Splines IV – B- spline Curves

Splines IV – B-spline Curvesbased on:

Michael Gleicher: Curves, chapter 15 inFundamentals of Computer Graphics, 3rd ed.

(Shirley & Marschner)Slides by Marc van Kreveld

1

Interpolation vs. approximation

• Interpolation means passing through given points, approximation means getting “close” to given points

• Bézier curves and B-spline curves

p3

p2p1

p0 p3

p2p1

p0

p1 and p2 are interpolated

p1 and p2 are approximated

2

B-spline curves

• Approximates all control points• Polynomials of any degree d• C d-1 continuous everywhere

3

B-spline curves

• A linear combination ofthe control points

• The bi(t) are the basis functionsand show how to blend the points

• The basis functions are splines themselves B-splines

n

iii tbt

1

)()( pf

t

bi(t)the contribution of point pi to the curve depending on the parameter value t

4

B-splines and B-spline curves

t

bi(t)bi-1(t) bi+1(t)

pi-1 pi

pi+1

t’0.15

0.55

0.3

5



pi-1 pi

pi+1

t’

bi(t’)=0.55

bi-1(t’)=0.15

bi+1(t’)=0.3

0.15

0.55

0.3

t6



pi-1 pi

pi+1

t’0.15

0.55

0.3

t

the blended point determined by pi-1, pi , pi+1 and their basis functions

7

B-spline curves

8

• 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

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

9

Uniform linear B-splines• Basis functions are piecewise linear, for example:

b1,2(t)1

54321

022 it

ittbi, )(

i t i+1

i +1 t i +2

otherwise

knots10

Uniform linear B-splines, knots• A knot (for a uniform linear B-spline, or any type of

B-spline) is a parameter value where the definition of the function of some B-spline changes

• Slight abuse: the corresponding point on the curve

b1,2(t)1

54321

knots11

Uniform linear B-splines, knots

b1,2(t)

b2,2(t)

1

54321

1

54321

b3,2(t)1

5432112

five knots in total

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

022 it

ittbi, )(

i t i +1

i +1 t i +2

otherwise

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)

b1,2(t)

b2,2(t)

1

543211

54321

b3,2(t)1

54321 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

t

b1,3(t)

54321

1

15


t

b1,3(t)

54321

1

01 2

2

2

,3

21

21

21

)( uuu

u

bi

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

0.5

u is set to be in [0,1] for the t-range

16

Quadratic B-spline curve

p1

p6

p2

p5

p4

p3

p7

control points defining this piece of curve between the knots

p2p3p4p1p2p3

p3p4p5

p4p5p6

p5p6p7

knot

17


• At the 4 knots, the left and right derivatives are equal C1 continuous B-splines C1 continuous B-spline curve

t54321

1

0.5b2,3(t)

b’2,3(t) (derivative)– 1

– 0.5

18


• 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

t54321

1

0.5b1,3(t)

b2,3(t) b3,3(t) b4,3(t) b5,3(t)

19


• Suppose n control points• Then we have 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


• What is the starting point and what is the ending point of the uniform quadratic B-spline curve using control points p1, p2, …, pn ?

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

t

b1,4(t)

54321

1

0.5

6

2/3

22


0133

4631333

23

23

23

3

,4

61

61

61

61

)(

)(

)()(

uuu

uu

uuu

u

tbi

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

if t [i+3, i+4] u = t – (i+3)

t

b1,4(t)

54321

1

0.5

6

2/3

23


• 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, and everywhere else too of course) a cubic B-spline curve is C2 continuous

24


• 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

b1,4(t)

54321

1

0.5

6

2/3

7 8

b2,4(t) b3,4(t) b4,4(t)

t54321

10.5

6

2/3

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 parameter 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 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

)()()( 11,1

1,1

, tbtttttb

tttttb ki

iki

kiki

iki

iki

01)(1, tbi

if ti t < ti+1

otherwise

• Cox-de Boor recurrence for defining B-splines with parameter k and knot vector t = [t1, … , tn+k]:

29

The recurrence shows that parameter k B-splines are a weighted interpolation of parameter k – 1 B-splines, with weights linearly dependent on t

Repeating control pointsin 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]

t54321

1

0.5b1,3(t)

b2,3(t) b3,3(t) b4,3(t) b5,3(t)

76 8

p1

p2 p3 p4 p5

30


• Consider quadratic B-spline curves, in the figure:– Suppose p1 is repeated– 9 knots, useful interval [3, 7]

t54321

1

0.5b1,3(t)

b2,3(t) b3,3(t) b4,3(t) b5,3(t)

76 8

p1

p2 p3 p4 p5

9

b*1,3(t)p1

31


• 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

t54321

1

0.5

b2,3(t) b3,3(t) b4,3(t) b5,3(t)

76 8

p1

p2 p3 p4 p5

9

b1,3(t) b*1,3(t)+

32


• Consider quadratic B-spline curves, in the figure:– Suppose p3 is repeated– 9 knots, useful interval [3, 7]

t54321

1

0.5b1,3(t)

b2,3(t)b3,3(t) b4,3(t) b5,3(t)

76 8

p1

p2p3 p4 p5

9

b*3,3(t)p3

33


• 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

t54321

1

0.5b1,3(t)

b2,3(t)b3,3(t) b4,3(t) b5,3(t)

76 8

p1

p2 p4 p5

9

b*3,3(t)p3

+

34


• 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


• 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]

t54321

1

0.5b1,3(t)

b2,3(t)b3,3(t) b4,3(t) b5,3(t)

76 8

p1

p2 p4 p5

9

b*3,3(t)p3

+

36


p1

p6

p2

p5p3

p7

quadratic B-spline curve

p1p2p3

p3p4p5

p5p6p7

knot 3

37

p2p3p4

p4p5p6

p4


p1

p6

p2

p5

p4

p3

p7

quadratic B-spline curvewith repeated control point

p2p3p4p1p2p3

p3p4

p4p5p6p5p6p7

knot 3

repeated control point p4

38

p4p5


• The quadratic B-spline curve remains C1 although it is only G0 at the repeated control point

t54321

1

0.5b1,3(t)

b2,3(t)b3,3(t) b4,3(t) b5,3(t)

76 8

p1

p2 p4 p5

9

b*3,3(t)p3

+

39


p1

p6

p2

p5

p4

p3

p7

quadratic B-spline curvewith repeated control point

p2p3p4p1p2p3

p3p4

p4p5p6p5p6p7

knot 3

repeated control point p4

40

p4p5

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 (because the B-splines must still be continuous and sum up to 1)

41




t54321

1

0.5b1,3(t)

b2,3(t) b3,3(t) b4,3(t) b5,3(t)

76 842




t54321

1

0.5b1,3(t)

b2,3(t)b3,3(t) b4,3(t) b5,3(t)

76 8=43




t54321

1

0.5b1,3(t)

b2,3(t)b3,3(t)b4,3(t) b5,3(t)

76 844


p1

p6

p2

p5

p4

p3

p7


p2p3p4p1p2p3

p3p4p5

p4p5p6p5p6p7

knot 3

repeat knot 5

45


p1

p6

p2

p5

p4

p3

p7

quadratic B-spline curvewith repeated knot

p2p3p4p1p2p3

p4p5p6p5p6p7

knot 3

knot 5 = 6

46


• 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


p1

p6

p2

p5

p4

p3

p7


p2p3p4p1p2p3

p3p4p5

p4p5p6p5p6p7

knot 3

repeat knot 5 twice

48


p1

p6

p2

p5

p4

p3

p7


p2p3p4p1p2p3

p5p6p7

knot 3

knot 5 = 6 = 7

49


• In cubic B-spline curves– a double knot gives C1 / G1

continuity– a triple knot gives C0 / 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


• 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

51


• 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):

52

)()()( 11,1

1,1

, tbtttttb

tttttb ki

iki

kiki

iki

iki

01)(1, tbi

if ti t < ti+1

otherwise

These recurrences also allow easy evaluation: get the point on a B-spline curve for a given t

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)

53

Questions1. Let p1 p2 p3 be the three points of a uniform linear

B-spline curve, so there are five knots 1, 2, 3, 4 and 5. The useful parameter range is from knot 2 to knot 4. Draw the curve from knot 1 to knot 5, so also outside the useful parameter range, where p1 = (1,2), p2 = (3,1), and p3 = (2,1)

2. Approximately draw a quadratic B-spline curve for the points (1,2), (4,2), (4,0), (6,3), and (4,4) in its useful para-meter range. Make sure the curve is correct at its knots

3. Same but now for a cubic B-spline curve4. Draw a cubic B-spline and its derivative in one figure, as was

done for quadratic B-splines in the slides5. Suppose a quadratic B-spline curve is based on 8 points.

Point 4 is repeated and so is point 6. How many polygonal pieces does the B-spline curve have? 54

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Splines IV – B- spline Curves

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