1. Splines IV – B-spline Curves
based on:
Michael Gleicher: Curves, chapter 15 in
Fundamentals of Computer Graphics, 3rd ed.
(Shirley & Marschner)
Slides by Marc van Kreveld

2. Interpolation vs. approximation
Interpolation means passing through given points, approximation means getting “close” to given points
Bézier curves and B-spline curves p2 p2 p1 p1 p0 p3 p0 p3
p1 and p2 are interpolated
p1 and p2 are approximated

3. B-spline curves Approximates all control points
Polynomials of any degree d
C d-1 continuous everywhere

4. B-spline curves A linear combination of the control points
The bi(t) are the basis functions and show how to blend the points
The basis functions are splines themselves  B-splines
B-splines stands for basis splines
the contribution of point pi to the curve depending on the parameter value t
bi(t) t

5. B-splines and B-spline curves
pi-1 pi
pi+1 t’
bi-1(t)
bi(t)
bi+1(t)
0.15
0.55
0.3 t

6. B-splines and B-spline curves
pi-1 pi
bi-1(t’)=0.15
bi(t’)=0.55
pi+1
bi-1(t)
bi(t)
bi+1(t)
0.15
0.55
0.3
bi+1(t’)=0.3 t t’

7. B-splines and B-spline curves
pi-1 pi
pi+1
bi-1(t)
bi(t)
bi+1(t)
0.15
0.55
0.3
the blended point determined by pi-1, pi , pi+1 and their basis functions t t’

8. B-spline curves A B-spline curve of n points and parameter value k
is C k-2 continuous
is made of polynomials of degree k – 1
has local control: any location on the curve is determined by only k control points
is bounded by the convex hull of the control points
has the variation diminishing property (any line intersects the B-spline curve at most as often as that line intersects the control polygon)

9. Types of B-splines Uniform linear B-splines
Uniform quadratic B-splines
Uniform cubic B-splines
Non-uniform B-splines
NURBS (non-uniform rational B-splines)
 To define the B-splines is to define the B-spline curve

10. Uniform linear B-splines
Basis functions are piecewise linear, for example:
i  t  i+1
i +1  t  i +2
otherwise 1
b1,2(t)
The 2 in the subscript of the b-function denotes the parameter = degree +1 1 2 3 4 5
knots

11. Uniform linear B-splines1
b1,2(t) 1 2 3 4 5 1
b2,2(t)
Three B-splines, each with 3 knots, 5 knots in total 1 2 3 4 5 1
b3,2(t) 1 2 3 4 5

12. Uniform linear B-splines
Basis functions are piecewise linear, for example:
The B-spline curve can be used for parameter values t  [2, n+1] (the t where the bi,2(t) sum up to 1), given points p1, p2, …, pn
The B-spline curve is just the polygonal line through the control points; only two B-splines are non-zero for any t
i  t  i +1
i +1  t  i +2
otherwise
The 2 in the subscript of the b-function denotes the parameter = degree +1

13. Uniform linear B-splines
Each B-spline has 3 knots and they are uniformly spaced  They are shifted copies of each other
In total there are n+2 knots (at 1, 2, …, n+2) 1
b1,2(t) 1 2 3 4 5 1
b2,2(t) 1 2 3 4 5 1
b3,2(t) 1 2 3 4 5

14. Uniform quadratic B-splines
Uniform quadratic B-splines bi,3(t) have knots at i, i+1, i+2, and i+3
They are shifted copies of each other 1
b1,3(t) 1 2 3 4 5 t

15. Uniform quadratic B-splines
if t [i, i+1] u = t – i
if t [i+1, i+2] u = t – (i+1)
if t [i+2, i+3] u = t – (i+2)
otherwise
u is set to be in [0,1] for the t-range 1
b1,3(t)
0.5 1 2 3 4 5 t

16. Quadratic B-spline curve
knot p6 p3 p5 p2
p5p6p7
p2p3p4
p1p2p3
p4p5p6
p3p4p5 p1 p7 p4
control points defining this piece of curve between the knots

17. Uniform quadratic B-splines
At the 4 knots, the left and right derivatives are equal  C1 continuous B-splines  C1 continuous B-spline curve 1
b2,3(t)
0.5 1 2 3 4 5 t
– 0.5
b’2,3(t) (derivative)
– 1

18. Uniform quadratic B-splines
At the 4 knots, the left and right derivatives are equal  C1 continuous B-splines  C1 continuous B-spline curve
Starting at t = 3, the B-splines sum up to exactly 1
Why is it important that the B-splines sum up to 1?
b2,3(t)
b3,3(t)
b4,3(t)
b5,3(t) 1
b1,3(t)
0.5 1 2 3 4 5 t

19. Uniform quadratic B-splines
Suppose n control points
Then n B-splines and n+3 knots
The B-spline curve has n – 2 quadratic pieces, starting at the 3rd knot and ending at the n+1st knot

20. Uniform quadratic B-splines
What is the starting point and what is the ending point of the uniform quadratic B-spline curve using control points p1, p2, …, pn ?
Go back two slides and look at the B-splines. The curve starts at the knot at t=3. Here the first and second B-spline both have value 0.5 and all other B-splines are 0, so the midpoint of p_1 and p_2 is the starting point of the B-spline curve.

21. Uniform cubic B-splines
Uniform cubic B-splines bi,4(t) have knots at i, i+1, i+2, i+3 and i+4
They are shifted copies of each other 1
b1,4(t)
2/3
0.5 1 2 3 4 5 6 t

22. Uniform cubic B-splines
if t [i, i+1] u = t – i
if t [i+1, i+2] u = t – (i+1)
if t [i+2, i+3] u = t – (i+2)
if t [i+3, i+4] u = t – (i+3)
otherwise 1
b1,4(t)
2/3
0.5 1 2 3 4 5 6 t

23. Uniform cubic B-splines
Every location on the B-spline curve is determined by four control points and their B-splines
Starting at the 4th knot, t = 4, the B-spline curve can be used because the B-splines sum up to 1
A cubic B-spline is C2 continuous (the 1st and 2nd derivatives from the left and right are the same at the knots)  a cubic B-spline curve is C2 continuous

24. Uniform cubic B-splines
At what height are the second and fourth knots of a cubic B-spline?
What weight of what control points determines the first point of a B-spline curve?

25. Closed uniform B-spline curves
Simply repeat the first k – 1 points as the last points
quadratic: p1, p2, …, pn  p1, p2, …, pn, p1, p2
cubic: p1, p2, p3, …, pn  p1, p2, …, pn, p1, p2, p3

26. Non-uniform B-splines
Uniform B-splines have knots at 1, 2, 3, …, but generally knots can have any value  non-uniform B-splines
The knot vector t = [t1, … , tn+k] is a sequence of non-decreasing values specifying where the knots for the parameter range t occur

27. Non-uniform B-splines
We can repeat control points and combine the consecutive B-splines into one  allows more local control
We can repeat knots and adapt the B-splines affected  allows more local control
When we have a sequence of control points and a useful parameter range k, ..., n+1, we may want to insert an extra point (and knot) in the middle without changing the parameter range near the ends

28. Cox-de Boor recurrence
Cox-de Boor recurrence for defining B-splines with parameter k and knot vector t = [t1, … , tn+k]:
if ti  t < ti+1
otherwise
Note the definition of b at the end of the recurrence: this is a piecewise constant B-spline that is not even C0 continuous. The B-spline curve is not a proper curve, because it jumps from control point to control point at the knots. It is just a sequence of points with a parameterization at which point we are. The recurrence allows evaluation at any t value for non-uniform B-splines.
The recurrence shows that parameter k B-splines are a weighted interpolation of parameter k – 1 B-splines, with weights linearly dependent on t

29. Repeating control points in B-spline curves
Consider quadratic B-spline curves, in the figure:
Five B-splines of control points
5+3 = 8 knots (t = 1, 2, ..., 8), useful in interval [3, 6] p2 p3 p4 p5 p1
b2,3(t)
b3,3(t)
b4,3(t)
b5,3(t) 1
b1,3(t)
0.5 1 2 3 4 5 6 7 8 t

30. Repeating control points in B-spline curves
Consider quadratic B-spline curves, in the figure:
Suppose p1 is repeated
9 knots, useful interval [3, 7] p1 p2 p3 p4 p5 p1
b’1,3(t)
b2,3(t)
b3,3(t)
b4,3(t)
b5,3(t) 1
Since b_{1,3} and b_{2,3} are coefficients of the same point p1, we can add up their functions
b1,3(t)
0.5 1 2 3 4 5 6 7 8 9 t

31. Repeating control points in B-spline curves
Consider quadratic B-spline curves, in the figure:
Suppose p1 is repeated
9 knots, useful interval [3, 7]
The B-spline curve starts at p1 at knot 3 p2 p3 p4 p5
b1,3(t)
b’1,3(t) +
b2,3(t)
b3,3(t)
b4,3(t)
b5,3(t) 1
Since b_{1,3} and b_{2,3} are coefficients of the same point p1, we can add up their functions p1
0.5 1 2 3 4 5 6 7 8 9 t

32. Repeating control points in B-spline curves
Consider quadratic B-spline curves, in the figure:
Suppose p3 is repeated
9 knots, useful interval [3, 7] p3 p3 p2 p4 p5 p1
b3,3(t)
b’3,3(t)
b2,3(t)
b4,3(t)
b5,3(t) 1
b1,3(t)
0.5 1 2 3 4 5 6 7 8 9 t

33. Repeating control points in B-spline curves
Consider quadratic B-spline curves, in the figure:
Suppose p3 is repeated
9 knots, useful interval [3, 7]
The B-spline curve passes through point p3 p3 p2 p4 p5
b3,3(t) +
b’3,3(t) p1
b2,3(t)
b4,3(t)
b5,3(t) 1
b1,3(t)
0.5 1 2 3 4 5 6 7 8 9 t

34. Repeating control points in B-spline curves
For a B-spline curve with parameter k (degree k – 1), it will pass through the first and last control points if each occurs k – 1 times
Similarly, we can make it pass through an intermediate control point by having k – 1 copies of that control point
The level of geometric continuity, G, at an intermediate repeated control point decreases

35. Repeating control points in B-spline curves
On the parameter interval [4,5], only p2 and p3 have a non-zero B-spline  the curve is a line segment! Also on parameter interval [5,6] p3 p2 p4 p5
b3,3(t) +
b’3,3(t) p1
b2,3(t)
b4,3(t)
b5,3(t) 1
b1,3(t)
0.5 1 2 3 4 5 6 7 8 9 t

36. Repeating control points in B-spline curves
p5p6p7
p4p5p6
p1p2p3
p2p3p4
p3p4p5
knot 3 p1 p7 p4
quadratic B-spline curve

37. Repeating control points in B-spline curves
p5p6p7
p4p5p6
p1p2p3
p2p3p4
knot 3 p1
p3p4
p4p5 p7 p4
repeated control point p4
quadratic B-spline curve
with repeated control point

38. Repeating control points in B-spline curves
The quadratic B-spline curve remains C1 although it is only G0 at the repeated control point p3 p2 p4 p5
b3,3(t) +
b’3,3(t) p1
b2,3(t)
b4,3(t)
b5,3(t) 1
All B-splines remain C1 so the B-spline curve remains C1 as well. However, the speed along the B-spline curve reduces to 0 in the double control point
b1,3(t)
0.5 1 2 3 4 5 6 7 8 9 t

39. Repeating control points in B-spline curves
p5p6p7
p4p5p6
p1p2p3
p2p3p4
knot 3 p1
p3p4
p4p5 p7 p4
repeated control point p4
quadratic B-spline curve
with repeated control point

40. Repeating knots in B-spline curves
Repeating a knot means that one sequence of control points no longer occurs
For example, for quadratic B-spline curves, the part between knots 4 and 5 (was based on points p2 p3 p4 )
The curve part before (based on points p1 p2 p3 ) directly connects to the curve part after (based on points p3 p4 p5 )
The B-splines get different shapes to accommodate the missing part

41. Repeating knots in B-spline curves
Repeating a knot means that one sequence of control points no longer occurs
For example, for quadratic B-spline curves, the part between knots 4 and 5 (was based on points p2 p3 p4 )
b2,3(t)
b3,3(t)
b4,3(t)
b5,3(t) 1
b1,3(t)
0.5 1 2 3 4 5 6 7 8 t

42. Repeating knots in B-spline curves
Repeating a knot means that one sequence of control points no longer occurs
For example, for quadratic B-spline curves, the part between knots 4 and 5 (was based on points p2 p3 p4 )
b3,3(t)
b2,3(t)
b4,3(t)
b5,3(t) 1
b1,3(t)
0.5 1 2 3 4 = 5 6 7 8 t

43. Repeating knots in B-spline curves
Repeating a knot means that one sequence of control points no longer occurs
For example, for quadratic B-spline curves, the part between knots 4 and 5 (was based on points p2 p3 p4 )
b3,3(t)
Since p1, p2, p3 have only one point in common with p3, p4, p5, the degree of continuity goes down. In particular, the B-spline b_{3,3}(t) is C0 where it passes through point p3 (at the double knot). Hence the B-spline curve is only C0 as well.
b2,3(t) 1
b4,3(t)
b1,3(t)
b5,3(t)
0.5 1 2 3 6 7 8 4 5 t

44. Repeating knots in B-spline curves
p5p6p7
p4p5p6
p1p2p3
p2p3p4
p3p4p5
knot 3
repeat knot 5 p1 p7 p4
quadratic B-spline curve

45. Repeating knots in B-spline curves
p5p6p7
p4p5p6
p1p2p3
p2p3p4
knot 3 p1 p7 p4
knot 5 = 6
quadratic B-spline curve
with repeated knot

46. Repeating knots in B-spline curves
Repeating a control point creates line segments on either side, but repeating a knot does not have this effect on quadratic B-spline curves
Repeating a knot twice (three same knots in a sequence) creates a discontinuity in a quadratic B-spline curve
Two consecutive curve parts have no point in common
The curve part before the two disappearing parts (e.g., based on points p1 p2 p3 ) is disconnected from the curve part after those parts (based on points p4 p5 p6 )

47. Repeating knots in B-spline curves
p5p6p7
p4p5p6
p1p2p3
p2p3p4
p3p4p5
knot 3
repeat knot 5 twice p1
Note: there is no longer a curve part defined by any same point: p2,p3,p4 and p5,p6,p7 have no point in common p7 p4
quadratic B-spline curve

48. Repeating knots in B-spline curves
p5p6p7
p1p2p3
p2p3p4
knot 3 p1
Note: there is no longer a curve part defined by any same point: p2,p3,p4 and p5,p6,p7 have no point in common p7 p4
knot 5 = 6 = 7
quadratic B-spline curve

49. Repeating knots in B-spline curves
In cubic B-spline curves
a double knot gives C1 and G1 continuity
a triple knot gives C0 and G0 continuity
a quadruple knot disconnects the curve
a triple knot at the start lets p1 be the start of the curve
a triple knot at the end lets pn be the end of the curve

50. Repeating knots in B-spline curves
For cubic B-spline curves:
a double knot gives C1 continuity
a triple knot gives C0 continuity
C2 continuity exists because the speed along the curve essentially becomes 0 at a triple control point, so the sharp turn happens at 0 speed which can have any C-continuity in the parameter

51. Repeating knots in B-spline curves
Cubic B-splines for a B-spline curve passing through its endpoints, knot vector [1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 8, 8]
The first three and last three B-splines are different from the standard cubic B-spline shape

52. Repeating knots in B-spline curves
The expressions for B-splines can be determined for any parameter k and knot vector t = [t1, … , tn+k] using the Cox-de Boor recurrence (also when knots are repeated):
if ti  t < ti+1
otherwise

53. Summary
B-spline curves are defined by B-splines and provide great flexibility in curve design
They exist of any order (degree) and continuity
Repeating control points or knots allows interpolation instead of approximation
B-spline surfaces are a direct generalization
Even more general are NURBS: Non-Uniform Rational B-Splines
defined using ratios of two polynomials
allow conic sections (circles, ellipses, hyperbolas)