chap 3 interpolating values animation(u), chap 3, interpolating values1
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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
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang
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
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang
5
Interpolation between key frames
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang
desired result undesired result
Need a smooth interpolation with user control
Interpolation between key frames
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
Interpolation functions
Interpolation vs. approximation
Interpolation Hermite, Catmull-Rom
approximation Bezier, B-spline
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang
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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
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang
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
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang
10
Interpolation functions
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang
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
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang
CS, NCTU, J. H. Chuang13
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
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang14
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
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang15
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
Animation(U), Chap 3, Interpolating Values
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
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang16
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
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang17
CS, NCTU, J. H. Chuang18
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)
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang19
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
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang20
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
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang21
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
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang22
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
CS, NCTU, J. H. Chuang23
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('
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang24
Hermite Curves in Matrix Form
ith segment incomposite curves
dcubuauuP 23)(
MBUuP T)(
parameter theis ]1 , , ,[ 23 uuuU T
matrixt coefficien theis M
ninformatio geometric theis B
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
P
P
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang25
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
)(
p
p
p
p
uu
uuu
uu
uu
up
BMUuP T )(
Animation(U), Chap 3, Interpolating Values
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.
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang26
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)
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang27
4
3
2
1
0001
0033
0363
1331
1)( 23
P
P
P
P
uuuup
Bezier Curves Another representation
Blend the control point position using Bernstein polynomials
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang28
)!(!
!),(
)1(),()(
spolynomialBernstein are )( where
)()(
,
,
0,
knk
nknC
uuknCuB
uB
PuBup
knknk
nk
n
kknk
CS, NCTU, J. H. Chuang29
Bezier Curves
Animation(U), Chap 3, Interpolating Values
Composite Bezier Curves
How to control the continuity between adjacent Bezier segment?
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang30
De Casteljau Construction of Bezier Curves
How to derive a point on a Bezier curve?
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang31
CS, NCTU, J. H. Chuang32
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
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang33
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.
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang34
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.
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang35
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.
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang36
Catmull-Rom Spline Matrix
Derived similarly to Hermite and Bezier s is the tension parameter; typically s=1/2
2
1
1
0020
0101
1452
1331
)(
,2/1When
i
i
i
i
T
P
P
P
P
Uup
s
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang37
Catmull-Rom Spline What about endpoint tangents?
Provided by users Several good answers: extrapolate, or use extra
control points p0, pn+1
Animation(U), Chap 3, Interpolating Values
)2(2
1
)))(((2
1
021
0121'
0
PPP
PPPPP
CS, NCTU, J. H. Chuang38
Catmull-Rom Spline Example
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang39
Catmull-Rom Spline Drawback
An internal tangent is not dependent on the position of the internal point relative to its two neighbors
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang41
Splines and Other Interpolation Forms
See Computer Graphics textbooks Review
Appendix B.4 in Parent
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang42
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
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang43
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.
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang44
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 )()()()(
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang45
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
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang46
Arc Length Reparameterization
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
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang47
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
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang48
Arc Length Reparameterization
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
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang49
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 Length
0.0 0.00
0.1 0.08
0.2 0.19
0.3 0.32
0.4 0.45
… …
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang50
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
… …
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang51
Arc Length Reparameterization Using Forward Differencing
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
u Arc Length
0.0 0.00
0.1 0.08
0.2 0.19
0.3 0.32
0.4 0.45
… …
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang52
Arc Length Reparameterization Using Forward Differencing
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
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang53
Arc Length Reparameterization Using Adaptive forward differencing
Animation(U), Chap 3, Interpolating Values
Arc Length Reparameterization Using Numerical Integral
Arc length integral
Simpson’s and trapezoidal integration using evenly spaced intervals
Gaussian quadrature using unevenly spaced intervals
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang54
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
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang55
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
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang56
s=ease(t)
Start at 0, slowly increasein value and gain speed until themiddle value andthen decelerate as it approaches to 1.
CS, NCTU, J. H. Chuang57
Constant Velocity Speed Curve
Moving at 1 m/s if meters and seconds are the units Too simple to be what we want
Animation(U), Chap 3, Interpolating Values
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)))
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang58
CS, NCTU, J. H. Chuang59
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
Animation(U), Chap 3, Interpolating Values
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
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang60
CS, NCTU, J. H. Chuang61
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
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang62
Ease-in/Ease-outSine Interpolation
-1
1
- /2p/2p
Time
Arc Length
22 ),sin(
10 ,2
1)
2( sin
2
1)(ease ttts
Animation(U), Chap 3, Interpolating Values
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.
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang63
CS, NCTU, J. H. Chuang64
Ease-in/Ease-outPiecing Curves Together for Ease In/Out
Time
Linear segmentArc Length
Sinusoidal segments
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang65
Ease-in/Ease-outPiecing Curves Together for Ease In/Out
Animation(U), Chap 3, Interpolating Values
CS, NCTU, J. H. Chuang66
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
Animation(U), Chap 3, Interpolating Values
Ease-in/Ease-outIntegrating to avoid the sine function
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
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang67
Ease-in/Ease-outIntegrating to avoid the sine function
In ease-in/ease-out function Velocity starts out at 0 and ends at 0
V1 = V2 inplies A*t1 = - D*(1-t2)
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang68
Ease-in/Ease-outIntegrating to avoid the sine function
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
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang69
Ease-in/Ease-outIntegrating to avoid the sine function
The result is Parabolic ease-in/ease-out More flexible than the sinusoidal ease-in/ease-out
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang70
CS, NCTU, J. H. Chuang71
Videos Bunny (Blue Sky Studios, 1998)
Animation(U), Chap 3, Interpolating Values
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
Review - Quaternions
Animation(U), Chap 2, Tech. Background CS, NCTU, J. H. Chuang73
If a is unit length, then q will be also
11
2sin
2cos
2sin
2cos
2sin
2cos
2sin
2sin
2sin
2cos
22222
22222
2222222
23
22
21
20
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
Animation(U), Chap 2, Tech. Background CS, NCTU, J. H. Chuang74
qzyxzyx
),,)(
2sin(),
2cos(Rot ),,(,
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
Animation(U), Chap 2, Tech. Background CS, NCTU, J. H. Chuang
q
2222 zyxsq
qq 0 ,0 ,0 ,1
2
1 ,
q
vsq
0 ,0 ,0 ,11 qq
q
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
Animation(U), Chap 2, Tech. Background CS, NCTU, J. H. Chuang
1)(' qvqvRotv q
Review - Compose Rotations
Rotating a vector v by first quaternion p followed by a quaternion q is like rotation using qp
Animation(U), Chap 2, Tech. Background CS, NCTU, J. H. Chuang78
)(
)()(
)())((
1
11
1
vRot
qpvqp
qqpvp
pvpRotvRotRot
qp
qpq
Review - Compose Rotations
To rotate a vector v by quaternion q followed by its inverse quaternion q-1
Animation(U), Chap 2, Tech. Background CS, NCTU, J. H. Chuang79
v
qqvqq
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
Animation(U), Chap 2, Tech. Background CS, NCTU, J. H. Chuang
81
Review - Linear Interpolation
Linear interpolation generates unequal spacing of points after projecting to circle
Animation(U), Chap 2, Tech. Background CS, NCTU, J. H. Chuang
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)
Animation(U), Chap 2, Tech. Background CS, NCTU, J. H. Chuang
)(cos and 1 to0 from goes where
sin
)sin(
sin
))1sin((),,(
211
2121
qqu
qu
qu
uqqslerp
83
Review – Spherical Linear Interpolation (slerp)
Animation(U), Chap 2, Tech. Background CS, NCTU, J. H. Chuang
2121
1
21
21
sin
)sin(
sin
))1sin((),,(
give to
,
,1
:givenfor solvecan We
Let
qu
qu
uqqslerp
uqq
q
qqq
84
Review – Spherical Linear Interpolation (slerp)
2q
1q
2q
021 qq
Animation(U), Chap 2, Tech. Background CS, NCTU, J. H. Chuang
85
Review – Spherical Linear Interpolation (slerp)
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
Animation(U), Chap 2, Tech. Background CS, NCTU, J. H. Chuang
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Review – Spherical Linear Interpolation (slerp)
Animation(U), Chap 2, Tech. Background CS, NCTU, J. H. Chuang
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.
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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
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang
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
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang88
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
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang89
bn =pn+(pn-an)
Interpolating a Series of Quaternions Interpolated points […, pn-1, pn, pn+1, ….] in 3D
End conditions
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang90
Interpolating a Series of Quaternions Interpolated points […, pn-1, pn, pn+1, ….] in 3D
Between any 2 interpolated points pn and pn+1,
a cubic Bezier curve segment is defined by pn, an, bn+1, and pn+1
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang91
Interpolating a Series of Quaternions
How to evaluate a point on Bezier curve? De Casteljau Construction
Constructing Bezier curve by multiple linear interpolation
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang92
u=1/3
p(1/3)
Interpolating a Series of Quaternions
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang93
Animation: Seehttp://en.wikipedia.org/wiki/File:Bezier_3_big.gif
Interpolating a Series of Quaternions
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
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang94
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
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang
),( nnn qadoubleb
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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
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang
Path Following
Issues about path following Arc length parameterization and speed control Changing the orientation Smoothing a path Path along a surface
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang97
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
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang98
Orientation Along a Curve
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang99
Orientation Along a CurveFrenet Frame
Frenet frame: a moving coordinate system determined by curve tangent and curvature
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang100
wuv
sPsPu
sPw
)(')("
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Orientation Along a CurveFrenet Frame
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
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang101
Orientation Along a CurveFrenet Frame
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang102
Orientation Along a CurveFrenet Frame
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang103
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
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang104
wuv
wu
w
y_axis
POSCOI
wuv
wu
sPsCw
y_axis
)( )(
wuv
sPsUwu
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
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang105
11
11
4
1
2
1
4
1
22'
iii
iii
i PPP
PPP
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Smoothing a Pathwith Linear Interpolation
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang106
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.
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang107
dcbaPP
dcbaPP
dcbaPP
dPP
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Smoothing a Pathwith Cubic Interpolation
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang108
Smoothing a Pathwith Cubic Interpolation
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang109
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)
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang110
Determine a Path on a Surface
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang111
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
Animation(U), Chap 3, Interpolating Values CS, NCTU, J. H. Chuang112