ + f
There are six unknown parameter vectors
• There are six unknown parameter vectors !{a, b, c, d, e, f } There are six unknown parameter vectors {a, b, c, d, e, f } Graphics Lecture 12: Slide 6!
6 / 35 Associating points and parameters Associating points and parameters " We can solve for the six vector unknowns by substituting in six
points at can solve for of µ six vector unknowns by substituting
• We known values the and ⌫ . in six points at known values of µ and ν. !
We might have an arrangement such as: such as: !
• We might have an arrangement P0
P1
P2
P3
P4
P5 µ
0
0
1
1
1/2
1/2 ⌫
0
1
0
1
0
1 Graphics Lecture 12: Slide 7!
7 / 35 Surface parameter equations Surface parameter equations"
Substituting thesethese values of ⌫ into thento theequation gives
• Substituting values of µ and µ and ν i patch patch
us these six equations these six equations !
equation gives us
P0 = f
P1 = d + 2 e + f
P2 = a + 2 c + f
P3 = a + 2 b + 2c + d + 2 e + f
P4 = a/4 + c + f
P5 = a/4 + b + c + d + 2e + f
The The are known and we can solve for the unknowns {a, . . . , f }
• P’s P’s are known and we can solve for the unknowns
using a, . . . , f} methods:
{ standard using standard methods!
Graphics Lecture 12: Slide 8! 8 / 35 Getting the edges from the surface equation
Getting the edges from the surface equation
P(µ, ⌫ ) = aµ2 + d⌫ 2 + 2bµ⌫ + 2cµ + 2e⌫ + f
(µ, ⌫ ) = aµ2 + d⌫ 2 + 2bµ⌫ + 2cµ + 2e⌫ + f
P
Getting the edges from the surface equation
µ and ⌫ are in the range [0, 1].
⌫ are in
range [0 1]
µ and and ν arethethe range ,[0, .1].
in
Thusµthe contours that bound
the contours that bound
ThusThus the contours that bound the
the patch can be found by
found by
the patch can be found by
patch can
substituting 0 or 1 for one of µ µ
for one of µ or
substituting
substituting 0 or01or 1 for one of oror
⌫ in the the patch equation.
ν in patch equation.
⌫ in the patch equation. ! !
P(0, ⌫ )
P(0, ⌫ )
P(1, ⌫ )
P(1, ⌫ )
P(µ, 0)
P(µ, 0)
P(µ, 1)
P(µ, 1) =
=
=
=
=
=
=
= Graphics Lecture 12: Slide 9! " 2
d⌫ 2 + 2e⌫ + f
d⌫ + 2e⌫ + f
2
a + 2(b + e)⌫ + 2c +...
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 Spring '14
 Surface, Parametric equation, Parametric surface

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