**Unformatted text preview: **Interactive Computer Graphics:
Lecture 12
"
Introduction to Surface Construction
! Teapot Subdivision: Russ Fish
Teapot Subdivision: Russ Fish " Graphics Lecture 12: Slide 2!
2 / 35 Non Parametric Surface"
• Surfaces can be constructed from Cartesian equations
directly, and this is acceptable for speciﬁc applications,
usually involving interpolation. !
• As before, using a simple polynomial surface is a quick
and easy approach. ! Graphics Lecture 12: Slide 3! Non Parametric Polynomial Surface
Non on Parametric Polynomial Surface "
N Parametric Polynomial Surface
00
11 011
0
a a b b c c d d xx
BB e f g CC BCC
b b e f C By C
g C By C 0
B
x x y y z z 1 1BB
@@ f h j AA @AA = 0
@z =
cc f h j
z
ddgg j j 11 11
whichwhich multiplies out to: !
• multiplies out to:
which multiplies out to:
2 ey 2 2 hz 2 2 2
ax2 ++ ey++ hz++bxy ++cxz ++f yz ++dx ++gy ++jz ++ 1 = 0
ax
2bxy 2 2cxz 2 2f yz 2 2dx 2 2gy 2 2jz 1 = 0 Because ofof theof the symmetryare 9 scalar 9 scalar inin the
• Because symmetry there 9 scalar unknowns the
Because the symmetry there arethere are unknownsunknowns in
equation equation !
the
equation • So we need to specify nine points through which the
So we need toto specify nine points through which the surface will
So we need will pass ! points through which the surface will
surface specify nine
pass
pass Graphics Lecture 12: Slide 4! As Before "
• This formulation suffers the same problems as the nonparametric spline curve. It is a ﬁxed surface for a given
set of nine points. !
• We need more ﬂexibility for the design of surfaces. ! Graphics Lecture 12: Slide 5! Simple Parametric surfaces
Simple Parametric surfaces Simple Parametric surfaces "
We can extend the formulation to simple parametric surfaces using
• can extend the equation:
WeWe can extendformulation to simplesimple parametric using
the vector the formulation to parametric surfaces
urfaces using
!
thesvector equation:the vector equation: 0
10 1
abc
µ
0
10 1
a 1) c
µ
P(µ, ⌫ ) = (µ, ⌫, b @b d eA @⌫ A
P(µ, ⌫ ) = (µ, ⌫, 1) @b d eA @⌫ A
cef
1
cef
1
2
2
P(µ, ⌫ ) = aµ + d⌫ + 2bµ⌫ + 2cµ + 2...

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- Spring '14
- Surface, Parametric equation, Parametric surface