Unformatted text preview: phics Lecture 11: Slide 30! Four point Bezier Curves and Cubic Patches Four point Bezier Curves and Cubic Patches "
• We can show their equivalence:
! We can show their equivalence:
Four point Bezier curve = Cubic patch going through the
ﬁrst and last knots (P0 and P3 ) It is possible to show their equivalence by • It is possible to show their equivalence by !
I Expanding the iterative blending equation !
– Expanding the iterative blending equation I Reversing the de Casteljau algorithm !
– Reversing the de Casteljau algorithm Graphics Lecture 11: Slide 31!
31 / 38 Expanding the blending equation
Expanding the blending equation
Expanding of fourblending equation "
the knots we can
Expanding the blending equationexpand the Bernstein blending
For the case Forfunction to get a knots we can expand the Bernstein blending
For the case of four knots we can expand the Bernstein blending
the case of four polynomial in µ:
function to get a polynomial in µ::we can expand the Bernstein
• For to get a polynomial in µ
function the case of four knots
3
X
blending3 function to get a polynomial in µ: !
X
P(µ) = 3
X W (3, i, µ)Pi
P(µ) =
W (3, i, µ)P
P(µ) =
iW (3, i, µ)Pii
=0
ii=0
==0(1 3µ)3 P0 + 3µ(1 2µ)2 P1 +23µ2 (1 µ)P2 + µ3 P3
= (1 µ) 3P0 + 3µ(1 µ) 2P1 + 3µ 2(1 µ)P2 + µ3 P3
= (1 µ) P0 + 3µ(1 µ) P1 + 3µ (1 µ)P2 + µ3 P3
•This can multiplied outoutout give equation of the the form:
Thisbe bebe multiplied giveto giveequation of form:
can multiplied to to an an an equation of the
This can be multiplied out to give an equation of the form:
This can
form: !
P(µ) = a3 µ3 + a2 µ2 + a1 µ + a0
P(µ) = a3 µ3 + a2 µ2 + a1 µ + a0
P(µ) = a3 µ3 + a2 µ2 + a1 µ + a0
where !
w
where here
a0 = P 0
where
a0 = P 0
a0 a = = 0 3 P
P
3P0
1
1
a 1 = 3 P 1 3P0
a1 a = = P3P 3P6P + 3P
31
0
2
2
1
0
a 2 = 3 P 2 6P1 + 3 P 0
a2 a = = PP 6PP++PP
32
3 1 2 3 3 0 1 P0
3
3
a3 = P3 3P2 + 3P1 P0
a3 = P3 3P2 + 3P1 P0
32 / 38 Expanding the blending equation
Expanding the blending equation "
Expanding the blending equation
These equations are linear
These equations are linear
• These equations are a
linear ! P
0=
0
!
a0 = P 0 3 P
a=
1 !
! a1 =
a2
a2 =
a 1 3P0 3 P 1 3 P P0 6P + 3 P
3
=
2
1
0
3P2 6P1 P 3+ 0 P
3+2 P 3 1 P0
3 = P3
a3 = P3 3P2 + 3P1 P0
Note that P0 P a P3 are the endpoints
• Note that P0 and and re the endpoints!
NoteRecallP0 and P33 are the endpoints
that the matrix form used for a cubic spline
• Recall the matrix form used for a cubic spline patchpatch
!
Recall the matrix form used for a cubic spline patch
!
0
10
10
1
a
1
0
0 10
P
0
10 0
0
10
B
a0 a1 C B 0
1
01
00
0 0 C B P0 C
P0
BC C = B
CB C0 C
B a1 a A @ 3
@C 2 B 0
12
0 3 C1 A @ C 3 A
0 C B P0 P
B
B0
=B
@ a2 A @ 3 2
21
3 2 A 1 P3 A 0
[email protected]
a3
P3
a3
2
1
2
1
P0
3
Graphics see
Lecture 11: Slide 33!
see
33 / 38 Expanding the blending equation
Expanding the blending equation "
Expanding the blending equation
These equations are linear
These equations are linear
• These equations are = P!
a0 linear 0
!
a0 = PP
a = 30
1 !
! 3P0
3P0 + 3 P
6P 3 P1
3 P2
1
0
3P2 36P1+ 33P0 P
P3
P2 + P1
3
0
a3...
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 Spring '14
 µ, cubic spline patch, Interpolating Splines

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