CPSC 441 Computer Graphics: Keyframe AnimationSmooth Curves

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CPSC 441 Computer Graphics Keyframe
Animation/Smooth Curves

• Jinxiang Chai

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Outline

• Keyframe interpolation
• Curve representation and interpolation
• - natural cubic curves
• - Hermite curves
• - Bezier curves
• Required readings HB 8-8,8-9, 8-10

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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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Animation topics

• Rigid body simulation
• - bouncing ball
• - millions of chairs falling down

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Animation topics

• Rigid body simulation
• - bouncing ball
• - millions of chairs falling down
• Natural phenomenon
• - water, fire, smoke, mud, etc.

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Animation topics

• Rigid body simulation
• - bouncing ball
• - millions of chairs falling down
• Natural phenomenon
• - water, fire, smoke, mud, etc.
• Character animation
• - articulated motion, e.g. full-body animation
• - deformation, e.g. face

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Animation topics

• Rigid body simulation
• - bouncing ball
• - millions of chairs falling down
• Natural phenomenon
• - water, fire, smoke, mud, etc.
• Character animation
• - articulated motion, e.g. full-body animation
• - deformation, e.g. face
• Cartoon animation

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Animation criterion

• Physically correct
• - rigid body-simulation
• - natural phenomenon
• Natural
• - character animation
• Expressive
• - cartoon animation

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Animation criterion

• Physically correct
• - rigid body-simulation
• - natural phenomenon
• Natural
• - character animation
• Expressive
• - cartoon animation

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Keyframe animation
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Outline

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

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Oldest keyframe animation

• Two conditions to make moving images in 19th
  century
• at least 10 frames per second
• a period of blackness between images

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2D Animation

• Highly skilled animators draw the key frames
• Less skilled (lower paid) animators draw the
  in-between frames
• Time consuming process
• Difficult to create physically realistic
  animation

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3D Animation

• Animators specify important key frames in 3D
• Computers generates the in-between frames
• Some dynamic motion can be done by computers
  (hair, clothes, etc)
• Still time consuming Pixar spent four years to
  produce Toy Story

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The Process of Keyframing

• Specify the keyframes
• Specify the type of interpolation
• - linear, cubic, parametric curves
• Specify the speed profile of the interpolation
• - constant velocity, ease-in-ease-out, etc
• Computer generates the in-between frames

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A Keyframe

• In 2D animation, a keyframe is usually a single
  image
• In 3D animation, each keyframe is defined by a
  set of parameters

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Keyframe parameters

• What are the parameters?
• position and orientation
• body deformation
• facial features
• hair and clothing
• lights and cameras

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Outline

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

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Inbetween frames

• Linear interpolation
• Cubic curve interpolation

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Linear interpolation

• Linearly interpolate the parameters between
  keyframes

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Cubic curve interpolation

• We can use three cubic functions to represent a
  3D curve
• Each function is defined with the range 0 lt t
  lt1

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Compact representation
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Compact representation
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Smooth Curves

• Controlling the shape of the curve

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Smooth Curves

• Controlling the shape of the curve

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Smooth Curves

• Controlling the shape of the curve

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Smooth Curves

• Controlling the shape of the curve

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Smooth Curves

• Controlling the shape of the curve

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Smooth Curves

• Controlling the shape of the curve

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Constraints on the cubics

• How many constraints do we need to determine a
  cubic curve?

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Constraints on the cubics

• How many constraints do we need to determine a
  cubic curve?

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Constraints on the cubic functions

• How many constraints do we need to determine a
  cubic curve?

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Constrains on the cubic functions

• How many constraints do we need to determine a
  cubic curve?

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Constrains on the cubic functions

• How many constraints do we need to determine a
  cubic curve? 4

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Natural cubic curves
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Interpolation

• Find a polynomial that passes through specified
  values

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Interpolation

• Find a polynomial that passes through specified
  values

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Interpolation

• Find a polynomial that passes through specified
  values

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Interpolation

• Find a polynomial that passes through specified
  values

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Interpolation

• Find a polynomial that passes through specified
  values

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Interpolation

• Perform interpolation for each component
  separately
• Combine result to obtain parametric curve

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Interpolation

• Perform interpolation for each component
  separately
• Combine result to obtain parametric curve

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Interpolation

• Perform interpolation for each component
  separately
• Combine result to obtain parametric curve

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Constraints on the cubic curves

• How many constraints do we need to determine a
  cubic curve? 4
• - does not provide local control of the curve

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Constraints on the cubic curves

• How many constraints do we need to determine a
  cubic curve? 4
• - does not provide local control of the curve
• Redefine C as a product of the basis matrix M and
  the control vector G C MG

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Constraints on the cubic curves

• How many constraints do we need to determine a
  cubic curve? 4
• - does not provide local control of the curve
• Redefine C as a product of the basis matrix M and
  the control vector G C MG

MG
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Constraints on the cubic curves

• How many constraints do we need to determine a
  cubic curve? 4
• - does not provide local control of the curve
• Redefine C as a product of the basis matrix M and
  the control vector G C MG

M
G
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Constraints on the cubic curves

• How many constraints do we need to determine a
  cubic curve? 4
• - does not provide local control of the curve
• Redefine C as a product of the basis matrix M and
  the control vector G C MG

M?
G?
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Outline

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

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Hermite curve

• A Hermite curve is determined by
• - endpoints P1 and P4
• - tangent vectors R1 and R4 at the
  endpoints

R1
P4
R4
P1
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Hermite curve

• A Hermite curve is determined by
• - endpoints P1 and P4
• - tangent vectors R1 and R4 at the
  endpoints
• Use these elements to control the curve, i.e.
  construct control vector

R1
P4
R4
P1
Mh
Gh
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Hermite basis matrix

• Given desired constraints
• - endpoints meet P1 and P4
• Q(0) 0 0 0 1 Mh Gh P1
• Q(1) 1 1 1 1 Mh Gh P4
• - tangent vectors meet R1 and R4

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Tangent vectors
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Tangent vectors
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Hermite basis matrix

• Given desired constraints
• - endpoints meet P1 and P4
• Q(0) 0 0 0 1 Mh Gh P1
• Q(1) 1 1 1 1 Mh Gh P4
• - tangent vectors meet R1 and R4
• Q(0) 0 0 1 0 Mh Gh R1
• Q(1) 3 2 1 0 Mh Gh R4

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Hermite basis matrix

• Given desired constraints
• - endpoints meet P1 and P4
• Q(0) 0 0 0 1 Mh Gh P1
• Q(1) 1 1 1 1 Mh Gh P4
• - tangent vectors meet R1 and R4
• Q(0) 0 0 1 0 Mh Gh R1
• Q(1) 3 2 1 0 Mh Gh R4

So how to compute the basis matrix Mh?
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Hermite basis matrix

• We can solve for basis matrix Mh

Mh
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Hermite basis matrix

• We can solve for basis matrix Mh

Mh
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Hermite basis matrix
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Hermite basis function

• Lets define B as a product of T and M
• Bh(t) indicates the weight of each element in Gh

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Bezier curve

• Indirectly specify tangent vectors by specifying
  two intermediate points

R1
P2
P3
P4
P1
R2
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Bezier curve

• Indirectly specify tangent vectors by specifying
  two intermediate points

R1
P2
P3
P4
P1
R4
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Bezier curve

• Indirectly specify tangent vectors by specifying
  two intermediate points

R1
P2
P3
P4
P1
R4
How to compute the basis matrix Mb?
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Bezier basis matrix

• Establish the relation between Hermite and Bezier
  control vectors

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Bezier basis matrix

• Establish the relation between Hermite and Bezier
  control vectors

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Bezier basis matrix

• Establish the relation between Hermite and Bezier
  control vectors

Mhb
Gb
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Bezier basis matrix

• For Hermite curves, we have
• For Bezier curves, we have

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Bezier basis matrix

• For Hermite curves, we have
• For Bezier curves, we have

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Bezier basis matrix
P2
P1
P4
P3
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Hermite basis function

• Lets define B as a product of T and M
• Bh(t) indicates the weight of each element in Gh

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Hermite basis function

• Lets define B as a product of T and M
• Bh(t) indicates the weight of each element in Gh

Whats function of this red curve?
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Hermite basis function

• Lets define B as a product of T and M
• Bh(t) indicates the weight of each element in Gh

Whats function of this red curve? 2t3-3t21
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Bezier basis functions

• Bezier blending functions are also called
  Bernstein polynomials

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Bezier basis functions

• Bezier blending functions are also called
  Bernstein polynomials

Whats function of this red curve?
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Bezier basis functions

• Bezier blending functions are also called
  Bernstein polynomials

Whats function of this red curve? -t33t2-3t1
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Bezier basis functions

• Bezier blending functions are also called
  Bernstein polynomials

Whats function of this red curve? -t33t2-3t1
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How to interpolate a 3D curve
y
x
o
z
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How to interpolate a 3D curve
y
x
o
z
Bezier curve
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Bezier java applet

• Try this online at
• - Move the interpolation point, see how the
  others (and the point on curve) move
• - Control points (can even make loops)

http//www.cse.unsw.edu.au/lambert/splines/
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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
• Why not use a high-order Bézier?
• - Wiggly curves
• - No local control

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Complex curves

• Suppose we want to draw a more complex curve
• Why not use a high-order Bézier?
• - Wiggly curves
• - No local control

• Instead, 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
• Why not use a high-order Bézier?
• - Wiggly curves
• - No local control

• Instead, 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
• Why not use a high-order Bézier?
• - Wiggly curves
• - No local control

• Instead, 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
• Why not use a high-order Bézier?
• - Wiggly curves
• - No local control

• Instead, well splice together a curve from
  individual segments that are cubic Béziers
• Why cubic?
• - Lowest dimension with control for the
  second derivative
• - Lowest dimension for non-planar
  polynomial curves

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Next lecture

• Spline curve and more key frame interpolation