chap 3 interpolating values

Chap 3Interpolating Values

Animation(U), Chap 3, Interpolating Values 1

2

Outline Interpolating/approximating curves Controlling the motion of a point along a curve Interpolation of orientation Working with paths

Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang

3

Interpolation between key frames Parameters to be interpolated

Position of an object Normal Joint angle between two joints Transparency attribute of an object


4

Interpolation between key frames Interpolation between frames is not trivial

Appropriate interpolating function Parameterization of the function Maintain the desired control of the interpolated values

over time Example

(-5,0,0) at frame 22, (5,0,0) art frame 67 Stop at frame 22 and accelerate to reach a max speed by

frame 34 Start to decelerate at frame 50 and come to a stop at

frame 67


5

Interpolation between key frames


desired result undesired result

Need a smooth interpolation with user control

Interpolation between key framesSolution

Generate a space curve Distribute points evenly along curve Speed control: vary points temporally

Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang6

A

B

C

D

Time = 0

Time = 10

Time = 35

Time = 60

7

Interpolation functions Interpolation vs. approximation

Interpolation Hermite, Catmull-Rom

approximation Bezier, B-spline


8

Interpolation functions Complexity => computational efficiency

Polynomials Lower than cubic

No inflection point, may not fit smoothly to some data points Higher than cubic

Doesn’t provide any significant advantages, costly to evaluate Piecewise cubic

Provides sufficient smoothness Allows enough flexibility to satisfy constraints such as end-

point position and tangents


9

Interpolation functions Continuity within a curve

Zero-order First order (tangential)

Suffices for most animation applications Second order

Important when dealing with time-distance curve Continuity between curve segments

Hermite, Catmull-Rom, cubic Bezier provide first order continuity between segments


10

Interpolation functions


CS, NCTU, J. H. Chuang11

Continuity

noneposition(0th order)

tangent(1st order)

curvature(2nd order)

At junction of two circular arcs

Animation(U), Chap 3, Interpolating Values

12

Interpolation functions Global vs. local control



Types of Curve Representation Explicit y = f(x)

Good for generate points For each input, there is a unique output

Implicit f(x,y) = 0 Good for testing if a point is on a curve Bad for generating a sequence of points

Parametric x = f(u), y = g(u) Good for generating a sequence of points Can be used for multi-valued function of x



Example: Representing Unit Circle Cannot be represented explicitly as a function

of x

Implicit form:f(x,y)=x2+y2=1

Parametric form:x=cos(u), y=sin(u), 0<u<2π u

f<0

f>0f=0



More on 3-D Parametric Curves

Parametric form: P(u) = (Px(u), Py(u), Pz(u)) x = Px(u), y = Py(u), z = Pz(u)

Space-curve P = P(u) 0.0 <=u<=1.0

u=1/3

u=2/3

u=0.0

u=1.0


Polynomial Interpolation

An n-th degree polynomial fits a curve to n+1 points Example: fit a second degree curve to three points

y(x)= a x2 + b x + c points to be interpolated

(x1, y1), (x2, y2), (x3, y3) solve for coefficients (a, b, c):

3 linear equations, 3 unknowns called Lagrange Interpolation


Polynomial Interpolation (cont.)

Result is a curve that is too wiggly, change to any control point affects entire curve (nonlocal) – this method is poor

We usually want the curve to be as smooth as possible minimize the wiggles high-degree polynomials are bad

Higher degree, higher the wiggles



Composite Segments Divide a curve into multiple segments Represent each in a parametric form Maintain continuity between segments

position Tangent (C1-continuity vs. G1-continuity) curvature

P1(u) P2(u) P3(u) Pn(u)



Splines: Piecewise Polynomials A spline is a piecewise polynomial - many low

degree polynomials are used to interpolate (pass through) the control points

Cubic polynomials are the most common lowest order polynomials that interpolate two

points and allow the gradient at each point to be defined - C1 continuity is possible

Higher or lower degrees are possible, of course



A Linear Piecewise PolynomialThe simple form for interpolating two points.

p1

p2

Each segment is of the form: (this is a vector equation)

u0 1

1

Two basis (blending) functions

u

21 )1()( puupup



Hermite Curves—cubic polynomial for two points

Hermite interpolation requires Endpoint positions derivatives at endpoints

To create a composite curve, use the end of one as the beginning of the other and share the tangent vector

Hermite Interpolation

)0('P

control points/knots

dcubuauuP 23)(

)0(P

)1('P)1(P

))(),(),(()( uPuPuPuP zyx



Hermite Curve Formation Cubic polynomial and its derivative

Given Px(0), Px(1), P’x(0), P’x(1), solve for a, b, c, d 4 equations are given for 4 unknowns

xxxxx ducubuauP 23)(

xxxx cubuauP 23)( 2'

xx dP )0(xxxxx dcbaP )1(

xx cP )0('

xxxx cbaP 23)1('Animation(U), Chap 3, Interpolating Values


Hermite Curve Formation (cont.) Problem: solve for a, b, c, d

Solution:

)0()0(

)1()0(2))0()1((3

)1()0())1()0((2

'

''

''

xx

xx

xxxxx

xxxxx

PdPc

PPPPb

PPPPa

xx dP )0(xxxxx dcbaP )1(

xx cP )0('

xxxx cbaP 23)1('



Hermite Curves in Matrix Form

ith segment incomposite curves

dcubuauuP 23)(

MBUuP T)(

parameter theis ]1 , , ,[ 23 uuuU T

matrixt coefficien theis Mninformatio geometric theis B

000101001233

1122

M

)1()0()1()0(

B

'

'

x

x

x

x

PPPP

'1

'1B

i

i

i

i

PPPP



Blending Functions of Hermite Splines Each cubic Hermite spline is a linear combination of

4 blending functions

geometric information

2

1

2

1

23

23

23

23

''2

32

132

)(

pppp

uu

uuu

uu

uu

up

BMUuP T )(


Composite Hermite Curve Continuity between connected segments is

ensured by using ending tangent vector of one segment as the beginning tangent vector of the next.


Bezier Curves Similar to Hermite Curve Instead of endpoints and tangents, four control

points are given points P1 and P4 are on the curve points P2 and P3 are used to control the shape p1 = P1, p2 = P4, p1' = 3(P2-P1), p2' = 3(P4 - P3)


4321

0001003303631331

1)( 23

PPPP

uuuup

Bezier Curves Another representation

Blend the control point position using Bernstein polynomials


)!(!!),(

)1(),()(

spolynomialBernstein are )( where

)()(

,

,

0,

knknknC

uuknCuB

uB

PuBup

knknk

nk

n

kknk


Bezier Curves


Composite Bezier Curves

How to control the continuity between adjacent Bezier segment?


De Casteljau Construction of Bezier Curves

How to derive a point on a Bezier curve?



Bezier Curves (cont.) Variant of the Hermite spline

basis matrix derived from the Hermite basis (or from scratch)

Gives more uniform control knobs (series of points) than Hermite

Scale factor (3) is chosen to make “velocity” approximately constant



Catmull-Rom Splines With Hermite splines, the designer must arrange

for consecutive tangents to be collinear, to get C1 continuity. Similar for Bezier. This gets tedious.

Catmull-Rom: an interpolating cubic spline with built-in C1 continuity.

Compared to Hermite/Bezier: fewer control points required, but less freedom.



Catmull-Rom Splines (cont.) Given n control points in 3-D: p1, p2, …, pn,

Tangent at pi given by s(pi+1 – pi-1) for i=2..n-1, for some s.

Curve between pi and pi+1 is determined by pi-1, pi, pi+1, pi+2.



Catmull-Rom Splines (cont.) Given n control points in 3-D: p1, p2, …, pn,

Tangent at pi given by s(pi+1 – pi-1) for i=2..n-1, for some s

Curve between pi and pi+1 is determined by pi-1, pi, pi+1, pi+2

What about endpoint tangents? (several good answers: extrapolate, or use extra control points p0, pn+1)

Now we have positions and tangents at each knot – a Hermite specification.



Catmull-Rom Spline Matrix

Derived similarly to Hermite and Bezier s is the tension parameter; typically s=1/2

2

1

1

002001011452

1331

)(

,2/1When

i

i

i

i

T

PPP

P

Uup

s



Catmull-Rom Spline What about endpoint tangents?

Provided by users Several good answers: extrapolate, or use extra

control points p0, pn+1


)2(21

)))(((21

021

0121'

0

PPP

PPPPP


Catmull-Rom Spline Example



Catmull-Rom Spline Drawback

An internal tangent is not dependent on the position of the internal point relative to its two neighbors



Splines and Other Interpolation Forms

See Computer Graphics textbooks Review

Appendix B.4 in Parent



Now What?

We have key frames or points We have a way to specify the space curve Now we need to specify velocity to traverse

the curve

Speed Curves



Speed Control The speed of tracing a curve needs to be under the

direct control of the animator Varying u at a constant rate will not necessarily

generate P(u) at a constant speed.



Generally, equally spaced samples in parameter space are not equally spaced along the curve

Non-uniformity in Parametrization

A

B

C

D

Time = 0

Time = 10

Time = 35

Time = 60

)()( 1212 ususuu

duuPuPuPus zyx222 )()()()(



Arc Length Reparameterization

To ensure a constant speed for the interpolated value, the curve has to be parameterized by arc length (for most applications)

Computing arc length Analytic method (many curves do not have, e.g., B-

splines) Numeric methods

Table and differencing Gaussian quadrature




Space curve vs. time-distance function Relates time to the distance traveled along the

curve, i.e., Relates time to the arc length along the curve



Arc Length Reparameterization Given a space curve with arc length

parameterization Allow movement along the curve at a constant speed by

stepping at equal arc length intervals Allows acceleration and deceleration along the curve by

controlling the distance traveled in a given time interval Problems

Given a parametric curve and two parameter values u1 and u2, find arclength(u1,u2)

Given an arc length s, and parameter value u1, find u2 such that arclength(u1,u2) = s




Converting a space curve P(u) to a curve with arc-length parameterization Find s=S(u), and u=S-1(s)=U(s) P*(s)=P(U(s))

Analytic arc-length parameterization is difficult or impossible for most curves, e.g., B-spline curve cannot be parameterized with arc length. Approximate arc-length parameterization



Forward Differencing Sample the curve at small intervals of the parameter Compute the distance between samples Build a table of arc length for the curve

u Arc Length0.0 0.00

0.1 0.08

0.2 0.19

0.3 0.32

0.4 0.45

… …



Arc Length Reparameterization Using Forward Differencing

Given a parameter u, find the arc length Find the entry in the table closest to this u Or take the u before and after it and interpolate arc

length linearlyu Arc Length

0.0 0.00

0.1 0.08

0.2 0.19

0.3 0.32

0.4 0.45

… …




Given an arc length s and a parameter u1, find the parametric value u2 such that arclength(u1, u2)=s Find the entry in the table closest to this u using binary search Or take the u before and after it and interpolate linearly


0.1 0.08

0.2 0.19

0.3 0.32

0.4 0.45

… …




Easy to implement, intuitive, and fast Introduce errors

Super-sampling in forming table 1000 equally spaced parameter values + 1000 ebtries in

each interval Better interpolation Adaptive forward differencing



Arc Length Reparameterization Using Adaptive forward differencing


Arc Length Reparameterization Using Numerical Integral

Arc length integral

Simpson’s and trapezoidal integration using evenly spaced intervals

Gaussian quadrature using unevenly spaced intervals


duuPuPuPus zyx222 )()()()(

Speed Control

Given a arc-length parameterized space curve, how to control the speed at which the curve is traced? By a speed-control functions that relate an equally

spaced time interval to arc length Input time t, output arc length: s=S(t) Linear function: constant speed control Most common: ease-in/ease-out

Smooth motion from stopped position, accelerate, reach a max velocity, and then decelerate to a stop position


Speed Control Function

Relates an equally spaced time interval to arc length Input time t, output arc length: s=S(t) Normalized arc length parameter


s=ease(t)

Start at 0, slowly increasein value and gain speed until themiddle value andthen decelerate as it approaches to 1.


Constant Velocity Speed Curve

Moving at 1 m/s if meters and seconds are the units Too simple to be what we want


Distance Time Functionand Speed Control

We have a space curve p=P(u) and a speed control function s=S(t).

For a given t, s=S(t) Find the corresponding value u=U(s) by looking

up an arc length table for a given s A point on the space curve with parameter u p=P(U(S(t)))



Speed Control


0.1 0.08

0.2 0.19

0.3 0.32

0.4 0.45

… …

t*

s*u*)( *uP

Speed Control Curve

Arc Length Table


Distance Time Function

Assumptions on distance time function The entire arc length of the curve is to be traversed

during the given total time Additional optional assumptions

The function should be monotonic in t The function should be continuous



Ease-in/Ease-out

Most useful and most common ways to control motion along a curve

(distance)

Time

Arc length

Equally spaced samples in time specify arc length required for that frame



Ease-in/Ease-outSine Interpolation

-1

1

p/2p/2

Time

Arc Length

22 ),sin( pp

10 ,21)

2( sin

21)(ease ttts pp


Ease-in/Ease-outSine Interpolation

The speed of s w.r.t. t is never constant over an interval but rather is always accelerating or decelerating.

Zero derivatives at t=0 and t=1 indicating smooth accelerating and deceleration at end points.



Ease-in/Ease-outPiecing Curves Together for Ease In/Out

Time

Linear segmentArc Length

Sinusoidal segments



Ease-in/Ease-outPiecing Curves Together for Ease In/Out



Ease-in/Ease-outIntegrating to avoid the sine function

acceleration

time

t1 t2

A

D

velocity

timet1 t2

v

arc length

time

V1 = V2 A*t1 = - D*(1-t2)

Find v such that area = 1



To avoid the transcendental function evaluation while providing a constant speed interval between ease-in and ease-out intervals Acceleration-time curve Velocity-time curve

Velocity-time curve is defined by integral of the acceleration-time curve

Distance-time curve is defined by integral of the velocity-time curve



In ease-in/ease-out function Velocity starts out at 0 and ends at 0

V1 = V2 inplies A*t1 = - D*(1-t2)



Total distance = 1 implies 1 = ½ V0 t + V0 (t2-t1)+1/2 V0 (1-t2) Users specifies any two of three variables t1, t2, and V0,

and the system can solve for the third



The result is Parabolic ease-in/ease-out More flexible than the sinusoidal ease-in/ease-out



Videos Bunny (Blue Sky Studios, 1998)


Review - Quaternions Similar to axis-angle representations

4-tuple of real numbers q=(s, x, y, z) or [s, v]s is a scalar; v is a vector

The quaternion for rotating an angle about an axis (an axis-angle rotation):

Animation(U), Chap 2, Tech. Background CS, NCTU, J. H. Chuang72

q

X

Y

Z

a=(ax, ay, az)

If a is unit length, then q will be also

),,()2

sin(),2

cos(

)),,(,(Rot

zyx

zyx

aaa

aaaq

qq

q

Review - Quaternions


If a is unit length, then q will be also

112

sin2

cos2

sin2

cos

2sin

2cos

2sin

2sin

2sin

2cos

22222

22222

2222222

23

22

21

20

qqqq

qq

qqqq

a

q

zyx

zyx

aaa

aaa

qqqq

Review - Quaternions Rotating some angle around an axis is the same as

rotating the negative angle around the negated axis


qzyxzyx

),,)(

2sin(),

2cos(Rot ),,(,

qqq

75

Review - Quaternion Math Addition

Multiplication

Multiplication is associative but not commutative

Animation(U), Chap 2, Tech. Background CS, NCTU, J. H. Chuang

21212211 , , , vvssvsvs

21122121212211 , , , vvvsvsvvssvsvs

1221 qqqq 321321 )()( qqqqqq

76

Review - Quaternion Math (cont.)

Multiplicative identity: [1, 0,0,0]

Inverse

Normalization for unit quaternion

2222 zyxsq

0 ,0 ,0 ,11 qq


qqq

2222 zyxsq

qq 0 ,0 ,0 ,1

2

1 ,q

vsq 0 ,0 ,0 ,11 qq

qqq

77

Review – Rotating Vectors Using Quaternion

A point in space, v, is represented as [0, v] To rotate a vector v using quaternion q

Represent the vector as v = [0, v] Represent the rotation as a quaternion q Using quaternion multiplication

The proof isn’t that hard Note that the result v’ always has zero scalar value


1)(' qvqvRotv q

Review - Compose Rotations Rotating a vector v by first quaternion p

followed by a quaternion q is like rotation using qp


)()()(

)())((

1

11

1

vRotqpvqp

qqpvp

pvpRotvRotRot

qp

qpq

Review - Compose Rotations To rotate a vector v by quaternion q followed

by its inverse quaternion q-1


vqqvqq

qvqRotvRotRotqqq

11

1)())(( 11

80

Review - Quaternion Interpolation A quaternion is a point on a 4D unit sphere

Unit quaternion: q=(s,x,y,z), ||q|| = 1 Form a subspace: a 4D sphere

Interpolating quaternion means moving between two points on the 4D unit sphere A unit quaternion at each step – another point on the 4D unit sphere Move with constant angular velocity along the greatest circle between the two points on the 4D unit sphere


81

Review - Linear Interpolation Linear interpolation generates unequal spacing

of points after projecting to circle


82

Want equal increment along arc connecting two quaternion on the spherical surface Spherical linear interpolation (slerp)

Normalize to regain unit quaternion

Review – Spherical Linear Interpolation (slerp)


)(cos and 1 to0 from goes where

sin)sin(

sin))1sin((),,(

211

2121

qqu

ququuqqslerp

q

qq

qq

83



2121

1

21

21

sin)sin(

sin))1sin((),,(

give to

,

,1 :givenfor solvecan We

Let

ququuqqslerp

uqqqq

q

qqq

qq

qq

qq

84


2q

1q

2q

021 qq


85


Recall that q and –q represent same rotation What is the difference between:

Slerp(u, q1, q2) and Slerp(u, q1, -q2) ? One of these will travel less than 90 degrees while

the other will travel more than 90 degrees across the sphere

This corresponds to rotating the ‘short way’ or the ‘long way’

Usually, we want to take the short way, so we negate one of them if their dot product is < 0


86



If we have an intermediate position q2, the interpolation from q1-->q2-->q3 will not necessarily follow the same path as the interpolation from q1 to q3.

87

Interpolating a Series of Quaternions As linear interpolation in Euclidean space, we

can have first order discontinuity problem when interpolating a series of orientations.

Need a cubic curve interpolation to maintain first order continuity in Euclidean space

Similarly, slerp can have 1st order discontinuity We also need a cubic curve interpolation in 4D

spherical space for 1st order continuity


Interpolating a Series of Quaternions

Interpolated points […, pn-1, pn, pn+1, ….] in 3D Between each pair of points, two control

points will be constructed. e.g., For pn an : the one intermediate after pn

bn : the one immediately before pn


pn+(pn-pn-

1)Take average

Interpolating a Series of Quaternions Interpolated points […, pn-1, pn, pn+1, ….] in 3D

Between each pair of points, two control points will be constructed. e.g., For pn

an : the one intermediate after pn

bn : the one immediately before pn


bn =pn+(pn-an)


End conditions



Between any 2 interpolated points pn and pn+1, a cubic Bezier curve segment is defined by pn, an, bn+1, and pn+1



How to evaluate a point on Bezier curve? De Casteljau Construction

Constructing Bezier curve by multiple linear interpolation


u=1/3

p(1/3)



Animation: Seehttp://en.wikipedia.org/wiki/File:Bezier_3_big.gif

http://en.wikipedia.org/wiki/File:Bezier_3_big.png


Bezier interpolation on 4D sphere? How are control points generated? How are cubic Beziers defined?

Control points are automatically generated as it is not intuitive to manually adjust them on a 4D sphere

Derive Bezier curve points by iteratively subdivision s on pn, an, bn+1, and pn+1


95

1

11 '

'),'(sec

nn

nnnn qq

qqqqtBi

Interpolating a Series of QuaternionsConnecting Segments on 4D Sphere

Automatically generating interior (spherical) control points

qn-1

qn

qn+1

qn’an

1. Double the arc

2. Bisect the span

111' )(2),( nnnnnnn qqqqqqdoubleq

bn


),( nnn qadoubleb

96

1nqnq

Interpolating a Series of QuaternionsDe Casteljau Construction on 4D Sphere

p1 slerp(qn ,an, 13)

p2 slerp(an ,bn1, 13)

p3 slerp(bn1,qn1, 13)

p12 slerp(p1, p2, 13)

p23 slerp(p2, p3, 13)

p slerp(p12, p23, 13)

u=1/3

P(1/3)

na1nb

1p

2p

3p


Path Following

Issues about path following Arc length parameterization and speed control Changing the orientation Smoothing a path Path along a surface


Orientation Along a Curve

A local coordinate system (u, v, w) is defined for an object to be animated Position along the path P(s), denoted as POS w-axis: the direction the object is facing v-axis: up vector u-axis: perpendicular to w and v (u, v, w): a right-handed coordinate system


Orientation Along a Curve


Orientation Along a CurveFrenet Frame

Frenet frame: a moving coordinate system determined by curve tangent and curvature


wuvsPsPu

sPw

)(')("

)('


Problems No concept of “up” vector Is undefined in segments that has no curvature;

i.e., P”(u)=0 Can be dealt with by interpolating a Frenet frame from

the Frenet frames at segment’s boundary, which differ by only a rotation around w

A discontinuity in the curvature vector Frenet frame has a discontinuous jump in orientation

Resulting motions are usually too extreme and not natural looking


Orientation Along a CurveCamera Path Following

COI: center of interest A fixed point in the scene Center point of an object in the scene Points along the path itself Points on a separate path in the scene Points on the interpolation between points in the

scene


wuvwu

w

y_axis

POSCOI

wuvwu

sPsCw

y_axis

)( )(

wuvsPsUwu

sPsCw

))()((

)( )(

Smoothing a Pathwith Linear Interpolation

Two points on either sides of a point are averaged, and this point is averaged with the original point.

Repeated application


11

11

41

21

41

22'

iii

iii

i PPP

PPPP

Smoothing a Pathwith Linear Interpolation


Smoothing a Pathwith Cubic Interpolation

Four points on either sides of a point are fitted by a cubic curve, and the midpoint of the curve is averaged with the original point.


dcbaPP

dcbaPP

dcbaPP

dPP

dcubuauuP

i

i

i

i

)1(43

169

6427)4/3(

41

161

641)4/1(

)0(

)(

2

1

1

2

23

Determine a Path on a Surface

If one object is to move across the surface of another object, we need to specify a path across the surface. Given start and destination points

Find a shortest path between the points – expensive Determine a plane that contain two points and is

generally perpendicular to the surface (averaged normal of two points)

Intersection of the plane with the surface (mesh)


Determine a Path on a Surface


Determine a Path on a Parametric Surface

Given start and destination points on parametric domain

Define the straight line connecting two points in the parametric space

The curve on the surface corresponding to this line


chap 3 interpolating values

Documents

interpolating valuescs

interpolating valuesanimationu

special effects chap

local control animationu

segments animationu

tangents animationu

paths animationu

object animationu