Thank you all who attended - (course notes are here)

Today

Computer Geometry
Coordinate Free
Projective Geometry
Geometric Numbers
Geometric Algebra
2D PGA
3D PGA

Computer Geometry

VECTOR AND MATRIX ALGEBRA

natural choice for ANALYTIC GEOMETRY
describe geometric objects with measurements
one bag of ℝ for each measurement
formulas, transformations describe what happens to the measurements.

PROJECTIVE GEOMETRIC ALGEBRA

natural choice for ``SYNTHETIC´´ GEOMETRY
new bag, each element in it IS a geometric object
formulas, transformations describe what happens to the objects
no coordinates, chirality, exceptions, conversions

\mathbb R^n

\mathbb R^n

\mathbb R^*_{n,0,1}

\mathbb R^*_{n,0,1}

Computer Geometry

ANALYTIC GEOMETRY : VECTOR AND MATRIX ALGEBRA

\mathbb R^n

\mathbb R^n

x = a \cos \alpha + b \cos \beta \cos \alpha

x = a \cos \alpha + b \cos \beta \cos \alpha

y = a \sin \alpha + b \cos \beta \sin \alpha

y = a \sin \alpha + b \cos \beta \sin \alpha

z = b \sin \beta

z = b \sin \beta

Computer Geometry

ANALYTIC GEOMETRY : VECTOR AND MATRIX ALGEBRA

SYNTHETIC GEOMETRY : PROJECTIVE GEOMETRIC ALGEBRA

\mathbb R^n

\mathbb R^n

\mathbb R^*_{n,0,1}

\mathbb R^*_{n,0,1}

x = a \cos \alpha + b \cos \beta \cos \alpha

x = a \cos \alpha + b \cos \beta \cos \alpha

y = a \sin \alpha + b \cos \beta \sin \alpha

y = a \sin \alpha + b \cos \beta \sin \alpha

z = b \sin \beta

z = b \sin \beta

{p} =\, \,e^{\beta_{xz}}\,\,\,\,e^{b_x}\,\,\,\,e^{\alpha_{xy}}\,\,\,\,e^{a_x}\,\,\,\,\mathbf{o}

{p} =\, \,e^{\beta_{xz}}\,\,\,\,e^{b_x}\,\,\,\,e^{\alpha_{xy}}\,\,\,\,e^{a_x}\,\,\,\,\mathbf{o}

Computer Geometry

ANALYTIC GEOMETRY : VECTOR AND MATRIX ALGEBRA

SYNTHETIC GEOMETRY : PROJECTIVE GEOMETRIC ALGEBRA

\mathbb R^n

\mathbb R^n

\mathbb R^*_{n,0,1}

\mathbb R^*_{n,0,1}

x = a \cos \alpha + b \cos \beta \cos \alpha

x = a \cos \alpha + b \cos \beta \cos \alpha

y = a \sin \alpha + b \cos \beta \sin \alpha

y = a \sin \alpha + b \cos \beta \sin \alpha

z = b \sin \beta

z = b \sin \beta

{p} =\, \,\curvearrowleft^{e^{\beta_{xz}}}\,\rightarrow^{e^{b_x}}\,\curvearrowleft^{e^{\alpha_{xy}}}\,\rightarrow^{e^{a_x}}\,\bullet

{p} =\, \,\curvearrowleft^{e^{\beta_{xz}}}\,\rightarrow^{e^{b_x}}\,\curvearrowleft^{e^{\alpha_{xy}}}\,\rightarrow^{e^{a_x}}\,\bullet

synthetic : tools are ruler $\rightarrow$ and compass $\curvearrowleft$

elements are points, lines, planes, ..

Computer Geometry

ANALYTIC GEOMETRY : VECTOR AND MATRIX ALGEBRA

SYNTHETIC GEOMETRY : PROJECTIVE GEOMETRIC ALGEBRA

\mathbb R^n

\mathbb R^n

\mathbb R^*_{n,0,1}

\mathbb R^*_{n,0,1}

x = a \cos \alpha + b \cos \beta \cos \alpha

x = a \cos \alpha + b \cos \beta \cos \alpha

y = a \sin \alpha + b \cos \beta \sin \alpha

y = a \sin \alpha + b \cos \beta \sin \alpha

z = b \sin \beta

z = b \sin \beta

{p} =\, \,\curvearrowleft^{e^{\beta_{xz}}}\,\rightarrow^{e^{b_x}}\,\curvearrowleft^{e^{\alpha_{xy}}}\,\rightarrow^{e^{a_x}}\,\bullet

{p} =\, \,\curvearrowleft^{e^{\beta_{xz}}}\,\rightarrow^{e^{b_x}}\,\curvearrowleft^{e^{\alpha_{xy}}}\,\rightarrow^{e^{a_x}}\,\bullet

synthetic : tools are ruler $\rightarrow$ and compass $\curvearrowleft$

elements are points, lines, planes, ..

Computer Geometry

ANALYTIC GEOMETRY : VECTOR AND MATRIX ALGEBRA

SYNTHETIC GEOMETRY : PROJECTIVE GEOMETRIC ALGEBRA

\mathbb R^n

\mathbb R^n

\mathbb R^*_{n,0,1}

\mathbb R^*_{n,0,1}

x = a \cos \alpha + b \cos \beta \cos \alpha

x = a \cos \alpha + b \cos \beta \cos \alpha

y = a \sin \alpha + b \cos \beta \sin \alpha

y = a \sin \alpha + b \cos \beta \sin \alpha

z = b \sin \beta

z = b \sin \beta

{p} =\, \,\curvearrowleft^{e^{\beta_{xz}}}\,\rightarrow^{e^{b_x}}\,\curvearrowleft^{e^{\alpha_{xy}}}\,\rightarrow^{e^{a_x}}\,\bullet

{p} =\, \,\curvearrowleft^{e^{\beta_{xz}}}\,\rightarrow^{e^{b_x}}\,\curvearrowleft^{e^{\alpha_{xy}}}\,\rightarrow^{e^{a_x}}\,\bullet

{

.. just as in topology .. circle times circle ..

Computer Geometry

ANALYTIC GEOMETRY : VECTOR AND MATRIX ALGEBRA

SYNTHETIC GEOMETRY : PROJECTIVE GEOMETRIC ALGEBRA

\mathbb R^n

\mathbb R^n

\mathbb R^*_{n,0,1}

\mathbb R^*_{n,0,1}

x = a \cos \alpha + b \cos \beta \cos \alpha + c \cos \gamma \cos \beta \cos \alpha

x = a \cos \alpha + b \cos \beta \cos \alpha + c \cos \gamma \cos \beta \cos \alpha

y = a \sin \alpha + b \cos \beta \sin \alpha + c \cos \gamma \cos \beta \sin \alpha

y = a \sin \alpha + b \cos \beta \sin \alpha + c \cos \gamma \cos \beta \sin \alpha

z = b \sin \beta + c \cos \gamma \sin \beta

z = b \sin \beta + c \cos \gamma \sin \beta

{p} =\, \,\curvearrowleft^{e^{\gamma_{xw}}}\,\rightarrow^{e^{c_x}}\,\curvearrowleft^{e^{\beta_{xz}}}\,\rightarrow^{e^{b_x}}\,\curvearrowleft^{e^{\alpha_{xy}}}\,\rightarrow^{e^{a_x}}\,\bullet

{p} =\, \,\curvearrowleft^{e^{\gamma_{xw}}}\,\rightarrow^{e^{c_x}}\,\curvearrowleft^{e^{\beta_{xz}}}\,\rightarrow^{e^{b_x}}\,\curvearrowleft^{e^{\alpha_{xy}}}\,\rightarrow^{e^{a_x}}\,\bullet

{

w = c \sin \gamma

w = c \sin \gamma

.. just as in topology .. circle times circle times circle ..

Computer Geometry

SYNTHETIC GEOMETRY : PROJECTIVE GEOMETRIC ALGEBRA

\mathbb R^*_{n,0,1}

\mathbb R^*_{n,0,1}

We want synthetic geometry.
Work with objects not measurements.
So how to write it down ? (..algebra..)
But first .. what do we want in our bag ?

GEOMETRIC OBJECTS : POINTS, LINES, PLANES, ROTATIONS, TRANSLATIONS, ...

The elements of Geometry - in our case Projective Geometry

NOTE : we will work first in 2D, but in a way that will generalize to nD.

Projective Geometry

Represent Euclidean points/lines/... in n-d using an (n+1)d space.

We embed a 2D plane in a 3D space and call it the projective plane.

Projective Geometry

Rotate around $\infty$ = Push at CoG

Rotate around CoG = Push at $\infty$

Rotate around CoG

Rotate around $\infty$

Push at $\infty$

Push at CoG

ALL FORCES ARE LINEAR, ALSO THE ANGULAR ONES, BOTH IN 2D AND 3D !

lines are special !

$\curvearrowleft$

$\uparrow$

$\curvearrowleft$

$\infty$

$\uparrow$

Projective Geometry

Represent Euclidean points/lines/... in n-d using an (n+1)d space.

all euclidean points,lines,planes, ..
all ideal (infinite) points,lines,planes, ..
robust, exception free incidence relations
translations are rotations around $\infty$
duality

Where are the formula's and numbers ? We have changed our difficult to write down arbitrary points and lines to slightly less difficult to write down entities through the origin. For lines through the origin we can use VECTORS, but lets think some more on the writing down .. enter Algebra.

Algebra

\,\,x^2 + 10x = 39\,\,

\,\,x^2 + 10x = 39\,\,

What picture do you get when you see this equation ?

Algebra

\,\,x^2 + 10x = 39\,\,

\,\,x^2 + 10x = 39\,\,

ax² = bx	ax² = c
bx = c	ax² + bx = c
ax² + c = bx	bx + c = ax²

Al-Khwarizmi's six problems

Today, one problem :

\quad\quad ax^2 + bx + c = 0\quad\quad

\quad\quad ax^2 + bx + c = 0\quad\quad

What was Al-Khwarizmi missing ?

no negative numbers, no zero

more numbers, less formulas, less code.

Algebra

What was Al-Khwarizmi missing ?

zero negative numbers

what are these crazy un-natural numbers ?

x \quad + \quad 1 \quad = \quad 0

x \quad + \quad 1 \quad = \quad 0

x \notin \mathbb N

x \notin \mathbb N

x^2 \in \mathbb N

x^2 \in \mathbb N

$\rightarrow$ They're not natural numbers !

$\rightarrow$ Yet they square to a natural number ??

Algebra

What was Al-Khwarizmi missing ?

zero negative numbers

maybe other numbers were hiding ?

x^2 \quad + \quad 1 \quad = \quad 0

x^2 \quad + \quad 1 \quad = \quad 0

x^2 = -1

x^2 = -1

\curvearrowleft

\curvearrowleft

x \notin \mathbb R

x \notin \mathbb R

e_- \notin \mathbb R, {e_-}^2=-1

e_- \notin \mathbb R, {e_-}^2=-1

Algebra

What was Al-Khwarizmi missing ?

zero negative numbers

maybe other numbers were hiding ?

x^2 \quad + \quad 1 \quad = \quad 0

x^2 \quad + \quad 1 \quad = \quad 0

x^2 = -1

x^2 = -1

\curvearrowleft

\curvearrowleft

x \notin \mathbb R

x \notin \mathbb R

e_- \notin \mathbb R, {e_-}^2=-1

e_- \notin \mathbb R, {e_-}^2=-1

Algebra

What was Al-Khwarizmi missing ?

zero negative numbers

maybe other numbers were hiding ?

x^2 \quad + \quad 1 \quad = \quad 0

x^2 \quad + \quad 1 \quad = \quad 0

x^2 = -1

x^2 = -1

\curvearrowleft

\curvearrowleft

x \notin \mathbb R

x \notin \mathbb R

e_- \notin \mathbb R, {e_-}^2=-1

e_- \notin \mathbb R, {e_-}^2=-1

e_+ \notin \mathbb R, {e_+}^2=1

e_+ \notin \mathbb R, {e_+}^2=1

Algebra

What was Al-Khwarizmi missing ?

zero negative numbers

maybe other numbers were hiding ?

x^2 \quad + \quad 1 \quad = \quad 0

x^2 \quad + \quad 1 \quad = \quad 0

x^2 = -1

x^2 = -1

\curvearrowleft

\curvearrowleft

x \notin \mathbb R

x \notin \mathbb R

e_- \notin \mathbb R, {e_-}^2=-1

e_- \notin \mathbb R, {e_-}^2=-1

e_+ \notin \mathbb R, {e_+}^2=1

e_+ \notin \mathbb R, {e_+}^2=1

Algebra

What was Al-Khwarizmi missing ?

zero negative numbers

maybe other numbers were hiding ?

x^2 \quad + \quad 1 \quad = \quad 0

x^2 \quad + \quad 1 \quad = \quad 0

x^2 = -1

x^2 = -1

\curvearrowleft

\curvearrowleft

x \notin \mathbb R

x \notin \mathbb R

e_+ \notin \mathbb R, {e_+}^2=1

e_+ \notin \mathbb R, {e_+}^2=1

e_0 \notin \mathbb R, {e_0}^2=0

e_0 \notin \mathbb R, {e_0}^2=0

BASIC INSIGHT : 0D,1D,2D,... numbers behave differently

e_- \notin \mathbb R, {e_-}^2=-1

e_- \notin \mathbb R, {e_-}^2=-1

Algebra

x^2 \quad + \quad 1 \quad = \quad 0

x^2 \quad + \quad 1 \quad = \quad 0

THE GEOMETRIC NUMBERS

these are of course the complex, hyperbolic and dual numbers. So why call them geometric numbers ? lets find out ..

e_- \notin \mathbb R, {e_-}^2=-1

e_- \notin \mathbb R, {e_-}^2=-1

e_+ \notin \mathbb R, {e_+}^2=1

e_+ \notin \mathbb R, {e_+}^2=1

e_0 \notin \mathbb R, {e_0}^2=0

e_0 \notin \mathbb R, {e_0}^2=0

Algebra - The Geometric Numbers

Addition and Scalar multiplication are trivial :

a lemon plus a lemon is almost always two lemons.

e_- + e_- = 2e_-

e_- + e_- = 2e_-

e_+ + e_+ = 2e_+

e_+ + e_+ = 2e_+

e_0 + e_0 = 2e_0

e_0 + e_0 = 2e_0

e_- \notin \mathbb R, {e_-}^2=-1

e_- \notin \mathbb R, {e_-}^2=-1

e_+ \notin \mathbb R, {e_+}^2=1

e_+ \notin \mathbb R, {e_+}^2=1

e_0 \notin \mathbb R, {e_0}^2=0

e_0 \notin \mathbb R, {e_0}^2=0

Algebra - The Geometric Numbers

Addition and Scalar multiplication are trivial :

a lemon plus a lemon is almost always two lemons.

e_0 + e_0 = 2e_0

e_0 + e_0 = 2e_0

But what about multiplication ? we'll just try ..

e_- + e_- = 2e_-

e_- + e_- = 2e_-

e_- \notin \mathbb R, {e_-}^2=-1

e_- \notin \mathbb R, {e_-}^2=-1

e_+ \notin \mathbb R, {e_+}^2=1

e_+ \notin \mathbb R, {e_+}^2=1

e_+ + e_+ = 2e_+

e_+ + e_+ = 2e_+

e_0 \notin \mathbb R, {e_0}^2=0

e_0 \notin \mathbb R, {e_0}^2=0

Algebra - The Geometric Numbers

e_- \notin \mathbb R, {e_-}^2=-1

e_- \notin \mathbb R, {e_-}^2=-1

e_+ \notin \mathbb R, {e_+}^2=1

e_+ \notin \mathbb R, {e_+}^2=1

e_0 \notin \mathbb R, {e_0}^2=0

e_0 \notin \mathbb R, {e_0}^2=0

(a + be_-)(c + de_-)

(a + be_-)(c + de_-)

We add $\mathbf e_-$ to our bag of numbers. Each element in it is now of the form (a + b $\mathbf e_-$ ).

Algebra - The Geometric Numbers

We add $\mathbf e_-$ to our bag of numbers. Each element in it is now of the form (a + b $\mathbf e_-$ ).

= ac + ade_- + bce_- + bd{e_-}^2 \\ = (ac - bd) + (ad + bc)e_-

= ac + ade_- + bce_- + bd{e_-}^2 \\ = (ac - bd) + (ad + bc)e_-

So, nothing special to remember ! No new multiplication, just a new element in the bag.

Easy to calculate - but what does it DO ?

e_- \notin \mathbb R, {e_-}^2=-1

e_- \notin \mathbb R, {e_-}^2=-1

e_+ \notin \mathbb R, {e_+}^2=1

e_+ \notin \mathbb R, {e_+}^2=1

e_0 \notin \mathbb R, {e_0}^2=0

e_0 \notin \mathbb R, {e_0}^2=0

(a + be_-)(c + de_-)

(a + be_-)(c + de_-)

Algebra - The Geometric Numbers

Easy to calculate - but what do they DO ?

\begin{aligned} 1*{e_-} &=& {e_-}^1 &=&{e_-} \\ {e_-}*{e_-} &=& {e_-}^2 &=&-1 \\ -1*{e_-} &=& {e_-}^3 &=&-{e_-} \\ -{e_-}*{e_-} &=& {e_-}^4 &=&\,\,1 \\ \end{aligned}

\begin{aligned} 1*{e_-} &=& {e_-}^1 &=&{e_-} \\ {e_-}*{e_-} &=& {e_-}^2 &=&-1 \\ -1*{e_-} &=& {e_-}^3 &=&-{e_-} \\ -{e_-}*{e_-} &=& {e_-}^4 &=&\,\,1 \\ \end{aligned}

Elements that square to -1 related to ROTATIONS

$\hookrightarrow$ this means $-e_-$ is the inverse of $e_-$

e_- \notin \mathbb R, {e_-}^2=-1

e_- \notin \mathbb R, {e_-}^2=-1

e_+ \notin \mathbb R, {e_+}^2=1

e_+ \notin \mathbb R, {e_+}^2=1

e_0 \notin \mathbb R, {e_0}^2=0

e_0 \notin \mathbb R, {e_0}^2=0

Algebra - The Geometric Numbers

Easy to calculate - but what do they DO ?

\begin{aligned} (1+\epsilon)^1 &= (1+\epsilon) \\ (1+\epsilon)^2 &= (1+2\epsilon) \\ (1+\epsilon)^3 &= (1+3\epsilon) \\ \end{aligned}

\begin{aligned} (1+\epsilon)^1 &= (1+\epsilon) \\ (1+\epsilon)^2 &= (1+2\epsilon) \\ (1+\epsilon)^3 &= (1+3\epsilon) \\ \end{aligned}

Elements that square to 0 related to TRANSLATIONS

$\rightarrow (1+\epsilon)^{1.5}$ ?

e_- \notin \mathbb R, {e_-}^2=-1

e_- \notin \mathbb R, {e_-}^2=-1

e_+ \notin \mathbb R, {e_+}^2=1

e_+ \notin \mathbb R, {e_+}^2=1

e_0 \notin \mathbb R, {e_0}^2=0

e_0 \notin \mathbb R, {e_0}^2=0

Algebra - The Geometric Numbers

ROTATIONS

HYPERBOLIC ROTATIONS

TRANSLATIONS

only around origin
$e_-^n$ is repeated multiplication
$e_-^n$ rotates in steps of 90°
need smooth : $e_-^x$ for $x \in \mathbb R$

only one axis
needs weird '+1'
is linear, so smooth.

e_- \notin \mathbb R, {e_-}^2=-1

e_- \notin \mathbb R, {e_-}^2=-1

e_+ \notin \mathbb R, {e_+}^2=1

e_+ \notin \mathbb R, {e_+}^2=1

e_0 \notin \mathbb R, {e_0}^2=0

e_0 \notin \mathbb R, {e_0}^2=0

Algebra - The Geometric Numbers

ROTATIONS

TRANSLATIONS

only one axis
needs weird '+1'
is linear, so smooth.

Also, we want translations and rotations in one bag. But first - we solve the "smooth" and "weird 1+" problems.

HYPERBOLIC ROTATIONS

e_- \notin \mathbb R, {e_-}^2=-1

e_- \notin \mathbb R, {e_-}^2=-1

e_+ \notin \mathbb R, {e_+}^2=1

e_+ \notin \mathbb R, {e_+}^2=1

e_0 \notin \mathbb R, {e_0}^2=0

e_0 \notin \mathbb R, {e_0}^2=0

only around origin
$e_-^n$ is repeated multiplication
$e_-^n$ rotates in steps of 90°
need smooth : $e_-^x$ for $x \in \mathbb R$

Algebra - The Exponential function

With $e_-^n$ where $n \in \mathbb N$ we can rotate in steps. What about $e_-^x$ with $x \in \mathbb R$ ?

${(4^2)}^2 = (4*4)^2 = (4*4)*(4*4) = 4^{2*2} = 4^4$

$4^x = {(2^2)}^x = 2^{2x}$

$a^x = {(b^{\alpha})}^x = b^{\alpha x}$

$\hookrightarrow$

All exponential functions are the same, with x scaled by some constant $\alpha$ .

So if we can calculate one smooth $a^x$ with $x \in \mathbb R$ , we're done.

$e^x = 1 + \frac x 1 + \frac {x^2} {2*1} + \frac {x^3} {3*2*1} + ...$

power can be ANYTHING

powers $\in \mathbb N$

So, if you see $e^x$ , the $e$ ONLY tells you its an exponential function, the $x$ has all the information. With our new numbers, $e^{\alpha e_-}$ are thus rotations and $e^{\alpha e_0}$ are translations.

Algebra - The Exponential function

$e^{\alpha \mathbf e_-}=\sum \limits_{n=0}^\infty \cfrac {(\alpha \mathbf e_-)^n} {n!} = \cos \alpha + \sin \alpha \mathbf e_-$

$\rightarrow$ Exponentiate $\alpha \mathbf e_-$ to get a rotation (these are our smooth rotations)

$e^{\alpha \mathbf e_0}=\sum \limits_{n=0}^\infty \cfrac {(\alpha \mathbf e_0)^n} {n!} = 1 + \alpha \mathbf e_0$

$\rightarrow$ Exponentiate $\alpha \mathbf e_0$ to get a translation. (this is that weird $+1$ )

With $e_-^n$ where $n \in \mathbb N$ we can rotate in steps. What about $e_-^x$ with $x \in \mathbb R$ ?

$\vec{x} = a\mathbf e_1 + b\mathbf e_2$

By the contraction axiom, any such vector must square to a real number :

$(a\mathbf e_1 + b\mathbf e_2)^2 = a^2\mathbf e_1^2 + ab\mathbf e_1\mathbf e_2 + ba\mathbf e_2\mathbf e_1 + b^2\mathbf e_2^2$

Geometric Algebra Vectors

THE CONTRACTION AXIOM :

We use our new $\mathbf e_i$ as basis vectors. (one for each $\perp$ axis)
Each vector squares to a scalar. ( $\vec x^2 \in \mathbb R$ )
We write all vectors as linear combinations of these basis vectors:

$\vec{x} = a\mathbf e_1 + b\mathbf e_2$

By the contraction axiom, any such vector must square to a real number :

$(a\mathbf e_1 + b\mathbf e_2)^2 = a^2\mathbf e_1^2 + ab\mathbf e_1\mathbf e_2 + ba\mathbf e_2\mathbf e_1 + b^2\mathbf e_2^2$

Since $a^2\mathbf e_1^2,b^2\mathbf e_2^2$ are both scalar, and $\mathbf e_1\mathbf e_2$ and $\mathbf e_2\mathbf e_1$ can not be simplified :

$ab\mathbf e_1\mathbf e_2 + ba\mathbf e_2\mathbf e_1 = 0$

$\leftrightarrow \mathbf e_1\mathbf e_2 = -\mathbf e_2\mathbf e_1$

Geometric Algebra Vectors

THE CONTRACTION AXIOM :

We use our new $\mathbf e_i$ as basis vectors. (one for each $\perp$ axis)
Each vector squares to a scalar. ( $\vec x^2 \in \mathbb R$ )
We write all vectors as linear combinations of these basis vectors:

use $\mathbf e_i$ as $\perp$ basis vectors

$\mathbf e_i\mathbf e_j = -\mathbf e_j\mathbf e_i$

$\mathbf e_i \notin \mathbb R$

$\mathbf e_i^2 \in \mathbb R$

Geometric Algebra Vectors

parallel vectors contract into scalar (+1, -1, 0)

orthogonal vectors anti-commute

$\mathbf e_i\mathbf e_j = \mathbf e_{ij}$

k-vector = product of k $\neq$ basis vectors

A general element of our GA is called a multivector and is the sum of a scalar, vector, bivector, ...

So, we start with two elements $\mathbf e_1, \mathbf e_2$ , and through multiplication we get another new element, $\mathbf e_{12}$ . It is the plane our two vectors lie in.

$\mathbb R_{2,0,0} :$

$\mathbb R$

$\mathbf e_1,\mathbf e_2$

$\mathbf e_{12}$

$\rightarrow$ 0-dimensional "scalar"

$\rightarrow$ 1-dimensional "vector"

$\rightarrow$ 2-dimensional "bivector"

a vector is a 1D oriented quantity. (line)

a bivector is a 2D oriented quantity. (plane)

a trivector is a 3D oriented quantity (volume)

Geometric Algebra Vectors

in practice, only two calculation rules ! $e_ie_i = \{+1,-1,0\},\quad e_ie_j = -e_je_i$

square any generator, replace with metric
swap two generators, change sign

Let's try a few examples, with two basis vectors, $\mathbf e_1,\mathbf e_2$ , both square to +1

$3\mathbf e_1 2\mathbf e_1 = 6$

$3\mathbf e_1 2\mathbf e_2 = 6 \mathbf e_1\mathbf e_2 = 6 \mathbf e_{12}$

$(3\mathbf e_1 + \mathbf e_2) \mathbf e_2 = 3\mathbf e_{12} + 1$

$\mathbf e_2 \mathbf e_{12} = \mathbf e_2 \mathbf e_1 \mathbf e_2 = - \mathbf e_1 \mathbf e_2 \mathbf e_2 = - \mathbf e_1$

$\mathbf e_{12} \mathbf e_{12} = \mathbf e_1 \mathbf e_2 \mathbf e_1 \mathbf e_2 = - \mathbf e_1 \mathbf e_1 \mathbf e_2 \mathbf e_2 = -1$

Geometric Algebra Vectors

in practice, only two calculation rules ! $e_ie_i = \{+1,-1,0\},\quad e_ie_j = -e_je_i$

square any generator, replace with metric
swap two generators, change sign

Let's try a few examples, with two basis vectors, $\mathbf e_1,\mathbf e_2$ , both square to +1

$3\mathbf e_1 2\mathbf e_1 = 6$

$3\mathbf e_1 2\mathbf e_2 = 6 \mathbf e_1\mathbf e_2 = 6 \mathbf e_{12}$

$(3\mathbf e_1 + \mathbf e_2) \mathbf e_2 = 3\mathbf e_{12} + 1$

$\mathbf e_2 \mathbf e_{12} = \mathbf e_2 \mathbf e_1 \mathbf e_2 = - \mathbf e_1 \mathbf e_2 \mathbf e_2 = - \mathbf e_1$

$\mathbf e_{12} \mathbf e_{12} = \mathbf e_1 \mathbf e_2 \mathbf e_1 \mathbf e_2 = - \mathbf e_1 \mathbf e_1 \mathbf e_2 \mathbf e_2 = -1$

Geometric Algebra Vectors

bivector rotates !

Geometric Algebra

We'll want more elements in the same bag, and names will get confusing, so let's first introduce some notation :

$\mathbb R_{positive,negative,zero}$

$\mathbb R_{0,1,0}\,\rightarrow$ one extra element that squares to $-1$

$\mathbb R_{0,0,1}\,\rightarrow$ one extra element that squares to $0$

$\mathbb R_{9,6,0}\,\rightarrow$ nine extra element that square to $1$

and six extra elements that square to $-1$

These elements are called generators, and are typically written $\mathbf e_0, \mathbf e_1, \mathbf e_2, ...$

so, only two calculation rules ! $e_ie_i = \{+1,-1,0\},\quad e_ie_j = -e_je_i$

square any generator, replace with metric
swap two generators, change sign

product of any two vectors in $\mathbb R_{3,0,0}$ :

$(a_1\mathbf e_1 + a_2\mathbf e_2 + a_3\mathbf e_3)(b_1\mathbf e_1 + b_2\mathbf e_2 + b_3\mathbf e_3) =$

$a_1b_1 + a_2b_2 + a_3b_3\\+ (a_1b_2-a_2b_1)\mathbf e_{12} + (a_1b_3-a_3b_1)\mathbf e_{13} + (a_2b_3-a_3b_2)\mathbf e_{23}$

The product of two VECTORS is a SCALAR + BIVECTOR

$a*b$ $=$ $a \cdot b$ $+$ $a \wedge b$

geometric product

inner product (dot)

outer product (wedge, 'cross')

$a*b$ $=$ $a \cdot b$ $+$ $a \times b$

$\mathbb R_{3,0,0}$

1 scalar
3 vectors $\mathbf e_1, \mathbf e_2, \mathbf e_3$
3 bivectors $\mathbf e_{12}, \mathbf e_{13}, \mathbf e_{23}$
1 trivector $\mathbf e_{123}$

Scalars + Bivectors = QUATERNIONS

Geometric Algebra $\mathbb R_{3,0,0}$

RECAP :

Start with $\mathbb R$
Add in some vectors $e_i$ - pick their metric.
Get free bivectors, trivectors, their metric cannot be picked.
an n-vector naturally contains the vectors in its name.
the standard product is the geometric product
if you only care about the 'similarity', calculate only the inner product
if you only care about the 'difference', calculate only the outer product
use scalar+bivector for transformations. (like quaternion multiplication)

Geometric Algebra

$ab$ = geometric product

$a \cdot b$ = inner product

$a \wedge b$ = outer product

$a^*$ = dual of $a$

$a \vee b$ = $(a^* \wedge b^*)^*$

$=$ regressive product

$ab$ = geometric product

$a \cdot b$ = inner product

$a \wedge b$ = outer product

$a^*$ = dual of $a$

$a \vee b$ = $(a^* \wedge b^*)^*$

$=$ regressive product

Geometric Algebra

We can now pick the Algebra that best fits our problem. For our current problem of computer geometry, we need it to store points, directions, lines, planes, translations and rotations.

We are now ready to create the bag we need for 2D PGA - The projective plane

2D Projective Geometric Algebra

We need a 1-up model, so start with three basis vectors :

$\mathbf e_0, \mathbf e_1, \mathbf e_2$

where $\mathbf e_0$ is the projective dimension. This results

in three bivectors $\mathbf e_{01}, \mathbf e_{20}, \mathbf e_{12}$ and one trivector $\mathbf e_{012}$

remembering that we need bivectors for isometries, we consider the metric for $\mathbf e_0$

${\mathbf e_0}^2 = +1 \quad \rightarrow \quad {\mathbf e_{01}}^2=-1, {\mathbf e_{20}}^2=-1, {\mathbf e_{12}}^2 = -1$

${\mathbf e_0}^2 = -1 \quad \rightarrow \quad {\mathbf e_{01}}^2=+1, {\mathbf e_{20}}^2=+1, {\mathbf e_{12}}^2 = -1$

${\mathbf e_0}^2 = 0 \quad \rightarrow \quad {\mathbf e_{01}}^2=0, {\mathbf e_{20}}^2=0, {\mathbf e_{12}}^2 = -1$

Only with $\mathbf e_0^2=0$ we get the two translations and one rotation we need for isometries in the 2D Euclidean plane.

We are now ready to create the bag we need for 2D PGA - The projective plane

2D Projective Geometric Algebra

We need a 1-up model, so start with three basis vectors :

$\mathbf e_0, \mathbf e_1, \mathbf e_2$

where $\mathbf e_0$ is the projective dimension. This results

in three bivectors $\mathbf e_{01}, \mathbf e_{20}, \mathbf e_{12}$ and one trivector $\mathbf e_{012}$

remembering that we need bivectors for isometries, we consider the metric for $\mathbf e_0$

${\mathbf e_0}^2 = +1 \quad \rightarrow \quad {\mathbf e_{01}}^2=-1, {\mathbf e_{20}}^2=-1, {\mathbf e_{12}}^2 = -1$

${\mathbf e_0}^2 = -1 \quad \rightarrow \quad {\mathbf e_{01}}^2=+1, {\mathbf e_{20}}^2=+1, {\mathbf e_{12}}^2 = -1$

${\mathbf e_0}^2 = 0 \quad \rightarrow \quad {\mathbf e_{01}}^2=0, {\mathbf e_{20}}^2=0, {\mathbf e_{12}}^2 = -1$

Only with $\mathbf e_0^2=0$ we get the two translations and one rotation we need for isometries in the 2D Euclidean plane.

$\rightarrow$ elliptic

$\rightarrow$ hyperbolic

$\rightarrow$ Euclidean

$\mathbb R^*_{2,0,1}$

1 scalar
3 vectors $\mathbf e_0, \mathbf e_1, \mathbf e_2$
3 bivectors $\mathbf e_{01}, \mathbf e_{20}, \mathbf e_{12}$
1 trivector $\mathbf e_{012}$

Scalars + Bivectors = Translate + Rotate

We are now ready to create the bag we need for 2D PGA - The projective plane

$e_1$ , $e_2$ represent the x=0,y=0 axis (square to +1)
$e_0$ for the projective axis. (squares to zero)
$e_{01}$ and $e_{20}$ square to zero, represent translations
$e_{12}$ squares to -1, represents rotation
Vectors are associated to PROJECTIVE LINES
Bivectors are associated to PROJECTIVE POINTS
$\wedge$ represents meet, $\vee$ represents join

1	e0	e1	e2	e01	e20	e12	e012

vector

bivector

scalar

trivec

projective LINES

projective POINTS

+1	0	+1	+1	0	0	-1	0

2D Projective Geometric Algebra

$\mathbb R^*_{2,0,1}$

1 scalar
3 vectors $\mathbf e_0, \mathbf e_1, \mathbf e_2$
3 bivectors $\mathbf e_{01}, \mathbf e_{20}, \mathbf e_{12}$
1 trivector $\mathbf e_{012}$

Scalars + Bivectors = Translate + Rotate

We are now ready to create the bag we need for 2D PGA - The projective plane

$e_1$ , $e_2$ represent the x=0,y=0 axis (square to +1)
$e_0$ for the projective axis. (squares to zero)
$e_{01}$ and $e_{20}$ square to zero, represent translations
$e_{12}$ squares to -1, represents rotation
Vectors are associated to PROJECTIVE LINES
Bivectors are associated to PROJECTIVE POINTS
$\wedge$ represents meet, $\vee$ represents join

1	e0	e1	e2	e01	e20	e12	e012

vector

bivector

scalar

trivec

projective LINES

projective POINTS

+1	0	+1	+1	0	0	-1	0

2D Projective Geometric Algebra

Unifies TRANSFORMATION and ELEMENT !! $e^\mathbf x =$ rotation around $\mathbf x$ !!

also in 3D ! "Plane based GA"

We are now ready to create the bag we need for 2D PGA - The projective plane

1	e0	e1	e2	e01	e20	e12	e012

vector

bivector

scalar

trivec

projective LINES

projective POINTS

Point at (x,y) :

Line between points $p_1, p_2$ :

Line with equation $a\mathbf x + b\mathbf y + c= 0$ :

Intersect lines $\ell_1, \ell_2$ :

Rotation of $\alpha$ around point $p$ :

Translate $\alpha$ with infinite point $p$ :

Angle between lines $\ell_1, \ell_2$ :

Project point on line :

$x\mathbf e_{20} + y\mathbf e_{01} + \mathbf e_{12}$

$p_1 \vee p_2$

$a\mathbf e_1 + b\mathbf e_2 + c\mathbf e_0$

$\ell_1 \wedge \ell_2$

$e^{\alpha p}$

$\cos^{-1}(\ell_1 \cdot \ell_2)$

$(\ell \cdot p) \ell$

+1	0	+1	+1	0	0	-1	0

2D Projective Geometric Algebra

Now that we can multiply points and lines, lets look at some interesting products :

2D Projective Geometric Algebra

The regressive product $\vee$

JOINS points into lines

$\ell=P_1 \vee P_2\\\,\,\,\,\,\,\,\,\,\,\,= (P_1^* \wedge P_2^*)^*$

The outer product $\wedge$

MEETS lines in points

$P=\ell_1 \wedge \ell_2$

Now that we can multiply points and lines, lets look at some interesting products :

2D Projective Geometric Algebra

$\ell$ = line (vector)

$P$ = point (bivector)

$\ell P$ = line through $P, \perp$ to $\ell$

$\ell P\ell$ = reflection of $P$ in $\ell$

$P\ell P$ = reflection of $\ell$ in $P$

$(\ell \cdot P)\ell$ = projection of $P$ on $\ell$

$(P \cdot \ell)P$ = projection of $\ell$ on $P$

Now that we can multiply points and lines, lets look at some interesting products :

2D Projective Geometric Algebra

$\ell_1 P\ell_1$ = reflection of $P$ in $\ell_1$

$\ell_2\ell_1 P\ell_1\ell_2$ = reflection of $P$ in $\ell_1$ then $\ell_2$

two reflections in lines (vectors)

= rotation around intersection point (bivector)

Just like the quaternions, a transformation $m = \ell_1\ell_2$ is applied on an element $\bf x$ using the "sandwich product" :

$m \mathbf x \tilde m$

where $\tilde m$ is the reverse of $m$ (here $\tilde m =\ell_2 \ell_1$ )

It is all exactly the same in 3D, just some more elements :

3D Projective Geometric Algebra

Still a one-up model.
4 vectors : $\mathbf e_0, \mathbf e_1, \mathbf e_2, \mathbf e_3$ are PLANES
6 bivectors : $\mathbf e_{01},\mathbf e_{02},\mathbf e_{03}$ (translate) $\mathbf e_{12}, \mathbf e_{31}, \mathbf e_{23}$ (rotate) are LINES
4 trivectors : $\mathbf e_{021},\mathbf e_{013},\mathbf e_{032}, \mathbf e_{123}$ are POINTS
1 quadvector : $\mathbf e_{0123}$ is the PSEUDO-SCALAR
exponentiate Euclidean line to generate a rotation
exponentiate Infinite line to generate a translation
scalar + bivectors + quadvector are now the DUAL QUATERNIONS
sandwich products, reflections, incidence, .. everything stays the same !

It is all exactly the same in 3D, just some more elements :

1	e0	e1	e2	e3	e01	e02	e03

Point at (x,y,z) :

Line between points $P_1, P_2$ :

Plane with equation $a\mathbf x + b\mathbf y + c\mathbf z + d = 0$ :

Intersect planes $p_1, p_2$ :

Rotation of $\alpha$ around line $L$ :

Translate $\alpha$ with infinite line $L$ :

Angle between planes $p_1, p_2$ :

Project point on plane :

$x\mathbf e_{023} + y\mathbf e_{013} + z\mathbf e_{012} + \mathbf e_{123}$

$P_1 \vee P_2$

$a\mathbf e_1 + b\mathbf e_2 + c\mathbf e_3 + d\mathbf e_0$

$p_1 \wedge p_2$

$e^{\alpha L}$

$\cos^{-1}(p_1 \cdot p_2)$

$(p \cdot P) p$

+1	0	+1	+1	+1	0	0	0

e12	e13	e23	e012	e013	e023	e123	e0123

scalar	vector	bivector	trivector	PSS

PROJECTIVE PLANES	PROJECTIVE LINES	PROJECTIVE POINTS

-1	-1	-1	0	0	0	-1	0

3D Projective Geometric Algebra

The regressive product $\vee$ JOINS points or lines in lines or planes

$\ell=P_1 \vee P_2$

$p = P_1 \vee P_2 \vee P_3$

The outer product $\wedge$ MEETS

planes or lines in lines or points

$\ell=p_1 \wedge p_2$

$P = p_1 \wedge p_2 \wedge p_3$

All incidence relations trivially work, and algebraic expressions with points, lines and planes can be simplified like any other.

3D Projective Geometric Algebra

PGA points, lines and planes are excellent for mesh vertices, edges and faces.

Edges $E$ and Faces $F$

$E_i = \widehat{P_1} \vee \widehat{P_2}\quad\quad\quad F_i = \widehat{P_1} \vee \widehat{P_2} \vee \widehat{P_3}$

Polygon circumference $c$ and area $a$ :

$c=\sum\lVert E_i\rVert\quad a=\frac 1 2 \lVert \sum E_i \rVert_\infty$

Mesh area $a$ and volume $v$

$a=\frac 1 2 \sum\lVert F_i\rVert\quad v=\frac 1 3 \lVert \sum F_i \rVert_\infty$

3D Projective Geometric Algebra

PGA points, lines and planes are excellent for mesh vertices, edges and faces.

Edges $E$ and Faces $F$

$E_i = \widehat{P_1} \vee \widehat{P_2}\quad\quad\quad F_i = \widehat{P_1} \vee \widehat{P_2} \vee \widehat{P_3}$

for DDG, similar simple formulas :

$\Omega_j = 2\pi - \sum \cos^{-1}({\widehat{E}_i\cdot \widehat{E}_{i+1}})$

$2\pi\chi = \sum \Omega_j$

note - no edge vectors needed ..

1 . 1

Thank you all who attended - (course notes are here )

SIGGRAPH2019 - GA for CGI

By Steven De Keninck

SIGGRAPH2019 - GA for CGI

SIGGRAPH2019 - gensub345b - Geometric Algebra for Computer Graphics

6 years ago
8,461

Steven De Keninck PRO

Mathematical Experimentalist

enkimute.github.io

SIGGRAPH2019 - GA for CGI

More from Steven De Keninck