1. Introduction to Computer Graphics with WebGL
Ed Angel
Professor Emeritus of Computer Science
Founding Director, Arts, Research, Technology and Science Laboratory
University of New Mexico
Angel and Shreiner: Interactive Computer Graphics 7E © Addison-Wesley 2015

2. Bezier and Spline Curves and Surfaces
Ed Angel
Professor Emeritus of Computer Science
University of New Mexico
Angel and Shreiner: Interactive Computer Graphics 7E © Addison-Wesley 2015

3. Objectives Introduce the Bezier curves and surfaces
Derive the required matrices
Introduce the B-spline and compare it to the standard cubic Bezier
Angel and Shreiner: Interactive Computer Graphics 7E © Addison-Wesley 2015

4. Bezier’s Idea (http://fr.wikipedia.org/wiki/Courbe_de_Bézier)
In graphics and CAD, we do not usually have derivative data
Bezier suggested using the same 4 data points as with the cubic interpolating curve to approximate the derivatives in the Hermite form
Bezier worked for Renault as an engineer (from 1933 to 1975) ; He developed UNISURF in 1971.
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5. Approximating Derivatives
p2 located at u=2/3
p1 located at u=1/3
slope p’(1)
slope p’(0) p3 p0 u
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6. Equations Interpolating conditions are the same p(0) = p0 = c0
p(1) = p3 = c0+c1+c2+c3
Approximating derivative conditions
p’(0) = 3(p1- p0) = c0
p’(1) = 3(p3- p2) = c1+2c2+3c3
Solve four linear equations for c and we obtain
c=MBp
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7. Bezier Matrix p(u) = uT c = uT MBp = b(u)T p blending functions
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8. Blending Functions
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9. Bernstein Polynomials
The blending functions are a special case of the Bernstein polynomials
These polynomials give the blending polynomials for any degree Bezier form
All zeros of the functions are at 0 and 1
For any degree they all sum to 1
They are all between 0 and 1 inside (0,1)
Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009

10. Convex Hull Property
The properties of the Bernstein polynomials ensure that all Bezier curves lie in the convex hull of their control points
Hence, even though we do not interpolate all the data, we cannot be too far away p1 p2
convex hull
Bezier curve p3 p0
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11. Bezier Patches
Using same data array P=[pij] as with interpolating form
Patch lies in
convex hull
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12. Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009
Analysis
Although the Bezier form is much better than the interpolating form, we STILL have that the derivatives are not continuous at join points
Need to align controls points to achieve continuity
(see 3rd example in
Can we do better?
Go to higher order Bezier
More work
Derivative continuity still only approximate
Apply different conditions
Tricky without letting order increase
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B-Splines
Splines :
Basis splines: use the data at p=[pi-2 pi-1 pi pi+1]T to define curve only between pi-1 and pi
Allows us to apply more continuity conditions to each segment
For derivation, see pages of reference manual
Cost is 3 times as much work for curves
Add one new point each time rather than three
For surfaces, we do 9 times as much work
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14. Cubic B-spline p(u) = uTMSp = b(u)Tp
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Cubic B-spline
Example :
(select B-spline and create 5 points…)
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16. Blending Functions p(u) = uTMSp = b(u)Tp convex hull property
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Joining B-Splines
Move the knots to see the effects
Middle knots can change the shape of 4 segments
End knots affects 1, 2 or 3 segments
Superimposing the first three knots forces the curve to begin at that position
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18. B-Spline Patches defined over only 1/9 of region
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19. Splines and Basis
If we examine the cubic B-spline from the perspective of each control (data) point, each interior point contributes (through the blending functions) to four segments
We can rewrite p(u) in terms of the data points as
defining the basis functions {Bi(u)}
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Splines and Basis
Let us study the 4th point of the curve shown in
For the leftmost segment, b3(u) multiplies that point. For the next segment, b2(u) multiplies that point. For the 3rd, it is b2(u). For the 4th, it is b0(u).
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21. Basis Functions
In terms of the blending polynomials (see link on slide 16)
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To start a B-spline:
We can repeat the first node 3 times
We can use a phantom control point
(see Figure 5.4 in - move P1 or P5 to see the effect)
We do similarly, at the end of a B-spline.
(try repeating last node in
Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009

23. Generalizing Splines We can extend to splines of any degree
Data and conditions do not have to be given at equally spaced values (the knots)
Nonuniform and uniform splines
Can have repeated knots
Can force spline to interpolate points
Cox-deBoor recursion gives method of evaluation
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24. NURBS
Nonuniform Rational B-Spline curves and surfaces add a fourth variable w to x,y,z
Can interpret as weight to give more importance to some control data
Can also interpret as moving to homogeneous coordinate
Requires a perspective division
NURBS act correctly for perspective viewing
Quadrics are a special case of NURBS
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NURBS
First, we note that B-spline basis functions can be computed using recurrence relations :
(t0, t1, …ti, tn+k) are called the “knots” and can be selected to fit particular needs
Equal spacing of the knots generates “regular” B-spline functions
Uneven spacing generates NURBS
(in the “second to last” figure of the link above, try moving knots above the basis functions).
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NURBS
Figure 7 : uniform basis functions
Figure 10 : non-uniform basis functions (obtained by modifying uniform basis functions – using the knots…)
In NURBS, repeating 3 times the first and the last three knots brings the curve right on the first and last control points
(see slides 18 and 39 in
Try it using a java applet in
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