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Unformatted text preview: ezier Curves"
• Bezier curves were developed as a method for CAD
design. They give very predictable results for small sets
of knots, and so are useful as spline patches.!
• The main characteristics of Bezier curves are!
– They interpolate the end points!
– The slope at an end is the same as the line joining the end point
to its neighbour! Graphics Lecture 11: Slide 21! A typical Bezier curve
A typical Bezier Curve" Graphics Lecture 11: Slide 22! Casteljau’s Algorithm"
• Bezier curves may be computed and visualised using a
geometric construction introduced by Paul de Casteljau. !
• Like a cubic patch, we need a parameter µ which is !
– 0 at the start of the curve !
– 1 at the end. ! • The construction !
– is recursive!
– can be made for any value of µ ! Graphics Lecture 11: Slide 23! Casteljau’s Algorithm Casteljau’shere is approximatelyµ = 0.25"
Construction 0.25
The value of µ Graphics Lecture 11: Slide 24! 24 / 38 Casteljau’s Algorithm Casteljau’s Construction µ = 0.6"
µ ⇡ 0.6 Graphics Lecture 11: Slide 25! 25 / 38 Casteljau’s Algorithm Casteljau.’µ ⇡ 0.9
s Construction µ = 0.9"
One more . . Graphics Lecture 11: Slide 26! 26 / 38 Bernstein Blending Function Bernstein Blending Function"
• Splines (including curves) can be can be formulated of
Splines (including BezierBezier curves)formulated as a blendas a
the blend of the knots.!
knots.
Consider the vectorvector line equation!
• Consider the line equation
P = (1 µ)P0 + µP1 • It is a linear ‘blend’ of two points, and could also be
It is considered the twopoint Bezier curve!!
a linear blend of two points.
Can also be considered as the 2 point Bezier curve!‡ Graphics Lecture 11: Slide 27!
‡ One level of recursion
27 / 38 Blending Equation
Blending Equation Blending Equation" Any Any pointthesplinespline is simply a ofall the other points.
Anypoint on the splineisissimply aablend ofblend of other points.
simply blend all the all the other
• point on on the
For Noints. knotsNwehave: we have:!
For p + 11knots we have:
N + For +1 knots
N
N
X
X
P(µµ)=
P( ) = W (N, i,i, µ) Pi
W (N, µ) Pi
i=0
i=0 where W isisW is Bernsteinblending functionfunction!
where W the Bernstein blending function
• where the the Bernstein blending
✓◆
✓◆
Ni
i
W (N, i,i, µ)= N µµ(1 µµNN i i
W (N, µ) = i
(1 ) )
i
✓◆
✓◆
N
N! !
N=
N
i i =(N i)!i!i!
(N i)!
Graphics Lecture 11: Slide 28!
28 / 38
28 / 38 Blending Equation: Expansions for di↵erent N Blending Equation: Expansions for different N
" N Expansion 1 (1 µ)P0 + µP1 2 (1 µ)2 P0 + 2µ(1 µ)P1 + µ2 P2 3
.
.
. (1
.
.
. µ)3 P0 + 3µ(1 µ)2 P1 + 3µ2 (1 µ)P2 + µ3 P3 Graphics Lecture 11: Slide 29!
29 / 38 Bezier Curves lack local control "
• Since all the knots of the Bezier curve all appear in the
blend they cannot be used for curves with ﬁne detail. !
• However they are very effective as spline patches. !
! Gra...
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 Spring '14

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