CPSC 441: Keyframe AnimationSmooth Curves Cont'

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CPSC 441 Keyframe Animation/Smooth Curves
(Cont.)

• Jinxiang Chai

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Key-frame Interpolation

• Given parameter values at key frames, how to
  interpolate parameter values for inbetween frames.

?
t
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Key-frame Interpolation

• Given parameter values at key frames, how to
  interpolate parameter values for inbetween frames.

?
t
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Key-frame Interpolation

• Given parameter values at key frames, how to
  interpolate parameter values for inbetween frames.

?
t
Nonlinear interpolation
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Review Natural cubic cruves
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Review Natural cubic cruves
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Review Natural cubic cruves
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Review Natural cubic curves
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Review Hermite Curves
P1 start position P4 end position R1 start
derivative R4 end derivative
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Review Hermite Curves
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Review Hermite Curves
Herminte basis matrix
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Review Hermite Curves
Herminte basis matrix
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Review Hermite Curves
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Review Hermite Curves
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Review Hermite Curves
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Review Hermite Curves
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Review Hermite Curves
Hermite basis functions
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Review Hermite Curves
basis function 1
basis function 1
basis function 1
basis function 1
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Review Hermite Curves
R1
R4
P1
P4




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Review Bezier Curves
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Review Bezier Curves
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Review Bezier Curves
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Review Bezier Curves
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Review Bezier Curves
v2
v3
v1
v0




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Computer animation

• Animation
• - making objects moving
• Compute animation
• - the production of consecutive images, which,
  when displayed, convey a feeling of motion.

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Review Different basis functions

• Cubic curves
• Hermite curves
• Bezier curves

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Complex curves

• Suppose we want to draw a more complex curve

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Complex curves

• Suppose we want to draw a more complex curve

How can we represent this curve?
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Complex curves

• Suppose we want to draw a more complex curve

• Idea well splice together a curve from
  individual segments that are cubic Béziers

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Complex curves

• Suppose we want to draw a more complex curve

• Idea well splice together a curve from
  individual segments that are cubic Béziers

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Splines

• A piecewise polynomial that has a locally very
  simple form, yet be globally flexible and smooth

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Splines

• There are three nice properties of splines wed
  like to have
• - Continuity
• - Local control
• - Interpolation

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Continuity

• C0 points coincide, velocities dont
• C1 points and velocities coincide
• Whats C2?
• - points, velocities and accelerations coincide

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Continuity

• Cubic curves are continuous and differentiable
• We only need to worry about the derivatives at
  the endpoints when two curves meet

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Local control

• Wed like our spline to have local control
• - that is, have each control point affect some
  well-defined neighborhood around that point

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Local control

• Wed like our spline to have local control
• - that is, have each control point affect some
  well-defined neighborhood around that point

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Local control

• Wed like our spline to have local control
• - that is, have each control point affect some
  well-defined neighborhood around that point

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Interpolation

• Bézier curves are approximating
• - The curve does not (necessarily) pass
  through all the control points
• - Each point pulls the curve toward it, but
  other points are pulling as well
• Instead, we may prefer a spline that is
  interpolating
• - That is, that always passes through every
  control point

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B-splines

• We can join multiple Bezier curves to create
  B-splines
• Ensure C2 continuity when two curves meet

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Derivatives at end points
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Derivatives at end points
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Derivatives at end points
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Derivatives at end points
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Derivatives at end points
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Derivatives at end points
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Continuity in B splines

• Suppose we want to join two Bezier curves (V0,
  V1, V2,V3) and (W0, W1, W2, W3) so that C2
  continuity is met at the joint

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Continuity in B splines

• Suppose we want to join two Bezier curves (V0,
  V1, V2,V3) and (W0, W1, W2, W3) so that C2
  continuity is met at the joint

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Continuity in B splines

• Suppose we want to join two Bezier curves (V0,
  V1, V2,V3) and (W0, W1, W2, W3) so that C2
  continuity is met at the joint

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Continuity in B splines

• Suppose we want to join two Bezier curves (V0,
  V1, V2,V3) and (W0, W1, W2, W3) so that C2
  continuity is met at the joint

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Continuity in B splines

• Suppose we want to join two Bezier curves (V0,
  V1, V2,V3) and (W0, W1, W2, W3) so that C2
  continuity is met at the joint

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Continuity in B splines

• Suppose we want to join two Bezier curves (V0,
  V1, V2,V3) and (W0, W1, W2, W3) so that C2
  continuity is met at the joint

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Continuity in B splines

• What does this derived equation mean
  geometrically?
• - What is the relationship between a, b and c,
  if a 2b - c?

w2v14v3-4v2
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de Boor points

• Instead of specifying the Bezier control points,
  lets specify the corners of the frames that
  forms a B-spline
• These points are called de Boor points and the
  frames are called A-frames

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de Boor points

• What is the relationship between Bezier control
  points and de Boor points?

Verify this by yourself!
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B spline basis matrix
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Building complex splines

• Constraining a Bezier curve made of many segments
  to be C2 continuous is a lot of work
• - for each new segment we have to add 3 new
  control point
• - only one of the control points is really
  free
• B-splines are easier (and C2)
• - First specify 4 vertices (de Boor points),
  then one per segment

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B splines properties

• v Continuity
• v Local control
• x Interpolation

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Catmull-Rom splines

• If we are willing to sacrifice C2 continuity, we
  can get interpolation and local control.
• If we set each derivative to be a constant
  multiple of the vector between the previous and
  the next controls, we get a Catmull-Rom spline

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Catmull-Rom splines
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Catmull-Rom splines
The segment is controlled by p1,p2,p3,p4
0.5
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Catmull-Rom splines
The segment is controlled by p1,p2,p3,p4
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Catmull-Rom splines

• The effect of t how sharply the curve bends at
  the control points

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Catmull-Rom splines
The segment is controlled by p1,p2,p3,p4
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Catmull-Rom splines
From Hermite curves
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Catmull-Rom splines
From Hermite curves
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Catmull-Rom splines
From Hermite curves
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Catmull-Rom splines
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Catmull-Rom splines
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Catmull-Rom splines
What do we miss?
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Catmull-Rom splines
?
?
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Catmull-Rom splines
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Catmull-Rom splines

• Catmull-Rom splines have C1 continuity (not C2
  continuity)
• Do not lie within the convex hull of their
  control points.

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Catmull-Rom splines

• Catmull-Rom splines have C1 continuity (not C2
  continuity)
• Do not lie within the convex hull of their
  control points.

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Catmull-Rom splines properties

• X Continuity (C2)
• v Local control
• v Interpolation

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Curves in the animation

• Interpolate each parameter separately
• Each animation parameter is described by a 2D
  curve
• Q(u) (x(u), y(u))
• Treat this curve as
• ?(u) y(u)
• t(u) x(u)
• where ? is a variable you want to animate. We can
• think of the results as a function ?(t)

?(u)
t(u)
u
u
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Keyframing interpolation

• is a two-step algorithm
• - Compute the spline parameter u corresponding
  to the time t

x(u)
u
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Keyframing interpolation

• is a two-step algorithm
• - Compute the spline parameter u corresponding
  to the time t
• how to evaluate the inverse function?

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Keyframing interpolation

• is a two-step algorithm
• Compute the spline parameter u corresponding to
  the time t
• how to evaluate the inverse function?
• - linear function easy (tu)
• - complex function finite difference or
  numerically root finding

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Keyframing interpolation

• is a two-step algorithm
• Compute the spline parameter u corresponding to
  the time t
• how to evaluate the inverse function?
• - linear function easy
• - complex function finite difference or
  numerically root finding
• Compute the ? corresponding to the spline
  parameter u
• ? y(u)

y(u)
u
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Outline

• Process of keyframing
• Key frame interpolation
• Hermite and bezier curve
• Splines
• Speed control

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Spline parameterization

• The spline is parameterized by u (0ltult1) and
  not by time t.

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Spline parameterization

• The spline is parameterized by u (0ltult1) and
  not by time t.
• Hard to deal with the questions like
• What is the location of point p at time t
• What is the velocity of point p at time t
• What is the acceleration a of a point p at time t

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Spline parameterization

• Solution reparameterize the curve in terms of t
• - express spline as a function of the arc
  length s ug(s)
• - express the arc length s as a function of t
  sf(t)

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Speed control

• Simplest form is to have constant velocity along
  the path

sf(t)
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Arc-length reparameterization

• Now we want a way to find u given a particular
  arclength s ug(f(t)
• Not possible analytically for most curves (e.g.
  B-splines)

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Finite difference

• Sample the curve at small intervals of the
  parameter and determine the distance between
  samples
• Use theses distances to build a table of
  arclength for this particular curve (ui,si)

Whats parametric value for the arc length .08?
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Finite difference

• Sample the curve at small intervals of the
  parameter and determine the distance between
  samples
• Use theses distances to build a table of
  arclength for this particular curve (ui,si)

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Finite difference

• Sample the curve at small intervals of the
  parameter and determine the distance between
  samples
• Use theses distances to build a table of
  arclength for this particular curve (ui,si)

Whats parametric value for the arc length .08?
89
Finite difference

• Sample the curve at small intervals of the
  parameter and determine the distance between
  samples
• Use theses distances to build a table of
  arclength for this particular curve (ui,si)

Whats parametric value for the arc length .08?
90
Finite difference

• Sample the curve at small intervals of the
  parameter and determine the distance between
  samples
• Use theses distances to build a table of
  arclength for this particular curve (ui,si)

Whats parametric value for the arc length .05? A
linear interpolation of 0.00 and 0.05
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Speed control recipe

• Given a time t, lookup the corresponding
  arclength S in the speed curve
• For S, look up the corresponding value of u in
  the reparameterization table
• Evaluate the curve at u to obtain the correct
  interpolated position for the animated object for
  the given time t