University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell

Interpolating curves

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 2

Reading

OptionalBartels, Beatty, and Barsky. An Introduction toSplines for use in Computer Graphics andGeometric Modeling, 1987. (See coursereader.)

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell

Parametric curve review

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 4

Parametric curvesWe use parametric curves, Q(u)=(x(u),y(u)),where x(u) and y(u) are cubic polynomials:

Advantages:easy (and efficient) to compute“well behaved”infinitely differentiable

We also assume that u varies from 0 to 1

HGuFuEuuy

DCuBuAuux

+++=

+++=

23

23

)(

)(

))0(),0(( yx))1(),1(( yx

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 5

Various ways to set A,B,C,D

0) Directly – non-intuitive; not very useful.1) Set positions and derivatives of endpoints: “Hermite Curve”2) Use “control points” that indirectly influence the curve:

“Bezier curve”: - interpolates endpoints - does not interpolate middle control points

“B-spline” - does not interpolate ANY control points

DCuBuAuux +++=23)(

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 6

Splines = join cubic curves

ConsiderationsWhat kind of continuity at join points (“knots”)?

C0 = valueC1 = first derivativeC2 = second derivative

How do control points work?

B-spline

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 7

Spline summary

Joined Hermite curves:C1 continuityInterpolates control points

B-splines:C2 continuityDoes not interpolate control points

Can we get…C2 continuityInterpolates control points

That’s what we’ll talk about towardsthe end of this lecture.But first, some other useful tips.

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell

Useful tips for Bézier curves

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 9

Displaying Bézier curvesHow could we draw one of these things?

It would be nice if we had an adaptive algorithm, thatwould take into account flatness.

DisplayBezier( V0, V1, V2, V3 )begin if ( FlatEnough( V0, V1, V2, V3 ) ) Line( V0, V3 ); else something;end;

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 10

Subdivide and conquer

DisplayBezier( V0, V1, V2, V3 )

begin if ( FlatEnough( V0, V1, V2, V3 ) ) Line( V0, V3 ); else Subdivide(V[]) ⇒ L[], R[] DisplayBezier( L0, L1, L2, L3 ); DisplayBezier( R0, R1, R2, R3 );end;

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 11

Testing for flatness

Compare total length of control polygon tolength of line connecting endpoints:

!

V0"V

1+ V

1"V

2+ V

2"V

3

V0"V

3

<1+ #

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell

Tips for B-splines

B-spline:- C2 continuity- does not interpolate any ctrl points

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 13

Endpoints of B-splinesWe can see that B-splines don’t interpolate the controlpoints.It would be nice if we could at least control the endpointsof the splines explicitly.There’s a trick to make the spline begin and end at controlpoints by repeating them.In the example below, let’s force interpolation of the lastendpoint: (use endpoint 3 times)

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell

Tips for animator project

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 15

What if we want a closed curve, i.e., a loop?With Catmull-Rom and B-spline curves,this is easy:

Closing the loop

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell

C2 interpolating curves

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 17

Simple interpolating splines

Join several Hermite curves:- Make derivatives match- You still have ability to pick what that matched derivative is.

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 18

Cardinal splinesIf we set each derivative to be some positive scalar multiple k of thevector between the previous and next controls, we get a Cardinalspline.This leads to:

for any two consecutive interior points pi and pi+1 (we can deal withthe endpoints separately if need be)

Think of τ as a parameter that controls the tension of the spline

!

pi

u= "(p

i+1 #pi#1)

pi+1

u= "(p

i+2 #pi)

τ(p2-p0)

τ(p3-p1)

τ(p4-p2)

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 19

Cardinal splines

!

p(u) = u3

u2

u 1[ ]

2 "2 1 1

"3 3 "2 "1

0 0 1 0

1 0 0 0

#

$

% % % %

&

'

( ( ( (

pi

pi+1

pi

u

pi+1

u

#

$

% % % %

&

'

( ( ( (

= u3

u2

u 1[ ]

2 "2 1 1

"3 3 "2 "1

0 0 1 0

1 0 0 0

#

$

% % % %

&

'

( ( ( (

pi

pi+1

) pi+1 "pi"1( )

) pi+2 "pi( )

#

$

% % % %

&

'

( ( ( (

= u3

u2

u 1[ ]

2 "2 1 1

"3 3 "2 "1

0 0 1 0

1 0 0 0

#

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% % % %

&

'

( ( ( (

0 1 0 0

0 0 1 0

") 0 ) 0

0 ") 0 )

#

$

% % % %

&

'

( ( ( (

pi"1

pi

pi+1

pi+2

#

$

% % % %

&

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( ( ( (

= u3

u2

u 1[ ] )

"1 2 /) "1 "2 /) +1 1

2 "3/) +1 3/) " 2 "1

"1 0 1 0

0 1/) 0 0

#

$

% % % %

&

'

( ( ( (

pi"1

pi

pi+1

pi+2

#

$

% % % %

&

'

( ( ( (

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 20

Catmull-Rom splinesIf we set τ = 1/2, we get a Catmull-Rom spline.So:

for any two consecutive interior points pi and pi+1 (again dealing withendpoints separately as needed)

!

p(u) = u3

u2

u 1[ ] "

#1 2 /" #1 #2 /" +1 1

2 #3/" +1 3/" # 2 #1

#1 0 1 0

0 1/" 0 0

$

%

& & & &

'

(

) ) ) )

pi#1

pi

pi+1

pi+2

$

%

& & & &

'

(

) ) ) )

= u3

u2

u 1[ ]1

2

#1 3 #3 1

2 #5 4 #1

#1 0 1 0

0 2 0 0

$

%

& & & &

'

(

) ) ) )

pi#1

pi

pi+1

pi+2

$

%

& & & &

'

(

) ) ) )

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 21

Catmull-Rom blending functions

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 22

C2 interpolating splinesHow can we keep the C2 continuity we get with B-splinesbut get interpolation, too?Again start with connected cubic curves.Each cubic segment is an Hermite curve for which we getto set the position and derivative of the endpoints.That leaves us with a spline that’s C0 and C1 such as aCatmull-Rom or Cardinal spline.But interestingly, there are other ways to choose the valuesof the (shared) first derivatives at the join points.Is there a way to set those derivatives to get other usefulproperties?

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 23

Find second derivativesSo far, we have:

C0, C1 continuity Derivatives are still free, as ‘D0…D4’

Compute second derivatives at both sidesof every join point:

For p1:For p2:…

!

" " Q 0(1) = 6p0 # 6p1 + 2D0 + 4D1 " " Q 1(0) = #6p1 + 6p2 # 4D1 # 2D2

" " Q 1(1) = 6p1 # 6p2 + 2D1 + 4D2" " Q 2(0) = #6p2 + 6p3 # 4D2 # 2D3

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 24

Match the second derivatives

Now, symbolically set the second derivatives tobe equal.For p1

For p2

…!

6p0 " 6p1 + 2D0 + 4D1 = "6p1 + 6p2 " 4D1 " 2D2

3(p2 "p0) =D0 + 4D1 +D2

!

6p1 " 6p2 + 2D1 + 4D2 = "6p2 + 6p3 " 4D2 " 2D3

3(p3 "p1) =D1 + 4D2 +D3

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 25

Not quite done yet

How many equations is this? m-1How many unknowns are we solving for? m+1We have two additional degrees of freedom, which we cannail down by imposing more conditions on the curve.There are various ways to do this. We’ll use the variantcalled natural C2 interpolating splines, which requiresthe second derivative to be zero at the endpoints.This condition gives us the two additional equations weneed.

At the P0 endpoint, it is:At the Pm endpoint, we have:

''

0 (0) 0Q =''

1(1) 0mQ

!=

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 26

Solving for the derivatives

Let’s collect our m+1 equations into a single linearsystem:

It’s easier to solve than it looks.See the notes from Bartels, Beatty, and Barsky fordetails.

!

2 1

1 4 1

1 4 1

O

1 4 1

1 2

"

#

$ $ $ $ $ $ $

%

&

' ' ' ' ' ' '

D0

T

D1

T

D2

T

M

Dm(1T

Dm

T

"

#

$ $ $ $ $ $ $

%

&

' ' ' ' ' ' '

=

3 p1(p

0( )T

3 p2(p

0( )T

3 p3(p

1( )T

M

3 pm(p

m(2( )T

3 pm(p

m(1( )T

"

#

$ $ $ $ $ $ $ $

%

&

' ' ' ' ' ' ' '

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 27

C2 interpolating spline

Once we’ve solved for the real Dis, we can plugthem in to find our Bézier or Hermite curves anddraw the final spline:

Have we lost anything?=> Yes, local control.

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 28

Next time: Subdivision curves

Basic idea:Represent a curve as an iterative

algorithm, rather than as an explicit function.Reading:

• Stollnitz, DeRose, and Salesin. Wavelets for Computer Graphics: Theory and Applications, 1996, section 6.1-6.3, A.5. [Course reader pp. 248-259 and pp. 273-274]