Bezier Curves and Splines

David Eno

MAT 499

Fall ‘06

Introduction

• In Engineering, one often wants a smooth curve through a set of known points.

• In Physics, a smooth curve is required to represent the shape of a deflected beam.

• Computer Aided Design and Manufacturing programs like lines and circular arcs.– Lots of things cannot be conveniently

described by lines and circular arcs.

Bézier Curves

• Bézier Curves were first developed in 1959 by Paul de Casteljau.

• They were popularized in 1962 by French engineer Pierre Bézier, who used them to design automobile bodies.

Quadratic Bézier Curves

• Given three points P0, P1, and P2, a quadratic Bézier curve is the path traced by the parabolic function:

10 where

)1(2)1()( 221

20

t

tPttPtPtx

Quadratic Bézier Curve

• Given points P0, P1, P2.

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5 3 3.5

P0

P1

P2

Quadratic Bézier Curve

• Given points P0, P1, P2.

• Construct a line from (1-t)P0 + tP1 to (1-t)P1 + tP2

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5 3 3.5

P0

P1

P2

(1-t)P0 + tP1

(1-t)P1 + tP2

Quadratic Bézier Curve

• Given points P0, P1, P2.

• Construct a line from (1-t)P0 + tP1 to (1-t)P1 + tP2

• The point, x, is on the curve.

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5 3 3.5

P0

P1

P2

(1-t)P0 + tP1

(1-t)P1 + tP2

(1-t)2P0 + 2t(1-t)P1 + t2P2

Quadratic Bézier Curve

• Given points P0, P1, P2.

• Construct a line from (1-t)P0 + tP1 to (1-t)P1 + tP2

• The point, x, is on the curve.

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5 3 3.5

P0

P1

P2

(1-t)P0 + tP1

(1-t)P1 + tP2

(1-t)2P0 + 2t(1-t)P1 + t2P2

Bézier Curve Advantages

• 3 points uniquely determine a parabola.

• It’s easy to calculate points.

• The numerical algorithm is stable. (i.e. given reasonable input, the algorithm won’t blow up.)

Cubic Bézier Curve• Extending this method to use four points, we can

construct a cubic curve.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3 3.5

P0

P1

P2

P3

(1-t)3P0 + 3t(1-t)2P1 + 3t2(1-t)P2 + t3P3

In General…

• To construct an nth degree Bezier curve, you need n+1 control points.

• The formula for a point on the curve is:

iini

n

i

Ptti

ntx

)1()(

0

Recursion

• Pseudo code for recursive technique:makeBezier(Control Points)

If points are collinear enoughOutput last point

ElseSubdivide points

makeBezier(Left Control Points)

makeBezier(Right Control Points)

End If

Bezier Splines

• We typically want a smooth curve that passes though a set of points.

• Problem: The first and last control points are the only ones guaranteed to be on a Bezier Curve.

• A Solution: Use Bezier Splines, which are composite (i.e. piecewise) Bezier Curves.– But then we need to compute control points.

Cubic Bezier Splines

• We could make splines from Bezier curves of any degree.

• We choose cubic (degree = 3) curves for the following reasons:– In modeling the bending of beams because of the "bending

moment" (which is related to equilibrium) is proportional to the second derivative (i.e. C2) of the displacement function and to be physically valid there must be bending moment continuity from one element to the next.

– Cubic polynomials allow for inflection points.– The computations aren’t horrible.

Composite Bezier Cubic Spline

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5

Node Pts Ctrl Pts Spline

P0

P1

P2

P3

P4

Composite Bezier Cubic Spline

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5

Node Pts Ctrl Pts Spline

P0=b0,0

P1=b0,3=b1,0

P2

P3

P4b0,1

b0,2

Computing Control Points

Condition: C0 continuity– Each curve has to start where the previous one ends.

In general,

Or

10 0,13, niPP ii

)0()1( 1 ii xx

Computing Control Points

Condition: C1 continuity– The slopes have to match at the common endpoints.– So set first derivatives equal.

...)1(3)(

)1(3)1(3)1()(2

31

22,

21,

3

tPtx

tPttbttbtPtx

ii

iiiii

iiiii Pbbxx 2)0()1( 1,12,1

Computing Control Points

Condition: C2 continuity– Provides better smoothing.– Gives us a unique solution.

022)0()1( 2,11,12,1,1 iiiiii bbbbxx

Arbitrarily let

02 and 02

0)1()0(

2,11,12,1,

10

nnii

n

bbbb

xx

Computing Control Points

0

0

2

0

2

0

2

0

21000000

12210000

01100000

00122100

00011000

00001221

00000110

00000012

3

2

1

1,3

1,3

1,2

1,2

1,1

1,1

2,0

1,0

P

P

P

b

b

b

b

b

b

b

b

NURBS

• Add a Weight coordinate.

Questions and Comments

• How do you prevent cusps?

• Can de Casteljau’s algorithm be modified to construct NURBS?

Further Direction(s)

• B-splines provide local control over the spline, we think.