bézier and b-spline curves for cadtangent vector of the parametric curve • differentiate with...

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Bézier and B-spline curvesfor CAD

Kenjiro T. MiuraShizuoka University

Based on Prof. Ohtake’s ppt :http://www.den.rcast.u-tokyo.ac.jp/~yu-ohtake/GeomPro/

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Objectives• Learn basics of the curves frequently used in

CG and CAD.• Learn offset and curvature of the parametric

curve.

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Outlines• What is a parametric curve？

• Bézier curve

• Tangent and normal

• Curvature

• B-spline curve

• Space curve

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Representation of plane curve• Explicit :

– Limited d.o.f.

• Parametric :– Easy to use, high d.o.f.– Easy to extend to space curves

• Implicit : – Also called iso-lines

)(xfy =

( ))(),()( tytxt =c

0),( =yxf

21 xy −=

)sin,(cos)( ttt =c

01 22 =−− yx

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Parametric

• Explicit representation for each coordinate

( ))(),()( tytxt =c

x

y

t

t

y

x

t

)(tx

)(ty

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Rendering of parametric curve• Use polyline

( ) ( )( )sec0,sec0 yx

( ) ( )( ) sec10sec0,, ≤≤ ttytx 　

( ) ( )( )sec5,sec5 yx

( ) ( )( )sec10,sec10 yx Points for 0.5 sec interval

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Outlines• What is a parametric curve？

• Bézier curve

• Tangent and normal

• Curvature

• B-spline curve

• Space curve

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Frequently used for CAD• Bézier curve• B-spline curve

Bézier curve B-spline closed curve

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Practical example

• Outline font– Always smooth with any zooming

Outline font(vector image)

Dot font(raster imags)

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Problem dealt by Bézier(For car design)

• Generate a curve based on a sequence of points in a plane.

• Change the shape according to the move of the points

Change of the design

Move “control points”

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Bézier’s idea• Add smooth functions to represent the whole

shape！Given data

Multiply

Add!

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Bernstein basis function• Probability of “Repeated trial”

– Probability for t trial, n times, get i times.

iniin ttC −− )1(

)(tBni0 < t < 1 と書く

)}(),(),(),({ 33

32

31

30 tBtBtBtB

100% lottry three times

Three wins: 100%

A lot : win a time among three tries,

Win one time.t

)(3 tBi

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∑=

=n

ii

ni xtBtx

0)()(

)(tBni i

ni xtB )(

∑=

n

i 0

=

=

∑

∑

=

=

i

n

i

ni

i

n

i

ni

ytBty

xtBtx

0

0

)()(

)()(

Same for y

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Cubic Bézier curve• The most frequently used curve

– # of control points = 4– Convenient for spedifying the start and end

points and tangent vectors there.– Cubic polynomials → Next page

),(:pointstart 00 yx ),(:point end 33 yx

),( 11 yx

),( 22 yxDirection of the curve

at the start point

Direction of the curveat the end point

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Cubic Bézier curve(x coordinate)

3332

321

310

30

3

0

3

)()()()(

)()(

xtBxtBxtBxtB

xtBtxi

ii

+++=

=∑=

2

13113

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)1(3)1()(

ttttCtB

−=

−= − cubic polynomial of t• t = 0 , 0• t = 1 , double roots• t = 1/3, max

t

)(3 tBi

),( 00 yx ),( 33 yx

),( 11 yx

),( 22 yx

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Properties of Bézier curve• Remains the same shape with a

coordinate system translation.• Inside the Convex hull

Translate coordinatesystem

Reason : point on the curve is a positive weighted sum of the control points

Origin

Origin

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Outlines• What is a parametric curve？

• Bézier curve

• Tangent and normal

• Curvature

• B-spline curve

• Space curve

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Tangent and normal vectors• Tangent vector : direction of the curve• Normal vector : perpendicular to tangent

forwardside※ look at the carfrom the top

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Tangent vector of the parametric curve

• Differentiate with respect to the parameter

=

dttdy

dttdxt )(,)()(t

timeelapsedlocationcurrent locationNext

timeelapsedntvectordisplacemeectorvelocity v

−=

=

slow

fasterPoints of the

same time interval

If the parameter is time, velocity vector

Make e-time to infinitesimal= differentiate

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Tangent vector of Bézier curve• Differentiate the basis functions

– Control points are coefficients

==

==

∑∑

∑∑

==

==

i

n

i

ni

i

n

i

ni

i

n

i

ni

i

n

i

ni

ydt

tdBytBdtd

dttdy

xdt

tdBxtBdtd

dttdx

00

00

)()()(

)()()(

)(tt

)31)(1(3)1(3)( 231 tttt

dtd

dttdB

−−=−=

First derivative of B1

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Bézier spline curve• Connect Bézier curves of low degree

(usually cubic)– Locate control points to be parallel at joints.

jointjoint

※ At the end points of a Bezier curve, tangent to the polylines made of the control points.

Make control points locate on the same line

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Normal vector• Rotate tangent vector by 90 degrees.

– Usually use a unit normal vector whose length=122 )()()(,)()(

+

−=

dttdy

dttdx

dttdx

dttdytn

Normal vectorTangent vector

),(vectoriseanticlockw degrees 90 rotate

),(vector

xy

yx

−↓

),(1vector

lengthunit ),(vector

22yx

yx

yx

+

↓

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Offset curve• Trajectory made by translating points on the curve

in the direction of norma vector– When machine with a ball-end mill

the path of its center is on the offset curve

Work’s shape

tool

)()( trt nc + r : radius of the machining tooln : unit normal vector

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Outlines• What is a parametric curve？

• Bezier curve

• Tangent and normal

• Curvature

• B-spline curve

• Space curve

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Curvature and radius of curvature• ROC : radius of the best fit circle

– If positive, the same direction of the normal vector• Curvature : reciprocal of ROC (0 if straight)

ROC (positive)

Normal vectorROC (negative)

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Application of radius of curvature• Cannot machine with a tool with a radius

llaeger the ROC.– Useful for tool selection

Tool

Work’ s shape Cannot machine(Self-intersection of

the offset curve)

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Calculation of curvature

ROC×rotation of tangent vector [radian] = length of the approximate arc

( ))(),()( tytxt =cκ1

=Rθd

ds

Consider small change

dsdθκ =

• Approximate the curve with a circle

: change of the angle

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Calculation of curvature

reference line for angle

Tangent line θ ( ) ( )∫ +=

=

t

ttdtdtdytdtdxts

dttdxdttdyt

0

ˆˆ)ˆ(ˆ)ˆ()(:length arc

)()()(tan:line tangent of slope

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θ

1)()()()()(

−

==

dttds

dttd

tdstdt θθκ

( ) ( )( ) 23

22

2222

)()(

)()()()(

)(dttdydttdx

dttyddttxddttdydttdx

t+

=κ

Use the equations below

※ need second derivative

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Evolute curve• Trajectory of the best fit circle circle

– Translate in the normal direction by the ROC. – Pass through points where the offset line breaks.

)()( vectornormalunit ROC

ttR n×

)(:curve tc

)()()(:Evolue

tntRtc +

Points where the offset curve breaks

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Approximate ROC with a polyline• Radius of the circle which passes through

3 points– Remind the sine theorem

c

bS

aCalculate ROC

at point Owith the side lengths

and area of triangle OAB

O

A B

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Outlines• What is a parametric curve？

• Bezier curve

• Tangent and normal

• Curvature

• B-spline curve

• Space curve

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Problems on Bezier curve• Degree increases with the number of control

points.• One movement will change the whole shape.点の移動

Of degree 30!

Change around here, too.

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B-spline curve(different from Bézier spline)

• Example of quadratic B-spline curve– The curve is tangent at the middle point of the

control point polyline.

( )∑ −= iitNt pc 2)(

Format is the sum of (basis func.×pos. of c.p.)

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B-spline basis function

otherwise1||

01||

)(1 ≤

+−

=tt

tN

≤≤

+−+−

=otherwise

2/3|| if else2/1|| if

08/)912||4(

4/3||)( 2

2

2 tt

ttt

tN

( )∑ −= ixitNtx 2)(

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How to make B-spline basis func.• Recursively defined by the following

convolution

otherwise1/2|t| if

01

)(0 ≤

=tN

∫ −=+ dsstNsNtN nn )()()( 01

s1/2-1/2

Area here is

t

( ))(1 tsN −−Reverse right and left

( )tN 2

1

-t

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Outlines• What is a parametric curve？

• Bezier curve

• Tangent and normal

• Curvature

• B-spline curve

• Space curve

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Space curve in parametric form

• Only add z coordinate.

Herical curve : ),sin,cos()( thtrtrt =c

( ))(),(),()( tztytxt =c

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Binormal and torsion (space curve)

• Drive a car on a space curve– tangent t : ongoing direciton– normal n : curve direction(cannot defined for

straight line )– binormal b : top direction for the driver

• Curvature κ : degree of curve

• Torsion τ: Vibration of head

tb

n

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Frenet-Serret formula (space curve)

)()()()()()()()(

)()()(

sskssssss

sss

tbnnb

nt

−=′−=′

=′

ττ

κ

tb

n

tb

n

κ

τ

Match the origins of the two frames.