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KIPA Game Engine Seminars

Jonathan Blow

Ajou University

December 13, 2002

Day 16

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Rotation in 2D

• Multiplication of complex numbers

• Complex number representation in polar coordinates (magnitude, angle)

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Euler’s Formula

• eiθ = cosθ + i sinθ

• Looking at the derivative of each side

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All rotations in 3D can be expressed as rotation by an angle

around an axis• Definition of the axis as an eigenvector of the

rotation

• Proof that in 3D all transforms have 1 or 3 (real) eigenvalues

• So we have 1 or 3 axes of rotation• 3 axes means it’s the identity rotation (maybe with

scaling)

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Quaternion

• As an axis/angle representation

• Re-derivation of qvq*

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Mathematical demonstrationthat the expression qvq* rotates

the vector v

• Work out the terms, breaking v into components parallel and orthogonal to the axis

• Show what qvq* does to each of these components

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Multiplying by a Non-Unit Quaternion

• Scales the vector

• (demonstration of this)

• (analogy to 2D case)

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Quaternion Powers

• What is q2?

• What is qt?

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SLERP

• As q0(1-t)q1

t

• Rewrite this as q0(q0-1q1)t

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Exponential Formof a Quaternion

• cosθ + y sinθ ; y is axis of rotation

• eyθ

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The Log of a Quaternion

• log eyθ = yθ

• But be careful, the additive log identity doesn’t work here!– It requires commutative multiplication– Demonstration of this on whiteboard– The identity does work if the axes are colinear

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The Exponential Map

• Is the log of a quaternion– Actually it can also mean the log of an arbitrary

transformation matrix, but here we are restricted to quaternions

• We can interpolate in this space• What are the properties of this

interpolation?

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The Exponential Map

• It interpolates at constant speed, but it is not the same function as slerp

• Therefore it cannot possibly follow the shortest path of interpolation– (otherwise they’d be the same function!)

• Indeed the further away the source orientations, the wider the path used by the exponential map

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The Exponential Map• But sometimes you care most about

constant speed, so the exponential map works well for that

• Provides a method of linearization that’s different from the linear interpolation in quaternion space discussed last week

• Also apparently there’s lots of other math you can do with these, which I don’t know, read some advanced robotics papers to find out

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Simple Cases For Quaternions

• qxq*

• qxq* dot x

• Convert quaternion to rotation matrix

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Discussion of my quaternion IK work

• (demo of app)

• People generally use Euler angles for IK but they are slow and unintuitive

• We can use quaternions instead to build a fast system