3-interpolatingvalues

8/2/2019 3-InterpolatingValues

1/111

Chap 3Interpolating Values

Animation(U), Chap 3, Interpolating Values 1


2/111

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


3/111

3

Interpolation between key frames

Parameters to be interpolated Position of an object

Normal

Joint angle between two joints Transparency attribute of an object



4/111

4


Interpolation between frames is not trivial Appropriate interpolating function

Parameterization of the function

Maintain the desired control of the interpolated valuesover 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



5/111

5



desired result undesired result

Need a smooth interpolation with user control


6/111


Solution 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


7/1117

Interpolation functions

Interpolation vs. approximation

Interpolation

Hermite, Catmull-Rom

approximation

Bezier, B-spline



8/1118


Complexity => computational efficiency Polynomials

Lower than cubic

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

Higher than cubic

Doesnt 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



9/1119


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



10/11110




11/111CS, NCTU, J. H. Chuang11

Continuity

noneposition(0th order)

tangent(1st order)

curvature(2nd order)

At junctionof twocircular arcs

Animation(U), Chap 3, Interpolating Values


12/11112


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 ofx




Example: Representing Unit Circle

Cannot be represented explicitly as a functionofx

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

Parametric form:x=cos(u),y=sin(u), 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


16/111

Polynomial Interpolation

An n-th degree polynomial fits a curve to n+1points

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



17/111

Polynomial Interpolation (cont.)

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

We usually want the curve to be as smooth aspossible

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 lowdegree polynomials are used to interpolate

(pass through) the control points

Cubic polynomials are the most common

lowest order polynomials that interpolate twopoints and allow the gradient at each point to bedefined - 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 avector equation)

u0 1

1

Two basis (blending) functions

u

21 )1()( puupup




Hermite Curvescubic 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



22/111

CS, NCTU, J. H. Chuang22

Hermite Curve Formation Cubic polynomial and its derivative

Given Px(0), Px(1),Px(0),Px(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('



23/111


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

Pd

Pc

PPPPb

PPPPa

xx dP )0(

xxxxx dcbaP )1(

xx cP )0('

xxxx cbaP 23)1('



24/111


Hermite Curves in Matrix Form

ith segment incomposite curves

dcubuauuP 23)(

MBUuPT)(

parametertheis]1,,,[ 23 uuuUT

matrixtcoefficientheisM

ninformatiogeometrictheisB

0001

0100

1233

1122

M

)1(

)0(

)1(

)0(

B

'

'

x

x

x

x

P

P

P

P

'

1

'

1B

i

i

i

i

P

P

PP



25/111


Blending Functions of Hermite Splines

Each cubic Hermite spline is a linear combination of4blending functions

geometric information

2

1

2

1

23

23

23

23

'

'2

32

132

)(

p

p

p

p

uu

uuu

uu

uu

up

BMUuPT )(



26/111

Composite Hermite Curve

Continuity between connected segments isensured by using ending tangent vector of one

segment as the beginning tangent vector of the

next.



27/111

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)


4

3

2

1

0001

0033

0363

1331

1)(23

P

P

P

P

uuuup


28/111

Bezier Curves

Another representation Blend the control point position using Bernstein polynomials


)!(!

!),(

)1(),()(

spolynomialBernsteinare)(where

)()(

,

,

0

,

knk

nknC

uuknCuB

uB

PuBup

knk

nk

nk

n

k

knk


29/111


Bezier Curves



30/111

Composite Bezier Curves

How to control the continuity betweenadjacent Bezier segment?


D C lj C i f B i


31/111

De Casteljau Construction of Bezier

Curves

How to derive a point on a Bezier curve?



32/111


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



33/111


Catmull-Rom Splines

With Hermite splines, the designer must arrangefor 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.



34/111


Catmull-Rom Splines (cont.)

Given n control points in 3-D:p1, p2, ,pn,

Tangent at pi given by s(pi+1pi-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.



35/111


Catmull-Rom Splines (cont.)

Given n control points in 3-D:p1, p2, ,pn, Tangent at pi given by s(pi+1pi-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.



36/111


Catmull-Rom Spline Matrix

Derived similarly to Hermite and Bezier

s is the tension parameter; typically s=1/2

2

1

1

0020

01011452

1331

)(

,2/1When

i

i

i

i

T

P

PP

P

Uup

s



37/111


Catmull-Rom Spline

What about endpoint tangents? Provided by users

Several good answers: extrapolate, or use extra

control points p0, pn+1


)2(2

1

)))(((2

1

021

0121

'

0

PPP

PPPPP


38/111


Catmull-Rom Spline

Example



39/111


Catmull-Rom Spline

Drawback An internal tangent is not dependent on the

position of the internal point relative to its two

neighbors



40/111


Splines and Other Interpolation Forms

See Computer Graphics textbooks

Review

Appendix B.4 in Parent



41/111


Now What?

We have key frames or points

We have a way to specify the space curve

Now we need to specify velocity to traversethe curve

Speed Curves



42/111


Speed Control

The speed of tracing a curve needs to be under thedirect control of the animator

Varying u at a constant rate will not necessarily

generate P(u) at a constant speed.



43/111


Generally, equally spaced samples inparameter 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 )()()()(



44/111


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



45/111



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



46/111



Given a space curve with arc lengthparameterization

Allow movement along the curve at a constant speed by

stepping at equal arc length intervals

Allows acceleration and deceleration along the curve bycontrolling the distance traveled in a given time interval

Problems

Given a parametric curve and two parameter values u1 andu2, find arclength(u1,u2)

Given an arc length s, and parameter value u1, find u2 such

that arclength(u1,u2) = s



47/111



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 arclength.

Approximate arc-length parameterization



48/111


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


A L th R t i ti U i


49/111


Arc Length Reparameterization Using


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 linearly u Arc Length0.0 0.00

0.1 0.08

0.2 0.19

0.3 0.32

0.4 0.45


A L th R t i ti U i


50/111




Given an arc length s and a parameter u1, findthe parametric value u2 such that

arclength(u1, u2)=s

Find the entry in the tableclosest to this u using

binary search

Or take the u before and after

it and interpolate linearly

u Arc Length

0.0 0.00

0.1 0.08

0.2 0.19

0.3 0.320.4 0.45


A L th R t i ti U i


51/111




Easy to implement, intuitive, and fast Introduce errors

Super-sampling in forming table

1000 equally spaced parameter values + 1000 ebtries ineach interval

Better interpolation

Adaptive forward differencing


A L th R t i ti U i


52/111



Adaptive forward differencing




53/111


Numerical Integral

Arc length integral

Simpsons and trapezoidal integration using evenly

spaced intervals

Gaussian quadrature using unevenly spaced

intervals


duuPuPuPus zyx222 )()()()(


54/111

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



55/111

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 gainspeed until themiddle value andthen decelerate asit approaches to 1.


56/111


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 Function


57/111


and 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)))



58/111


Speed Control

u Arc Length

0.0 0.00

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



59/111


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



60/111


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-out


61/111


Ease in/Ease out

Sine Interpolation

-1

1

p/2

p/2

Time

Arc Length

22),sin(

p

p

10,2

1)

2(sin

2

1)(ease ttts pp


Ease-in/Ease-out


62/111

Ease in/Ease out

Sine 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-out


63/111


Ease in/Ease out

Piecing Curves Together for Ease In/Out

Time

Linear segmentArc Length

Sinusoidal segments


Ease-in/Ease-out


64/111


Ease in/Ease out

Piecing Curves Together for Ease In/Out


Ease-in/Ease-out


65/111


Ease in/Ease out

Integrating to avoid the sine function

acceleration

time

t1 t2

A

D

velocity

timet1 t2

v

arc length

time

V1 = V2A*t1 = - D*(1-t2)

Find v such that area = 1


Ease-in/Ease-out


66/111

Ease in/Ease out


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


Ease-in/Ease-out


67/111

Ease in/Ease out


In ease-in/ease-out function

Velocity starts out at 0 and ends at 0

V1 = V2inplies A*t1 = - D*(1-t2)


Ease-in/Ease-out


68/111

Ease in/Ease out


Total distance = 1 implies

1 = V0t + 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


Ease-in/Ease-out


69/111

ase / ase out


The result is Parabolic ease-in/ease-out

More flexible than the sinusoidal ease-in/ease-out


Vid


70/111


Videos

Bunny (Blue Sky Studios, 1998)


R i Q t i


71/111

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

R i Q t i


72/111



Ifa is unit length, then q will be also

11

2sin

2cos

2sin

2cos

2sin

2cos

2sin2sin2sin2cos

22222

22222

2222222

2

3

2

2

2

1

2

0

qqqq

qq

qqqq

a

q

zyx

zyx

aaa

aaa

qqqq

R i Q t i


73/111


Rotating some angle around an axis is the same asrotating the negative angle around the negated axis


qzyxzyx

),,)(

2

sin(),

2

cos(Rot ),,(,qq

q

R i Q t i M th


74/111

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

R i Q t i M th ( t )


75/111

76

Review - Quaternion Math (cont.)

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

Inverse

Normalization for unit quaternion

2222zyxsq

0,0,0,11 qq


q

qq

2222 zyxsq

qq 0,0,0,1

21 ,

qvsq 0,0,0,11 qq

q

qq

Review


76/111

77

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 isnt that hard

Note that the result valways has zero scalar value


1

)('

qvqvRotvq

Re ie Compo e Rotation


77/111

Review - Compose Rotations

Rotating a vector v by first quaternionpfollowed by a quaternion q is like rotation

using qp


)()()(

)())((

1

11

1

vRotqpvqp

qqpvp

pvpRotvRotRot

qp

qpq

Review Compose Rotations


78/111

Review - Compose Rotations

To rotate a vector v by quaternion q followedby its inverse quaternion q-1


v

qqvqq

qvqRotvRotRot qqq

11

1

)())(( 11

Review Quaternion Interpolation


79/111

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 stepanother point

on the 4D unit sphere

Move with constant angular velocity

along the greatest circle between the

two points on the 4D unit sphere


Review Linear Interpolation


80/111

81

Review - Linear Interpolation

Linear interpolation generates unequal spacingof points after projecting to circle


Review


81/111

82

Want equal increment along arc connecting twoquaternion on the spherical surface

Spherical linear interpolation (slerp)

Normalize to regain unit quaternion

Spherical Linear Interpolation (slerp)


)(cosand1to0fromgoeswhere

sin

)sin(

sin

))1sin((),,(

21

1

2121

qqu

qu

qu

uqqslerp

q

q

q

q

q

Review


82/111

83



2121

1

21

21

sin

)sin(

sin

))1sin((),,(

giveto

,

,1

:givenforsolvecanWe

Let

qu

qu

uqqslerp

uqq

qq

q

qqq

q

q

q

q

q

q

Review


83/111

84


2q

1q

2q

021 qq


Review


84/111

85


Recall that q andq 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 whilethe other will travel more than 90 degrees acrossthe 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


Review


85/111

86



If we have an intermediate position q2, theinterpolation from q1-->q2-->q3 will not

necessarily follow the same path as the

interpolation from q1 to q3.

Interpolating a Series of Quaternions


86/111

87


As linear interpolation in Euclidean space, wecan 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




87/111


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



88/111


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)



89/111



End conditions




90/111



Between any 2 interpolated points pn and pn+1,

a cubic Bezier curve segment is defined by pn,

an, bn+1, and pn+1




91/111


How to evaluate a point on Bezier curve?

De Casteljau Construction

Constructing Bezier curve by multiple linear

interpolation


u=1/3

p(1/3)



92/111



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

http://bezier_3_big.gif/http://bezier_3_big.gif/http://bezier_3_big.gif/http://bezier_3_big.gif/http://bezier_3_big.gif/http://en.wikipedia.org/wiki/File:Bezier_3_big.png


93/111


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

How are cubic Beziers defined?

Control points are automatically generated as itis not intuitive to manually adjust them on a 4Dsphere

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




94/111

95

1

11

'

'),'(sec

nn

nnnn

qq

qqqqtBi

Connecting 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



95/111

96

1nqnq

De Casteljau Construction on 4D Sphere

p1 slerp(qn ,an, 13)

p2 slerp(a

n,b

n1, 1

3)

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)

na 1nb

1p

2p

3p


Path Following


96/111

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


97/111


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




98/111





99/111

Frenet Frame

Frenet frame: a moving coordinate systemdetermined by curve tangent and curvature


wuv

sPsPu

sPw

)(')("

)('



100/111

Frenet Frame

ProblemsNo 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 segments 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 lookingAnimation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang101


F F


101/111

Frenet Frame



F F


102/111

Frenet Frame



C P h F ll i


103/111

Camera 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


wuv

wu

w

y_axis

POSCOI

wuv

wu

sPsCw

y_axis

)()(

wuv

sPsUwu

sPsCw

))()((

)()(

Smoothing a Path

i h Li I l i


104/111

with Linear Interpolation

Two points on either sides of a point areaveraged, and this point is averaged with the

original point.

Repeated application


11

11

41

21

41

22'

iii

iii

i PPP

PPP

P

Smoothing a Path

i h Li I l i


105/111

with Linear Interpolation


Smoothing a Path

ith C bi I t l ti


106/111

with Cubic Interpolation

Four points on either sides of a point are fittedby 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(

4

3

16

9

64

27)4/3(

4

1

16

1

64

1)4/1(

)0(

)(

2

1

1

2

23

Smoothing a Path

ith C bi I t l ti


107/111



Smoothing a Path

ith C bi I t l ti


108/111



Determine a Path on a Surface


109/111

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 pointsexpensive

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


110/111


Determine a Path on a Parametric

S f


111/111

Surface

Given start and destination points on parametricdomain

Define the straight line connecting two points in the

parametric space

The curve on the surface corresponding to this line

3-interpolatingvalues

Documents