Fundamentals of Computer Animation

University of Victoria Graphics Lab. CSC 473 2010 page 1

Fundamentals ofComputer Animation

Object Deformation

• Many objects are not rigid– jello– mud– gases/liquids– etc.

• Two main techniques:– Geometric deformations– Physically-based methods

Object Deformation

• Assume general geometric objects• Morphing one object into another

– If the same number of points, same edgeconnectivity (simple)

• Simple shape modifications– General affine transformation

• Non-uniform scale• Shearing• Warping

Geometric Deformations

• Deform the object’s geometry directly• Two main techniques:

– control point / vertex manipulation– space warping

Control Point /Vertex Manipulation

Edit the surface vertices or control points directly

Space Warping

Deform the object by deforming the space it is in

Two main techniques: Nonlinear Deformation Free Form Deformation (FFD)

Independent of object representation

Nonlinear Global Deformation

Objects are defined in a local object space

Deform this space using a combination of: Non-uniform Scaling Tapering Twisting Bending

Combine these to produce complex shapes

Can produce quite interesting results

Non-uniform Scale

Transformation matrix - diagonal elements

Non-uniform Scale

Good for modeling [Barr 87]

Animation is harder

Global Deformations

Transformation matrix elements - functions of coordinates

f(x,y,z) g(x,y,z)

Global Deformations - taper

x’ = xy’ = f(x)

x’y’

1 00 f(x)

=x’y’

Tapered ObjectOriginal Object

Global Deformations - taper

Global Deformations - twist

x’ = x*cos(f(y)) – z*sin(f(y))y’ = yz’ = x*sin(f(y)) + z*cos(f(y))

Global Deformations - twist

Global Deformations - bend

The Jacoboan MatrixThe Jacobian Matrix:

The Jacobian of a functionrepresents the tangent planeto the function at a given point

E.g. Transformation from spherical polar coordinates to cartesian

(x1,x2 ,x3)

(r,",#)

The gradient is the derivative of a scalarfunction. The Jacobian is the derivative of amultivariate function.

if (x2,y2) = f(x1,y1) is used to transform an image, the Jacobian of f, J(x1,y1)describes how much the image in the neighborhood of (x1,y1) is stretched in thex, y, and xy directions.

x1 = rsin" cos#x2 = rsin" sin#x3 = rcos"

' ' ' ' ' ' '

* * * * * * *

' ' ' ' ' ' '

* * * * * * *

sin# cos$ rcos# cos$ +rsin# sin$sin# sin$ rcos# sin$ rsin# cos$cos# +rsin# 0

Jacobian

For a square matrix the determinant is also called the Jacobian.The Jacobian determinant is used when making achange of variables when integrating a function over its domain.determinant of the above = r2 sin θ

Using Jacobian to find Gradient of Barr WarpGlobal twist given by:

X = xC" # yS"Y = xS" + yC"

" = f (z)C" = cos(")S" = sin(")

% % % % % % %

( ( ( ( ( ( (

C) *S) *zS) f '(z) * yS) f '(z)S) C) zC) f '(z) * yS) f '(z)0 0 1

gives the normal transformation matrix.Determinant is unity so the twisttransform preserves volume.

Tangent

J "1T =

C# "S# 0S# C# 0

yf '(z) "xf '(z) 1

Global Deformations - compound

Barr Deformations apply to implicits too

Implicits

Animating Shape Change

• Per-vertex animation curves– beginning and end known

• Simply change parameters with time– Twist angle– Scaling constant– Seed vertex position

• All standard rules apply– Curve smoothness, motion controls, etc.

Free Form Deformation (FFD)

Deform space by deforming a lattice around anobject

The deformation is defined by moving the controlpoints

Imagine it as if the object were encased in rubber

Grid Deformation

2D technique used in the film HUNGER

Overlay 2D grid on top of object

Map object vertices to grid cells (create local coordinate system)

User distorts 2D grid vertices

Object vertices are remapped to local coordinate systemof 2D grid by using bilinear interpolation

Grid Deformation

For each vertexIdentify cellLocal u,v coorindate

Grid Deformation

Coordinate grid deformation

2D Bilinear interpolation

P01P11

1110)1(01)1(00

)1()1()1(

PPPPvuP

vPPvPuPPuPuPPuP

uvvuvuuv

+!+!+!

u,v correspond to the coordinateson the rectangular grid before thelattice is deformed.

e.g. point at u=0.4 v=0.6

Grid Deformation

Free-Form Deformations

(not necessarily mutually perpendicular)

Define local coordinate system for deformation

As for Burtnyk technique but higher order. Typically cubic - Bezier in Sederberg original.Local grid over objects.f(object P) --> P’(grid vertices)user manipulates grid changes coordinates of object.

FFD - register point in cell

))/(())(( 0 SUTPPUTsrrrrrrr

!"!=P(s,t,u) e.g.determined

vertex P

• Local coord system: (S,T,U)• Point P coordinate along S:

• Equiv. for T,U• Algorithm:

– Introduce fine grid– Deform grid points– Use Bezier interpolation (or other) to get new position

• Treat new grid points as control points

))/(())(( 0 SUTPPUTsrrrrrrr

FFD - register point in cell

s = (TxU) . (P-P0) / ((TxU) . S)

((TxU) . S) (TxU) . (P-P0)

P = P0 + sS + tT + uU

s = (TxU).(P-P0) / ((TxU).S)t = (UxS).(P-P0) / ((UxS).T)u = (SxT) .(P-P0) / ((SxT).U)

Essentially this is the projection of S onto TxU.

0<=s<=1 is the proportion of the range that the point will map to.

Could be non-cubic e.g. S,U,T 4x3x2 I x j x kPijk=P0 +(i/l) S + (j/m )T + (k/m)U0<=i<=l 0 <=j<=m 0<=k<=n

P151 text bookinterpolation function as per Bezier - Bernstein

Non-Cubic Lattice

FFD - create control grid

(not necessarily mutually perpendicular)

FFD - move and reposition

Move control grid points

Usually tri-cubic interpolation is used with FFDs

Originally Bezier interpolation was used.

B-spline and Catmull-Romm interpolation havealso been used (as well as tri-linear interpolation)(see Parent Animation Book Appendix )

FFD - extensions

Hierarchical FFDs

Animated FFD

Static FFD that object moves through

Non-parallelpiped FFD

FFD - films and videos

examples

Boppin’ in Bean Town by John Chadwick

Facit demo by Beth Hofer

Balloon Guy by Chris Wedge

Bezier solids

• Bezier curves: 1D entity• Bezier patches: 2D

– Have a surface connectivity control mesh– Repeat DeCasteljau twice, get a point

• Bezier solids (3D): Same, but three times• Each (s,t,u) point can be written through

control points– Cubic polynomial in s, t, u (for cubic Bezier)

• New position:– same expression, same s,t,u– new c.p. positions

The lattice defines a Bezier volume

ijk wBvBuBwvu )()()(),,( pQ

Compute lattice coordinates

Alter the control points

Compute the deformed points

),,( wvu

),,( wvuQ

),,( wvu

FFD Example

Animation with FFD

• Hierarchical FFD– Coarse level FFD modifies vertices and finer FFD grids

• Moving object through deformation tool– Can move the tool itself

• Modifying control points of FFD– Any technique applies

• Key frame• Physics based

• Example:– FFD is relative to wire skeleton– Moves of skeleton re-position FFD grid– Skin position is computed within new FFD

Ad-hoc muscle

Animate a reference and a deformed lattice

reference deformed morphed

FFD Animation

Animate the object through the lattice

reference deformed morphed

Factor Curves

Modify the transformation applied to the object based on where andwhen it is applied

)()(0 tfwf tw!! =

Factor Curves

Scripted animation can lead to complex motions(depending on animator skill)

Deformations can be nested

Show animation of :1. Deform control points over time2. Move Object through deformed space3. Deformation over time

Sample animations

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