View
239
Download
3
Category
Preview:
Citation preview
Numerical Computation
Lecture 14: Bezier Curves
United International College
Last Time
• During the last class period we covered:– Cubic Splines– Readings: • Pav, section 6.2 (omit 6.2.1)
Today
• We will cover:– Bezier Curves – Readings: • These Slides !!
Review
• So far we have looked at the question of finding a polynomial (or set of polynomials) to interpolate a set of data points {xk ,yk}
• We have come up with two ways to find a single interpolating polynomial for the data: Lagrange method and Newton’s method.
• We have computed spline curves that interpolate the data over a partition.
Drawbacks to These Methods
• These methods are quite useful for a variety of applications, but they have two major drawbacks:
• 1) They cannot model data that have duplicate y – values
Drawbacks to These Methods
• 2) The final shape of the curve is completely dependent on the data. It would be nice to be able to better control the shape of the interpolating curve without having to move data points.
Bezier Curves
• Bezier Curves can solve both of these problems !• Definition: A Bezier curve is a parametric curve
described by polynomials based on control points.– Any number of control points may be used. (usually 4)– Degree of polynomials = number of points – 1. (usually =3)– Several Bezier curves can easily be glued together in a way
that makes the curve as a whole smooth.– A Bezier curve passes through its first and last control
points, but, in general, no others.
Parametric Curves
• Definition: A parametric curve is a vector function of a single argument (t). The variable t is called the parameter.
• Example: p(t) = (x(t), y(t)) is a parametric curve in the plane.
• Example: p(t) = (x(t), y(t), z(t)) is a parametric curve in space.
• Example: p(t) = (cos(t), sin(t)) for 0 ≤ t ≤ π produces a circle.
Parametric Curves
• Example: x(t) = t2–2t, y(t) = t–1, t in [0,3].
• By using parametric curves we can create curves with multiple y-values for a given x-value. This solves the first problem discussed above as Drawback 1).
x
y
t = 0 (start)
(t2–2t, t–1)
t = 3 (end)
Parametric Curves
• Other Benefits of using Parametric Curves:– Every curve can be described as a parametric
curve.– A parametric curve can be easily computed.– A parametric curve is a useful description of
motion. We think of the parameter as representing time (thus, “t”).
Parametric Curves - Motion
• Given a parametric curve p(t) = (x(t), y(t)) p’(t) = (x’(t), y’(t)) velocity vector points in
direction of motionp’(t)/||p’(t)|| unit tangent (velocity) vector||p’(t)|| speed
t = 0t = 1
p(t) p’(t)
Bezier Curves
• Bezier curves are parametric curves, where each of the parametric functions for x(t) and y(t) are polynomials in t (typically degree 2 or 3).
• Quadratic Bezier Curve: x(t) and y(t) are quadratic polynomials in t. (0 ≤ t ≤ 1)
• Cubic Bezier Curve: x(t) and y(t) are cubic polynomials in t. (0 ≤ t ≤ 1)
x(t) = at3 + bt2 + ct + dy(t) = et3 + ft2 + gt + h
Bezier Curves
• Cubic Bezier Curve: x(t) = at3 + bt2 + ct + d y(t) = et3 + ft2 + gt + h (0 ≤ t ≤ 1)• A Bezier curve passes through its first and last control points, so
x0 = x(0) y0 = y(0) x3 = x(1) y3 = y(1)
How do we find the other points (x(t),y(t)) on the curve?
p0=(x0,y0)(x(t),y(t))
•
p1=(x1,y1)
p3=(x3,y3)
p2=(x2,y2)
de Casteljau Algorithm
• The de Casteljau algorithm calculates the points on a Bezier Curve as a recursive series of linear interpolations. (“Lerps”)• Step 1:
p0
q0
p1
p2
p3
q2
q1
32322
21211
10100
)1(,,
)1(,,
)1(,,
ppppq
ppppq
ppppq
tttLerp
tttLerp
tttLerp
de Casteljau Algorithm
• Step 2: q0
q2
q1
r1
r0
21211
10100
)1(,,
)1(,,
qqqqr
qqqqr
tttLerp
tttLerp
de Casteljau Algorithm
• Step 3:
r1p(t)=(x(t),y(t))
r0•
1010 )1(,,)( rrrrp tttLerpt
de Casteljau Algorithm
• Expanding this recursive formula gives:
3221
211010
3221211
2110100
32322
21211
10100
111
1111,,)(
111,,
111,,
1,,
1,,
1,,
pppp
pppprrp
ppppqqr
ppppqqr
ppppq
ppppq
ppppq
ttttttt
ttttttttLerpt
tttttttLerp
tttttttLerp
tttLerp
tttLerp
tttLerp
de Casteljau Algorithm
• Collecting powers of t and simplifying, we get
• Note: p(0) = p0 and p(1) = p3, so this curve does go through the first and last control points, as was required.
3
32
23
123
023
33
22
12
03
33
363133)(
,
13131)(
pp
ppp
ppppp
ttt
ttttttt
or
ttttttt
Bezier Curve
p(t)•
p0
p1
p2
p3
Bernstein Polynomials
• The polynomials that appear as coefficients of the powers of t are called Bernstein Polynomials.
3
32
23
123
023
33
22
12
03
33
363133)(
,
13131)(
pp
ppp
ppppp
ttt
ttttttt
or
ttttttt
Bernstein Polynomials
33
3
2332
2331
2330
3332
321
310
30
33
223
123
023
33
363
133
)(
33
363133)(
ttB
tttB
ttttB
ttttB
tBtBtBtBt
ttt
ttttttt
ppppp
pp
ppp
Bernstein Polynomials
B03(t)
B13(t) B2
3(t)
B33(t)
Bernstein Polynomials
!!
!
13
3
12
3)1(333
11
3)1(3363
10
3)1(133
30333
2122332
1222331
0332330
ini
n
i
n
ttttB
tttttttB
ttttttttB
tttttttB
• Definition: The i-th Bernstein polynomial of degree n is defined as
• Note: If we add up all of the Bernstein polynomials of degree n, we always get 1, no matter what t is.
Bernstein Polynomials
iinni tt
i
ntB
1
1 tBni
• Bernstein Formula for a Bezier curve of degree n:
Bezier Curves
n
ii
ni
iinni
tBt
tti
ntB
0
1
pp
• Bernstein Formula for a cubic Bezier curve:
Bezier Curves
3
0
3
iii tBt pp
• So far we have looked at two methods for computing the Bezier curve: de Casteljau’s Algorithm and Bernstein Polynomials. Now, we will consider a third method, which is the most efficient method. This method involves matrix calculations.
• Recall the equations for the cubic Bezier curve:
Matrix Form of Bezier Curves
3
32
23
123
023
33
363133)(
pp
ppp
ttt
ttttttt
• If we re-group the terms, so that we have all multipliers of powers of t, we get:
Matrix Form of Bezier Curves
133
36333)(
010
2210
33210
ppp
pppppppp
t
ttt
Matrix Form of Bezier Curves
0
10
210
3210
23
010
2210
33210
33
363
33
)(
133
36333)(
pd
ppc
pppb
ppppa
dcbap
ppp
pppppppp
tttt
t
ttt
Matrix Form of Bezier Curves
0
10
210
3210
23
33
363
33
)(
pd
ppc
pppb
ppppa
dcbap
tttt
3
2
1
0
23
0001
0033
0363
1331
1)(
p
p
p
p
d
c
b
a
d
c
b
a
p tttt
Matrix Form of Bezier Curves
yx
yx
yx
yx
pp
pp
pp
pp
tttt
tttt
33
22
11
00
23
3
2
1
0
23
0001
0033
0363
1331
1)(
0001
0033
0363
1331
1)(
p
p
p
p
p
p
function p = bezier(t, p0, p1, p2, p3 )% bezier computes the bezier curve value p(t)% defined by control points p0, p1, p2, p3u=[t^3 t^2 t 1];M = [-1, 3,-3, 1; 3,-6, 3, 0; -3, 3, 0, 0; 1, 0, 0, 0]C = [p0;p1;p2;p3]p=u*M*Cend
Matlab Function for Bezier Curves
function plot_bezier(p0,p1,p2,p3)%plot_bezier plots the bezier curve through the control % points p0, p1, p2, p3 tvector = 0.0:0.05:1; pts = zeros(length(tvector),2); for i = 1:length(tvector) pts(i,:) = bezier(tvector(i), p0, p1, p2, p3); end C = [p0;p1;p2;p3]; plot(C(:,1), C(:,2),'o',pts(:,1),pts(:,2),'-');end
Matlab Function for Bezier Curves
Matlab Function for Bezier Curves
• Bezier surfaces are an extension of Bezier curves to 3 dimensions
• Instead of the curve being parameterized by a single variable t, we use two variables, s and t
• By definition, we choose to have s and t range from 0 to 1
Bezier Surfaces
s
t
0,0
1,1
1,0
0,1
• Consider a bicubic Bezier surface (bicubic means that it is a cubic function in both the s and t parameters)
• A cubic curve has 4 control points, and a bicubic surface has a grid of 16 control points, p0 through p15
Control Mesh
p0 p1
p2
p3
p4 p5
p6
p7
p8 p9
p10
p11
p12 p13
p14 p15
s
t
• Recall that for a Bezier curve we had a formula in terms of the Bernstein polynomials:
• It seems reasonable to define the Bezier surface in a similar fashion so that the surface is really a family of Bezier curves:
Bernstein Polynomial
ijjiji
tBsBts pp )()(, 333
0
3
0
3
0
3
iii tBt pp
• Recall that the matrix form for a Bezier curve is
• A parametric surface will have three component functions of s and t – one for each of the three dimensions: p(s,t) = (x(s,t), y(s,t), z(s,t))
Matrix Form
yx
yx
yx
yx
pp
pp
pp
pp
tttt
33
22
11
00
23
0001
0033
0363
1331
1)(p
• To simplify the calculation of the parametric surface, we will consider the matrix equation for each of the x, y, and z components separately
• For example, to evaluate the x component of a Bezier curve, we have:
Matrix Form
x
x
x
x
p
p
p
p
tttx
3
2
1
0
23
0001
0033
0363
1331
1
• If we expand the Bernstein formula
and isolate the x component of p(s,t) we will get (after a lot of arithmetic!)
Matrix Form
ijjiji
tBsBts pp )()(, 333
0
3
0
1
1
,
23
23
ttt
sss
tsx TTBezxBez
t
s
tBGBs
Matrix Form
1
1
,
23
23
ttt
sss
ts
TBezxBezx
Tz
Ty
Tx
t
s
BGBC
tCs
tCs
tCs
x
xxxx
xxxx
xxxx
xxxx
x
TBezBez
pppp
pppp
pppp
pppp
151173
141062
13951
12840
0001
0033
0363
1331
G
BB
• We would have similar formulas for y(s,t) and z(s,t). • Notes:– Cx stores the coefficients of the bicubic equation for x
– Gx stores the geometry (x components of the control points)
– BBez is the basis matrix (Bezier basis)
– s and t are the vectors formed from the exponents of s and t
– The matrix form can also take advantage of 4x4 matrix support which is built into modern computer graphics hardware
Matrix Form
• Your Programming Assignment is to create two Matlab functions:– bezier_surf(s,t,G)
• This will take two parameters s and t and a 4x4x3 matrix G and will return the vector p(s,t) = (x(s,t), y(s,t), z(s,t))
– bezier_surf_plot(G)• This will take one parameter – a 4x4x3 matrix G and will plot the
Bezier surface defined by these points.
Programming Assignment
• To assist you in this work, I have included code on the lecture web page that shows how to input and work with a 4x4x3 matrix and gives an example you should use for the matrix G. It also shows what kind of output you should expect.
• The surface plot should look like the figure on the next page:
Programming Assignment
Programming Assignment
Recommended