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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

An Introduction to Computer Animation

Michiel van de Panne


Overview
Overview

  • Creating Animations

  • Representing Rotations


1 creating animations
(1) Creating Animations

algorithm

(parameterized models,physics, optimization)

user

data

(motion capture)

(keyframes)


Representing motion
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


2 representing rotations

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
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
Rotation DOFs

  • 2D: 1 DOF

  • 3D: 3 DOF

  • 4D: 6 DOF


Transformations
Transformations

  • Rotation

glRotatef(angle,x,y,z);

glRotated(angle,x,y,z);



3x3 rotation matrix1
3x3 Rotation Matrix

  • 9 elements

  • 6 constraints

  • renormalization algorithms

  • extracting pure rotational component (polar decomp)

... and determinant = 1


Rotations
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


Rotations1
Rotations

  • orientation vs rotation?

  • how to specify?

  • how to interpolate?

  • 2D vs 3D


Fixed angle representations

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
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
Quaternions

  • review of complex numbers

  • quaternions


Quaternions1
Quaternions

  • rotation of a vector

  • two successive rotations


Quaternions2
Quaternions

RH rule

  • unit quaternions

  • addition

  • multiplication