1. Lecture Notes #11 Curves and Surfaces II
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Lecture Notes #11
Curves and Surfaces II

2. Uniform Non-rational B-Splines.
For each i  4 , there is a knot between Qi-1 and Qi at t = ti.
Initial points at t3 and tm+1 are also knots. The following illustrates an example with control points set P0 … P9:
Knot.
Control point.
m=9 (10 control points)
m-1 knots
m-2 knot intervals.
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Lecture Notes #11

3. Uniform Non-rational B-Splines.
First segment Q3 is defined by point P0 through P3 over the range t3 = 0 to t4 = 1. So m at least 3 for cubic spline.
Knot.
Control point. P1 P2 P3 P0 Q3
m=9 (10 control points)
m-1 knots
m-2 knot intervals.
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4. Uniform Non-rational B-Splines.
Second segment Q4 is defined by point P1 through P4 over the range t4 = 1 to t5 = 2.
Knot.
Control point. Q4 P1 P3 P4 P2
m=9 (10 control points)
m-1 knots
m-2 knot intervals.
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5. Uniform Non-Rational B-Splines
The unweighted spline set will look as follows, 10 control points, 10 splines, but only 8 knots and 7 knot intervals.
You can see why t3 to t4 is the first interval with a curve since it is the first with all four B-Spline functions.
t9 to t10 is the last interval, t3 to tm+1 are the knots. t 8 6 m
m+1 3 4
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6. Cubic B-Splines
We have talked about a bi-infinite set of splines. What does this look like?
The unweighted B-Splines are shown for clarity.
How does a weighted set affect the shape of the curve? t 4 8 12
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7. Generating a curve
X(t) t
Opposite we see an example of a shape to be generated.
Here we see the curve again with the weighted B-Splines which generated the required shape.
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8. B-Spline example
A closed curve is rather like a bi-infinite set – a periodic set of N knots in which: for all i, ti+N = ti and ti = i for i = 0..N-1
Consider a closed curve with 5 control points with 5 knots. N=5, ie. x–1 x4, etc.
Control points are: x0 = (2,0); x1 = (1,1); x2 = (-1,1); x3 = (-1,-1) and x4 = (1,-1).
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9. B-Spline example
Draw the interval: t = [4..5] , the others are handled the same way. x1 x4 x0 x1 x2 x3 x4
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10. How smooth is a B-Spine? Smoothness increases with order k in Bi,k
Quadratic, k = 3, gives up to C1 order continuity.
Cubic, k = 4 gives up to C2 order continuity.
However, we can lower continuity order too with Multiple Knots, ie. ti = ti+1= ti+2 = … Knots are coincident and so now we have non-uniform knot intervals.
A knot with multiplicity m is continuous to the (k-1-m)th derivative.
A knot with multiplicity k has no continuity at all, ie. the curve is broken at that knot.
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11. Non-Uniform Non-Rational B-Splines
Note: as m increases by 1, an extra B-Spline function is attached to that knot: qiBi,k
B-Splines have special forms at multiple knots.
Obtained from a recursive formula defining how B-Splines are built, by setting m successive knots to be equal: tk+1 = tk+2 = … = tk+m
No point in having m>k
Thus for endpoints to be on curve, they must have a multiplicity of k, multiplicity of 4 in cubic case.
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12. B-Splines at multiple knots
The multiple-knot B-Splines are as follows:
B0,4(t)
B1,4(t)
B2,4(t)
B3,4(t) t
i=0,1,2,
B0,4 is totally discontinuous at t = t0 , B1,4 is position continuous,
B2,4 is gradient continuous, B3,4 is curvature continuous.
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13. B-Spline continuity example.
First knot shown with 4 control points, and their convex hull.
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14. B-Spline continuity example.
First two curve segments shown with their respective convex hulls.
Centre Knot must lie in the intersection of the 2 convex hulls.
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15. Repeated control point.
P1=P2 P0 P3 P4
First two curve segments shown with their respective convex hulls.
Knot is forced to lie on the line that joins the 2 convex hulls.
Curve is only C1 continuous
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16. Triple control point.
First two curve segments shown with their respective convex hulls.
Both convex hulls collapse to straight lines – all the curve must lie on these lines.
Curve is only C0 continuous
(Curiously it is actually C2 continuous because tangent vector magnitude falls to zero at join. )
P1=P2=P3 P0 P4
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17. Non-uniform non-rational B-splines.
Parametric interval between knots does not have to be equal.
Blending functions no longer the same for each interval.
Multiple knots may have to be spaced apart.
Advantages
Continuity at selected control points can be reduced to C1 or lower – allows us to interpolate a control point without side-effects.
Can interpolate start and end points.
Easy to add extra knots and control points.
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18. Summary of B-Splines.
Functions that can interpolate a series of control points with C2 continuity and local control.
Don’t pass through their control points, although can be forced.
note that if an order k B-Spline has k points locally colinear then a straight-line section will result midway in the set, and will touch (tangentially) the convex hull for (k – 1) colinear control points.
Uniform
Knots are equally spaced in t.
Non-Uniform
Knots are unequally spaced
Allows addition of extra control points anywhere in the set.
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19. Summary cont. For interactive curve modelling
B-Splines are very good.
To interpolate a number of positions
Catmull-Rom splines are best.
To interpolate with tangent control
Hemite or Bézier forms are useful and most often used.
To draw a spline.
Brute force method : Evaluate matrix to get parametric expressions for coordinates. Then for small increments of t, join with line segments.
Better : use method of forward differences to express spline in t form.
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20. Surfaces – a simple extension
Easy to generalise from cubic curves to bicubic surfaces.
Surfaces defined by parametric equations of two variables, s and t.
ie. a surface is approximated by a series of crossing parametric cubic curves
Result is a polygon mesh and decreasing step size in s and t will give a mesh of small near-planar quadrilateral patches and more accuracy.
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21. Example Bézier surface
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22. Control of surface shape
Control is now a 2D array of control points.
The two parameter surface function, forming the tensor product with the blending functions is:
Use appropriate blending functions for Bézier and B-Spline surface functions.
Convex Hull property is preserved since bicubic is still a weighted sum (1).
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23. Bézier example Matrix formulation is as follows:
Substitute suitable values for s and t (20 in the above ex.)
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24. B-Spline surfaces
Break surface into 4-sided patches choosing suitable values for s and t.
Points on any external edges must be multiple knots of multiplicity k.
Lot more work than Bézier.
There are other types of spline systems and NURBS modelling packages are available to make the work much easier.
Use polygon packages for display, hidden-surface removal and rendering. (Bézier too)
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25. Continuity of Bicubic patches.
Hermite and Bézier patches
C0 continuity by sharing 4 control points between patches.
C1 continuity when both sets of control points either side of the edge are collinear with the edge.
B-Spline patch.
C2 continuity between patches.
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26. Displaying Bicubic patches.
Can calculate surface normals to bicubic surfaces by vector cross product of the 2 tangent vectors.
Normal is expensive to compute
Formulation of normal is a biquintic (two-variable,fifth-degree) polynomial.
Display.
Can use brute-force method – very expensive !
Forward differencing method very attractive.
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27. Reading for this lecture
Foley at al., Chapter 11, sections , , , , 11.3 and 11.5.
Introductory text, Chapter 9, sections 9.2.4, 9.2.5, 9.2.7, and 9.3.
Other texts: look for B-Splines UNRBS, NUNRBS, something on forward difference method, comparison of systems, and Bézier and B-Spline surface modelling.
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