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Unformatted text preview: = P3 3P2 + 3P1 P0
So So we get the directionsat the endpoints by using P P1 and.!P2 .
• we get the directions at the endpoints by using 1 and P2
Notehave shown the 3 are the equation is the same as a cubic patch
and
We that Pshown P blending endpointsis the same as a cubic patch !
• We have 0
the blending equation
Recall the matrix form used for a cubic spline patch
0
10
10
1
0
0
0 10
P
0 a0 1 0 1
10
0
0
0 C B PP1 3P0 C
B a0 C B 1
a1 C B 0
1
0
0 C B 30
B
C
B a1 C = B 0
1
0
0 C B P0 C 3
@
A@
A@
A
2
3
1 CB 0 C
P
B a2 C = B 3
@ a2 A @ 3
2
3
1 A @ PP3A 3P2
a3
2
1
2
1
33
a3
2
1
2
1
P0
3
a1
a2
a2
a =
=
=
= 1 Graphics Lecture 11: Slide 34!
see 34 / 38 Reversing the de Casteljau algorithm "
Reversing the de Casteljau algorithm
We start from the point P3,0 and
work in reverse to express it in
terms of its construction line. P3,0 = (1
= (1
= (1
= (1 µ)P2,0 + µP2,1
µ) {(1 µ)P1,0 + µP1,1 } + µ {(1 µ)2 P1,0 + 2µ(1
µ)2 {(1 +2µ(1 +µ2 {(1 µ)P1,1 + µ2 P1,2 µ)P0,0 + µP0,1 } µ) {(1 µ)P0,1 + µP0,2 } µ)P0,2 + µP0,3 } µ)P1,1 + µP1,2 } Reversing the de Casteljau algorithm Reversing the de Casteljau algorithm "
. . . continuing the expansion, we can drop the ﬁrst subscript
. . . continuing the expansion, we get:
(which indicates the recursion level) tocan drop the ﬁrst subscript (which indicates the recursion level) to get: ! P(µ) = (1 µ)2 {(1 +2µ(1 +µ2 {(1 = (1 µ)P0 + µP1 } µ) {(1 µ)P1 + µP2 } µ)P2 + µP3 } µ)3 P0 + 3µ(1 µ)2 P1 + 3µ2 (1 µ)P2 + µ3 P3 This the same as as expanded Bernstein blending polynomial
This is is the same the the expanded Bernstein blending
polynomial already shown is equivalent to a is equivalent to
which we havewhich we have already shown cubic spline patcha
cubic spline patch !
!see Graphics Lecture 11: Slide 36! 36 / 38 Control Points "
Control Points
• We can summarise the four four point Bezier Curve by that it has
We can summarise the point Bezier Curve by saying saying
thattwohas ! that are interpolated (P0 , P3 )
I it points
– two points that are interpolated (P0, P3)
two control points (P1 , P2 )
– two control points (P1, P2) I • The curve starts at at0P0 and endsP3 P3 and shape can be
The curve starts P and ends at at and its its shape can
determined by moving control controlP1 , P2 . P , P .!
be determined by moving points points 1 2
This could be done interactively using a mouse.
• This could be done interactively using a mouse. ! Graphics Lecture 11: Slide 37! In summary . . . …"
In summary
In summary . . . • simplest and most ↵ective way to way to draw smooth
The The simplest andemost effective draw a smoothacurve
curveset
of points is to
throughsimplestof pointssetto usewaycubic patch.a cubic patch. !
a through a is↵ective a to drawuse
The
and most e
a smooth curve
through a set of points is to use a cubic patch. No interaction needed?
No interaction needed?!
I No interaction gradients by the
setting the needed?
!
I setting the gradientsby the
setting
gradients by !
central di↵erence
difference !
central
1 the centralerence
(Pi+1 di↵i 1 ) is e↵ective.
P
2 !1
2 (Pi+1
! Pi 1) is e↵ective. is effective. !
!
!
! User wants interactive shape
User interactive shape
User wants wants interactive !
adjustment?
shape
adjustment? adjustment? !
I ! the four point Bezier
I the four point Bezier
The four point is ideal!
formulation Bezier
formulation is ideal
formulation is ideal !
!
! Graphics Lecture 11: Slide 38!
38 / 38...
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This document was uploaded on 03/26/2014.
 Spring '14

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