An introduction to computer animation
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An Introduction to Computer Animation. Michiel van de Panne. Overview. Creating Animations Representing Rotations. (1) Creating Animations. algorithm. (parameterized models, physics, optimization). user. data. (motion capture). (keyframes). Representing motion. DOF vs time

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An Introduction to Computer Animation

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An Introduction to Computer Animation

Michiel van de Panne


Overview

  • Creating Animations

  • Representing Rotations


(1) Creating Animations

algorithm

(parameterized models,physics, optimization)

user

data

(motion capture)

(keyframes)


Representing motion

  • DOF vs time

  • cubic polynomial curves

x

smooth

t

fast

in-out

linear

  • alternative for motion through space:

y

+

v

apply arc-length

reparameterization

t

x


world

torso

RUarm

RLarm

LUarm

Rhand

LLarm

Lhand

RUleg

RLleg

LUleg

LLleg

Rfoot

Lfoot

head

trans(0.30,0,0) rot(z, )

(2) Representing Rotations


Transformation Hierarchies

glTranslate3f(x,y,0);

glRotatef( ,0,0,1);

DrawBody();

glPushMatrix();

glTranslate3f(0,7,0);

DrawHead();

glPopMatrix();

glPushMatrix();

glTranslate(2.5,5.5,0);

glRotatef( ,0,0,1);

DrawUArm();

glTranslate(0,-3.5,0);

glRotatef( ,0,0,1);

DrawLArm();

glPopMatrix();

... (draw other arm)

  • Example

y

x


Rotation DOFs

  • 2D: 1 DOF

  • 3D: 3 DOF

  • 4D: 6 DOF


Transformations

  • Rotation

glRotatef(angle,x,y,z);

glRotated(angle,x,y,z);


3x3 Rotation Matrix


3x3 Rotation Matrix

  • 9 elements

  • 6 constraints

  • renormalization algorithms

  • extracting pure rotational component (polar decomp)

... and determinant = 1


Rotations

  • SO(3)

    • rotations do not commute

    • require at least 4 parameters for a smooth parameterization

      • analogy: surface of the earth

        • 2D surface, 3 params

    • combing the hairy ball

      • camera orientation: view object from any dir


Rotations

  • orientation vs rotation?

  • how to specify?

  • how to interpolate?

  • 2D vs 3D


pitch

yaw

roll

Fixed Angle Representations

  • fixed sequence of 3 rotations

    • RPY orientation:

  • can use many ordering of axes

  • Euler angles:


Euler’s Rotation Theorem

  • can always go from one orientation to anotherwith one rotation about a single axis

180 deg about line in xy-plane

where


Quaternions

  • review of complex numbers

  • quaternions


Quaternions

  • rotation of a vector

  • two successive rotations


Quaternions

RH rule

  • unit quaternions

  • addition

  • multiplication


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