Lecture 8
Animation
Dr. Amy Zhang
Reading
Hill, Chapters 5 / 7 / 10
Red Book, Chapter 3, “Viewing”
Red Book, Chapter 12, “Evaluators and NURBS”
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Outline
Computer-aided animation
Keyframing and interpolation of position
Hierarchical modeling.
Interpolation of rotation
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Conventional Animation
Draw each frame of the animation
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great control
tedious
Reduce burden with cel animation
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Layer, keyframe, inbetween
cel panoramas (Disney’s Pinocchio)
...
ACM © 1997 “Multiperspective panoramas for cel animation”
Computer-Assisted Animation
Keyframing
 automate the inbetweening
 good control
 less tedious
 creating a good animation still
requires considerable skill and
talent
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Computer-Assisted Animation
Procedural animation
 describes the motion algorithmically
 express animation as a function of small number of
parameters
 Example: a clock with second, minute and hour hands
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Example 2: A bouncing ball
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hands should rotate together
express the clock motions in terms of a “seconds” variable
the clock is animated by varying the seconds parameter
Abs(sin(ωt+Θ))*e-kt
Computer-Assisted Animation
Physically Based Animation
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Assign physical properties to objects (masses, forces, inertial
properties)
Simulate physics by solving equations
Realistic but difficult to control
Computer-Assisted Animation
Motion Capture
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Captures style, subtle nuances and realism
You must observe someone
Motion Capture
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Motion Replay
http://www.youtube.com/watch?v=IJ4tndpwL-o
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Outline
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Computer-aided animation
Keyframing and interpolation of position
Hierarchical modeling.
Interpolation of rotation
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Keyframing
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Describe motion of objects as a function of time
from a set of key object positions. In short, compute
the inbetween frames.
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Interpolating Positions
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Given positions:
find curve
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such that
Linear Interpolation
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Simple problem: linear interpolation between first
two points assuming
:
The x-coordinate for the complete curve in the figure:
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Polynomial Interpolation
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A polynomial is a function of the form:
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n is the degree of the polynomial
The order of the polynomial is the number of coefficients
it has
Always: order = degree + 1
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Polynomial Interpolation
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What degree polynomial can interpolate n+1 points?
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An n-degree polynomial can interpolate any n+1 points
wikipedia: Lagrange interpolation
The resulting interpolating polynomials are called Lagrange
polynomials
We already saw the Lagrange formula for n = 1
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Spline Interpolation
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Lagrange polynomials of small degree are fine but high
degree polynomials are too wiggly
Technical term: overfitting
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Spline Interpolation
Spline (piecewise cubic polynomial) interpolation
produces nicer interpolation
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t
x(t)
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t
t
Spline Curves
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The word spline comes from ship building with wood
A wooden plank is forced between fixed posts, called
“ducks”
Real-world splines are still being used for designing
ship hulls, automobiles, and aircraft fuselage and wings
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Splines
www.abm.org
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Spline Curves
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Spline curve - any composite curve formed with piecewise
parametric polynomials subject to certain continuity
conditions at the boundary of the pieces.
Huh?
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Parametric Polynomial Curves
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A parametric polynomial curve is a parametric curve
where each function x(t), y(t) is described by a polynomial:
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Polynomial curves have certain advantages:
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Easy to compute
Indefinitely differentiable
Why Parametric Curves?
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Parametric curves are very flexible
They are not required to be functions
Curves can be multi-valued with respect to any
coordinate system
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Piecewise Curves
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To avoid overfitting we will want to represent a curve as
a series of curves pieced together
A piecewise parametric polynomial curve uses different
polynomial functions for different parts of the curve
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Advantages: Provides flexibility
Problem: How do we fit the pieces together?
Parametric Continuity
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C0: Curves are joined “watertight”
curve / mesh
C1: First derivative is continuous,
d/dt Q(t) = velocity is the same.
“looks smooth, no facets”
C2: Second derivative is continuous,
d2/dt2 Q(t) = acceleration is the
same (important for animation and
shading)
Specifying Splines
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Control Points - a set of
points that influence the
curve's shape
Hull - the lines that
connect the control points
Interpolating – curve passes
through the control points
Approximating – control
points merely influence
shape
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Parametric Cubic Curves
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In order to assure C2 continuity our functions must be of
at least degree three.
Here's what a 2D parametric cubic function looks like:
In matrix form:
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Parametric Cubic Curves
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This is a cubic function in 3D:
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To avoid the dependency on the dimension we will use
the following notation:
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Parametric Cubic Curves
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What does the derivative of a cubic curve look like?
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Hermite Splines
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We want to support general constraints: not just smooth
velocity and acceleration. For example, a bouncing ball does
not always have continuous velocity:
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Solution: specify position AND velocity at each point
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Hermite Splines
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Given:
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Two control points (P1, P2).
Tangents (derivatives) at the knot points (P’1, P’2):
“Control knob” vector: [P1, P2, P’1, P’2]
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Hermite Spline
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4 segments, (P1, P2) and (P’1, P’2) for each segment
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Building it up…
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Cubic curve equations:
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Boundary constraints:
or:
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Solve for the c’s
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Cubic Hermite Curves
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Curve equation in matrix form:
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T = Power basis
B = Spline basis, characteristic matrix
G = Geometry (control points)
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Bézier Curves
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Developed simultaneously by Bézier (at Renault) and de
Casteljau (at Citroen), circa 1960.
Without specify consecutive tangents
A cubic Bézier curve has four control points, two of which are
knots (P1, P4).
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Bézier Specification
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Four control points (P1, P2, P3, P4).
Gradients (tangents) at the knot points (P’1, P’4) are the
line segments between adjacent control points:
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Tangents: P’1 = 3 ( P2 - P1 ); P’2 = 3 ( P4 - P3 )
Scale factor (3) is chosen to make “velocity” approximately
constant
Endpoint interpolation: P(0) = P1 and P(1) = P4
P2 & P3 control the magnitude and orientation of the
tangent vectors at the end points
Geometry vector G: [P1, P2, P3, P4]
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Cubic Bézier Curves
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Repeating the same derivation gives:
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Bézier Curves
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Consider the cubic curve:
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The basis functions
Bi,3(u)are given by:
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Bézier Curves
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De Casteljau representation:
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where r= 1,…,k; i= 0,…,k–r; pi0(u)=pi
A point on the curve with parameter value u is given
by
,i.e.,
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Bézier Curves
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Using the de Casteljau algorithm, a Bézier curve can be
subdivided anywhere along its length into 2 smaller curves
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Note that the convex hull of the smaller section lies closer to
the curve than the original convex hull, and therefore repeated
subdivision will produce a closer approximation to the curve
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Bézier Curves
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Subdivision formulae about midpoint (i.e.,u=0.5):
Let original control points be pi
Let the control points of the small sections be qi& ri
The 2 small curve segments join at
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Properties of Bézier Curves
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Convex hull: Curve is contained in the hull [P1, P2, P3, P4]
Affine Invariance:
 Transforming the control points is computationally more
efficient than transforming each point on the curve
 Useful in ensuring smoothness while scaling a curve segment
Symmetry: P(t); [P1, P2, P3, P4] ≡ P(1-t); [P4, P3, P2, P1]
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Catmull-Rom Splines
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With built-in C1 continuity.
Compared to Hermite/Bezier: fewer control points
required, but less freedom.
Given n control points in 3-D: p1, p2, …, pn,
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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
Hungry for more?
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You got your Hermite and Bézier splines, CatmullRom splines
Once you know it's a piecewise cubic, you know what
game you're playing — putting conditions on the
segments and how they fit together
You should know the basic properties of Hermite and
Bézier splines
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Interpolating Key Frames
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Interpolation is not fool proof. The splines may
undershoot and cause interpenetration. The animator
must also keep an eye out for these types of sideeffects.
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Outline
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Computer-aided animation
Keyframing and interpolation of position
Hierarchical modeling
Interpolation of rotation
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Articulated Models
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Articulated models:
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rigid parts
connected by joints
They can be animated by specifying the joint angles as
functions of time.
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Forward Kinematics
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Describes the positions of the body parts as a function of
the joint angles
1 DOF: knee
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2 DOF: wrist
3 DOF: arm
Skeleton Hierarchy
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Each bone transformation described relative to the
parent in the hierarchy:
Tree traversal
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Forward Kinematics
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Transformation matrix for an effecter vs is a
matrix composition of all joint transformation
between the effecter and the root of the
hierarchy
Inverse Kinematics
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Forward Kinematics
 Given the skeleton parameters p and the
position of the effecter in local coordinates
vs, what is the position of the sensor in the
world coordinates vw?
Inverse Kinematics
 Given the position of the effecter in local
coordinates vs and the desired position vw in
world coordinates, what are the skeleton
parameters p?
Much harder; requires solving the inverse
Underdetermined problem with many
solutions
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Outline
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Computer-aided animation
Keyframing and interpolation of position
Hierarchical modeling
Interpolation of rotation
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3D rotations
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How many degrees of freedom for 3D orientations?
3 degrees of freedom:
e.g. direction of rotation and angle
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3D Rotation Representations
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Rotation Matrix
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Fixed Angle
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orthornormal columns/rows
bad for interpolation
rotate about global axes
bad for interpolation, gimbal lock
Euler Angle
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rotate about local axes
bad for interpolation, gimbal lock
Fixed Angle
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(0,90,0) in x-y-z order
(0,90,0)
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(90,45,90) in x-y-z order
(90,45,90)
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Interpolation Problem
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The rotation from (0,90,0) to (90,45,90) is a 45-degree xaxis rotation
Directly interpolating between (0,90,0) and (90,45,90)
produces a halfway orientation (45, 67.5, 45)
Desired halfway orientation is (90, 22.5, 90)
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Euler Angles
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An Euler angle is a rotation about a single axis. Any
orientation can be described composing three
rotation around each coordinate axis.
Roll, pitch and yaw
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Euler Angles
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Gimbal lock: two or more axis align resulting in a
loss of rotation dof.
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http://www.fho-emden.de/~hoffmann/gimbal09082002.pdf
Demo
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See
http://www.gamedev.net/reference/programming/featu
res/qpowers/page7.asp
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See also
http://www.cgl.uwaterloo.ca/GALLERY/image_html/gi
mbal.jpg.html
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3D Rotation Representations
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Axis angle
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rotate about A by , (Ax,Ay,Az, )
good interpolation, no gimbal lock
cannot compose rotations efficiently
Solution: Quaternion Interpolation
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4-tuple of real numbers
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q=(s,x,y,z) or [s,v]
s is a scalar; v is a vector
Same information as axis/angle but in a different form
Right-hand rotation of Θ radians about v is:
q = {cos(Θ/2); v sin(Θ /2)}
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Often noted (s, v) or {a; b, c, d}
Quaternions
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q = {cos( /2); v sin( /2)}
What if we use -v?
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q{1; 0; 0; 0}= q
Is there exactly one quaternion per rotation?
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around –v
The quaternion of the identity rotation: qi = {1; 0; 0; 0}
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Rotation of –
No.
q={cos( /2); v sin( /2)} is the same rotation as
-q={cos(( +2 )/2); v sin(( +2 )/2)}
The inverse q-1 of quaternion q {a, b, c, d}: {a, -b, -c, -d}
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qq-1 = {1; 0; 0; 0}
Quaternion Algebra
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Multiplication:
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http://www.genesis3d.com/~kdtop/QuaternionsUsingToRepresentRotation.htm
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Quaternions are associative:
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But not commutative:
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Rotation Example
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To rotate 3D point/vector p by a rotation q, compute:
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Compose Rotations
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Quaternion Interpolation
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A quaternion is a point on a 4D unit sphere
Unit quaternion: q=(s,x,y,z), ||q|| = 1
Interpolating rotations means moving on 4D sphere
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Quaternion Interpolation
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Linear interpolation generates unequal spacing of
points after projecting to circle
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Quaternion Interpolation
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Spherical linear interpolation (slerp)
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interpolats along the arc lines instead of the secant lines.
produce equal increment along arc connecting two quaternions
on the spherical surface
Useful Analogies
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Euclidean Space
Position
Linear interpolation
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• 4D Spherical Space
• Orientation
• Spherical linear
interpolation
(slerp)