Steven De Keninck

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Projective Geometric Algebra

Projective Geometric Algebra

The Red Pill of Computer Geometry

Computer Geometry

Computer Geometry

point

direction

line

plane

Computer Geometry

point

direction

line

plane

Computer Geometry

point

direction

line

plane

position

rotation

Computer Geometry

point

direction

line

plane

position

rotation

matrix

Computer Geometry

point

direction

line

plane

position

rotation

matrix

quaternion

Computer Geometry

point

direction

line

plane

position

rotation

matrix

quaternion

velocity

force

tensor

Computer Geometry

point

direction

line

plane

position

rotation

matrix

quaternion

velocity

force

tensor

dual quaternion

Computer Geometry

point

direction

line

plane

position

rotation

matrix

quaternion

velocity

force

tensor

dual quaternion

VECTOR AND MATRIX ALGEBRA

Computer Geometry

point

direction

line

plane

position

rotation

matrix

quaternion

velocity

force

dual quaternion

VECTOR AND MATRIX ALGEBRA

VECTOR

• natural choice for ANALYTIC GEOMETRY
• lots of design choices (axes, chirality, rotation order)
• lots of coordinates
• lots of exceptions
• lots of conversions (quaternion,euler,matrix,...)
• lots of programming/testing
• feels like bookkeeping

Geometric Objects are represented by choosing axes, taking measurements and writing down coefficients. Formulas produce and functions transform these measurements.

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.

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

Computer Geometry

ANALYTIC GEOMETRY : VECTOR AND MATRIX ALGEBRA

Computer Geometry

ANALYTIC GEOMETRY : VECTOR AND MATRIX ALGEBRA

SYNTHETIC GEOMETRY : PROJECTIVE GEOMETRIC ALGEBRA

Computer Geometry

ANALYTIC GEOMETRY : VECTOR AND MATRIX ALGEBRA

SYNTHETIC GEOMETRY : PROJECTIVE GEOMETRIC ALGEBRA

Computer Geometry

ANALYTIC GEOMETRY : VECTOR AND MATRIX ALGEBRA

SYNTHETIC GEOMETRY : PROJECTIVE GEOMETRIC ALGEBRA

Computer Geometry

ANALYTIC GEOMETRY : VECTOR AND MATRIX ALGEBRA

SYNTHETIC GEOMETRY : PROJECTIVE GEOMETRIC ALGEBRA

2D

3D

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

Computer Geometry

ANALYTIC GEOMETRY : VECTOR AND MATRIX ALGEBRA

SYNTHETIC GEOMETRY : PROJECTIVE GEOMETRIC ALGEBRA

2D

3D

4D

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

Computer Geometry

SYNTHETIC GEOMETRY : PROJECTIVE GEOMETRIC ALGEBRA

• 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 ?

Computer Geometry

SYNTHETIC GEOMETRY : PROJECTIVE GEOMETRIC ALGEBRA

• 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

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

We associate each projective point in 2D with a line through the origin in 3D

Projective Geometry

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

Lines that do not intersect the projective plane are associated with infinite points

Projective Geometry

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

#points_in_plane + #infinite_points = #lines_through_origin_in_3D

What about lines ?

Projective Geometry

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

We associate each projective line in 2D with a plane through the origin in 3D

Projective Geometry

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

The one plane parallel to the projective plane is associated with 1 infinite line

#points_in_plane + #infinite_points = #lines_through_origin_in_3D

#lines_in_plane + 1 = #planes_through_origin_in_3D

This works the same way in n-d and is called the Projective Map.

Projective Geometry

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

The planes always intersect, and so do their corresponding lines .. even at infinity.

Projective Geometry

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

The planes always intersect, and so do their corresponding lines .. even at infinity.

This means your code to intersect two lines has no exceptions, and the code that uses it has no if statements

What happens when you rotate around an infinite point ?

Projective Geometry

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

#points_in_plane + #infinite_points = #lines_through_origin_in_3D

#lines_in_plane + 1 = #planes_through_origin_in_3D

#lines_through_origin_in_3D = #planes_through_origin_in_3D

each 3D line is orthogonal to exactly one 3D plane so we also have :

that also means the same amount of projective points and lines !

so we can relate projective lines and planes leading to Duality

Projective Geometry

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

JOIN of two points is dual to MEET of their two dual lines

DUALITY : associate each projective point with one projective line

Projective Geometry

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

JOIN of two points is dual to MEET of two lines

This means your code to intersect two lines can also be used to join two points.

point

lies on

line

intersect

Works for all your functions !!!!!

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
• 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

What picture do you get when you see this equation ?

Algebra

What picture do you get when you see this equation ?

Algebra

Algebra

Al-Khwarizmi's six problems

Today, one problem :

What was Al-Khwarizmi missing ?

Algebra

Al-Khwarizmi's six problems

Today, one problem :

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

maybe other numbers were hiding ?

Algebra

What was Al-Khwarizmi missing ?

zero            negative numbers

maybe other numbers were hiding ?

Algebra

What was Al-Khwarizmi missing ?

zero            negative numbers

maybe other numbers were hiding ?

Algebra

What was Al-Khwarizmi missing ?

zero            negative numbers

maybe other numbers were hiding ?

Algebra

What was Al-Khwarizmi missing ?

zero            negative numbers

maybe other numbers were hiding ?

Algebra

What was Al-Khwarizmi missing ?

zero            negative numbers

maybe other numbers were hiding ?

Algebra

What was Al-Khwarizmi missing ?

zero            negative numbers

maybe other numbers were hiding ?

Algebra

What was Al-Khwarizmi missing ?

zero            negative numbers

maybe other numbers were hiding ?

Algebra

What was Al-Khwarizmi missing ?

zero            negative numbers

maybe other numbers were hiding ?

Algebra

THE GEOMETRIC NUMBERS

these are ofcourse the complex, hyperbolic and dual numbers. So why call them this ? lets find out ..

Algebra - The Geometric Numbers

Addition and Scalar multiplication are trivial : 

a lemon plus a lemon is almost always two lemons.

Algebra - The Geometric Numbers

Addition and Scalar multiplication are trivial : 

a lemon plus a lemon is almost always two lemons.

But what about multiplication ? - well lets find out .. 

Algebra - The Geometric Numbers

We add  i  to our bag of numbers. Each element in it is now of the form (a + bi), where a or b can be zero.

Algebra - The Geometric Numbers

We add  i  to our bag of numbers. Each element in it is now of the form (a + bi), where a/b could be zero.

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

Easy to calculate - but what does it DO ?

Algebra - The Geometric Numbers

Easy to calculate - but what do they DO ?

Elements that square to -1 related to ROTATIONS

\(\hookrightarrow\) this means \(-i\) is the inverse of \(i\)

Algebra - The Geometric Numbers

Easy to calculate - but what do they DO ?

Elements that square to +1 related to REFLECTIONS

\(\hookrightarrow\) this means \(j\) is its own inverse !

Algebra - The Geometric Numbers

Easy to calculate - but what do they DO ?

Elements that square to 0 related to TRANSLATIONS

Algebra - The Geometric Numbers

ROTATIONS

REFLECTIONS

TRANSLATIONS

• only around origin
• \(i^n\) is repeated multiplication
• \(i^n\) rotates in steps of 90°
• need smooth : \(i^x\) for \(x \in \mathbb R\)

• only around 1 line
• smooth exists, hyperbolic rotations
   

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

Algebra - The Geometric Numbers

ROTATIONS

REFLECTIONS

TRANSLATIONS

• only around origin
• \(i^n\) is repeated multiplication
• \(i^n\) rotates in steps of 90°
• need smooth : \(i^x\) for \(x \in \mathbb R\)

• only around 1 line
• smooth exists, hyperbolic rotations
   

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

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

Algebra - The Exponential function

With \(i^n\) where \(n \in \mathbb N\) we can rotate in steps. What about \(i^x\) with \(x \in \mathbb R\) ?

\(4^3 = 4*4*4\)
\(4^3 * 4^2 = (4 * 4 * 4) * (4 * 4) = 4^{3+2} = 4^5\)

\(\cfrac {4^3} {4^2} = \cfrac {4 * 4 * 4} {4 * 4} = 4^{3-2} = 4^1  = 4\)

\(\cfrac {4^2} {4^3} = \cfrac {4 * 4 } {4 * 4 * 4} = 4^{2-3} = 4^{-1} = \cfrac 1 4 \)

\(4^0 = 4^{1-1} = \cfrac 4 4 = 1\)

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

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

\( a^n * a^m = a^{n+m} \)

\(\cfrac {a^n} {a^m} = a^{n-m} \)

\(a^1=a,\,a^0=1,\,a^{-1}=\cfrac 1 a\)

\({(a^n)}^m = a^{n*m}\)

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

\(\hookrightarrow\)

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.

Algebra - The Exponential function

With \(i^n\) where \(n \in \mathbb N\) we can rotate in steps. What about \(i^x\) 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.

Thanks to Euler, we know an easy way to calculate one of these functions, he discovered the following series :

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

power can be ANYTHING

powers \(\in \mathbb N\)

Algebra - The Exponential function

With \(i^n\) where \(n \in \mathbb N\) we can rotate in steps. What about \(i^x\) 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, \(e^{\alpha i}\) are thus rotations and \(e^{\alpha \epsilon}\) are translations.

Algebra - The Exponential function

With \(i^n\) where \(n \in \mathbb N\) we can rotate in steps. What about \(i^x\) with \(x \in \mathbb R\) ?

\(e^{\alpha i}=\sum \limits_{n=0}^\infty \cfrac {(\alpha i)^n} {n!} = \cos \alpha + i \sin \alpha\)

\(\rightarrow\) Exponentiate \(\alpha i\) to get a rotation (these are our smooth rotations)

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

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

\(e^x\) when \(x \notin \mathbb R\) are called the roots of unity. (multiplying with them wont change scale, i.e. : \({(e^x)}^n=1\)

Algebra - The Geometric Numbers

Recap :

• Three new types of numbers
• They square to a real number
• Connected to translations, rotations, reflections

We define one last property, when we add more than one of these,

lets call them \(\mathbf e_1\) and \(\mathbf e_2\), we demand that they anticommute :

\(\mathbf e_1 \mathbf e_2 = - \mathbf e_2 \mathbf e_1\)

Let's now use these numbers to properly write down vectors ..

Algebra - in the bag.

So far, we've always added just one element to our bag of numbers \(\mathbb R\)

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 \(\mathbf e_0, \mathbf e_1, \mathbf e_2, ...\)

Algebra - in the bag.

Let's try with two generators - we take \(\mathbb R\) and add two positive generators \(\mathbf e_1, \mathbf e_2\) to represent our X and Y axis

\(\mathbb R_{2,0,0}\) : \(\quad\mathbb R \,{\scriptstyle\oplus}\, \mathbf e_1 \,{\scriptstyle\oplus}\, \mathbf e_2,\quad\mathbf e_1^2=+1,\,\mathbf e_2^2=+1\)

\(\vec a\vec b = (a_1{\color{green}\mathbf e_1} + a_2{\color{blue}\mathbf e_2}) (b_1{\color{green}\mathbf e_1} + {b_2\color{blue}\mathbf e_2}) \)

\(= a_1b_1{\color{green}\mathbf e_1^2} + a_2b_2{\color{blue}\mathbf e_2^2}\)

  \(+ a_1b_2{\color{green}\mathbf e_1}{\color{blue}\mathbf e_2} + a_2b_1{\color{blue}\mathbf e_2}{\color{green}\mathbf e_1}\)

\(\rightarrow\) these are scalar \(\in \mathbb R\)

\(\rightarrow\) new type ! no scalar, no vector !

Grassmann : different vectors ANTICOMMUTE : \(e_1e_2 = -e_2e_1\)

Any vector is a linear combination of \(\mathbf e_1, \mathbf e_2\), the product of two vectors is :

\(\rightarrow\) these are vectors

Algebra - in the bag.

Let's try with two generators - we take \(\mathbb R\) and add two positive generators \(\mathbf e_1, \mathbf e_2\) to represent our X and Y basis vectors

\(\mathbb R_{2,0,0}\) : \(\quad\mathbb R \,{\scriptstyle\oplus}\, {\color{green}\mathbf e_1} \,{\scriptstyle\oplus}\, {\color{blue}\mathbf e_2},\quad{\color{green}\mathbf e_1}^2=+1,\,{\color{blue}\mathbf e_2}^2=+1\)

\(\vec a\vec b = (a_1{\color{green}\mathbf e_1} + a_2{\color{blue}\mathbf e_2}) (b_1{\color{green}\mathbf e_1} + {b_2\color{blue}\mathbf e_2}) \)

\(= a_1b_1 + a_2b_2\)

  \(+ (a_1b_2-a_2b_1){\color{green}\mathbf e_1}{\color{blue}\mathbf e_2}\)

\(\rightarrow\) these are scalar \(\in \mathbb R\)

\(\rightarrow\) new type ! no scalar, no vector !

  bivector

Grassmann : different vectors ANTICOMMUTE : \(e_1e_2 = -e_2e_1 = e_{12} = -e_{21}\)

Any vector is a linear combination of \(\mathbf e_1, \mathbf e_2\), the product of two vectors is :

\(\rightarrow\) these are vectors

Algebra - in the bag.

Let's try with two generators - we take \(\mathbb R\) and add two positive generators \(\mathbf e_1, \mathbf e_2\) to represent our X and Y basis vectors

\(\mathbb R_{2,0,0}\) : \(\quad\mathbb R \,{\scriptstyle\oplus}\, {\color{green}\mathbf e_1} \,{\scriptstyle\oplus}\, {\color{blue}\mathbf e_2},\quad{\color{green}\mathbf e_1}^2=+1,\,{\color{blue}\mathbf e_2}^2=+1\)

\(\vec a\vec b = (a_1{\color{green}\mathbf e_1} + a_2{\color{blue}\mathbf e_2}) (b_1{\color{green}\mathbf e_1} + {b_2\color{blue}\mathbf e_2}) \)

\(= a_1b_1 + a_2b_2\)

  \(+ (a_1b_2-b_1a_2){\color{red}\mathbf e_{12}}\)

\(\rightarrow\) these are scalar \(\in \mathbb R\)

\(\rightarrow\) new type ! no scalar, no vector !

  bivector

Grassmann : different vectors ANTICOMMUTE : \(e_1e_2 = -e_2e_1 = e_{12} = -e_{21}\)

Any vector is a linear combination of \(\mathbf e_1, \mathbf e_2\), the product of two vectors is :

\(\rightarrow\) these are vectors

Algebra - in the bag.

So, we add 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)

Algebra - in the bag.

so, only two calculation rules !  \(e_xe_x = \{+1,-1,0\},\quad e_xe_y = -e_xe_y\)

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

Let's try a few examples, we are still in \(\mathbb R_{2,0,0}\). So \(\mathbf e_1,\mathbf e_2\), both square to +1

\(3\mathbf e_1 2\mathbf e_1 = \)

\(3\mathbf e_1 2\mathbf e_2 = \)

\((3\mathbf e_1 + \mathbf e_2) \mathbf e_2 =\)

\(\mathbf e_2 \mathbf e_{12} = \)

\(\mathbf e_{12} \mathbf e_{12} =  \)

Algebra - in the bag.

so, only two calculation rules !  \(e_xe_x = \{+1,-1,0\},\quad e_xe_y = -e_xe_y\)

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

Let's try a few examples, we are still in \(\mathbb R_{2,0,0}\). So \(\mathbf e_1,\mathbf e_2\), both square to +1

\(3\mathbf e_1 2\mathbf e_1 = 6\)

\(3\mathbf e_1 2\mathbf e_2 = \)

\((3\mathbf e_1 + \mathbf e_2) \mathbf e_2 =\)

\(\mathbf e_2 \mathbf e_{12} = \)

\(\mathbf e_{12} \mathbf e_{12} =  \)

Algebra - in the bag.

so, only two calculation rules !  \(e_xe_x = \{+1,-1,0\},\quad e_xe_y = -e_xe_y\)

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

Let's try a few examples, we are still in \(\mathbb R_{2,0,0}\). So \(\mathbf e_1,\mathbf e_2\), both square to +1

\(3\mathbf e_1 2\mathbf e_1 = 6\)

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

\((3\mathbf e_1 + \mathbf e_2) \mathbf e_2 =\)

\(\mathbf e_2 \mathbf e_{12} = \)

\(\mathbf e_{12} \mathbf e_{12} = \)

Algebra - in the bag.

so, only two calculation rules !  \(e_xe_x = \{+1,-1,0\},\quad e_xe_y = -e_xe_y\)

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

Let's try a few examples, we are still in \(\mathbb R_{2,0,0}\). So \(\mathbf e_1,\mathbf e_2\), both square to +1

\(3\mathbf e_1 2\mathbf e_1 = 6\)

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

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

\(\mathbf e_2 \mathbf e_{12} =\)

\(\mathbf e_{12} \mathbf e_{12} = \)

\(3\mathbf e_1 2\mathbf e_1 = 6\)

Algebra - in the bag.

so, only two calculation rules !  \(e_xe_x = \{+1,-1,0\},\quad e_xe_y = -e_xe_y\)

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

Let's try a few examples, we are still in \(\mathbb R_{2,0,0}\). So \(\mathbf e_1,\mathbf e_2\), both square to +1

\(3\mathbf e_1 2\mathbf e_1 = 6\)

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

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

\(\mathbf e_2 \mathbf e_{12} = \mathbf e_2 \mathbf e_1 \mathbf e_2 = - \mathbf e_1 \mathbf e_2 \mathbf e_2  = - \mathbf e_1\)

\(\mathbf e_{12} \mathbf e_{12} = \)

Algebra - in the bag.

so, only two calculation rules !  \(e_xe_x = \{+1,-1,0\},\quad e_xe_y = -e_xe_y\)

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

Let's try a few examples, we are still in \(\mathbb R_{2,0,0}\). So \(\mathbf e_1,\mathbf e_2\), both square to +1

\(3\mathbf e_1 2\mathbf e_1 = 6\)

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

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

\(\mathbf e_2 \mathbf e_{12} = \mathbf e_2 \mathbf e_1 \mathbf e_2 = - \mathbf e_1 \mathbf e_2 \mathbf e_2  = - \mathbf e_1\)

\(\mathbf e_{12} \mathbf e_{12} = \mathbf e_1 \mathbf e_2 \mathbf e_1 \mathbf e_2 = - \mathbf e_1 \mathbf e_1 \mathbf e_2 \mathbf e_2 = -1 \)

Algebra - in the bag.

so, only two calculation rules !  \(e_xe_x
 \{+1,-1,0\},\quad e_xe_y = -e_xe_y\)

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

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

\((a\mathbf e_1 + b\mathbf e_2 + c\mathbf e_3)(d\mathbf e_1 + e\mathbf e_2 + f\mathbf e_3) = ...\)

\(=ad + be + cf + (ae-bd)\mathbf e_{12} + (af-cd)\mathbf e_{13} + (bf-ce)\mathbf e_{23}\)

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

Algebra - in the bag.

RECAP :

• Start with \(\mathbb R\)
• Add in some vectors \(e_x\) - 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)

\(ab\) = geometric product

\(a \cdot b\) = inner product

\(a \wedge b\) = outer product

\(a^*\) = dual of \(a\)

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

Algebra - in the bag.

\(\mathbb R_{2,0,1}\)

• 1 scalar
• 3 vectors \(\mathbf e_0, \mathbf e_1, \mathbf e_2\)
• 3 bivectors \(\mathbf e_{01}, \mathbf e_{02}, \mathbf e_{12}\)
• 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 !!!! 

• We need 2+1D, so we pick three \(e_x\) elements
• \(e_1\),\(e_2\)  represent the X,Y axis (both square to +1)
• \(e_0\) for the projective axis. (squares to zero)
• \(e_{01}\) and \(e_{02}\) square to zero, represent translations
• \(e_{12}\) squares to -1, represents rotation
• Duality amounts to swapping vectors/bivectors and trivector/scalar !
• Vectors are associated to PROJECTIVE LINES
• Bivectors are associated to PROJECTIVE POINTS
• \(\wedge\) represents meet, \(\vee\) represents join

vector

bivector

scalar

trivec

projective LINES

projective POINTS

Algebra - in the bag.

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

vector

bivector

scalar

trivec

projective LINES

projective POINTS

Point at (x,y) :

Line between points \(p_1, p_2\) :

Line with equation \(a\mathbf x + b\mathbf y = c\) :

Intersect lines \(\ell_1, \ell_2\) :

Rotation of \(\alpha\) around point \(p\) :

Translate \(\alpha\) with infinite point \(p\) :

Angle between lines \(L_1, L_2\) :

Project point on line :

\(x\mathbf e_{02} + y\mathbf e_{01} + \mathbf e_{12}\)

\(p_1 \vee p_2\)

\(a\mathbf e_1 + b\mathbf e_2 - c\mathbf e_0\)

\(L_1 \wedge L_2\)

\(e^{\alpha p}\)

\(e^{\alpha p}\)

\(\cos^{-1}(L_1 \cdot L_2)\)

\( (L \cdot p) L\)

Algebra - in the bag.

Algebra - in the bag.

Algebra - in the bag.

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

Point at (x,y,z) :

Line between points \(p_1, p_2\) :

Plane with equation \(a\mathbf x + b\mathbf y + c\mathbf z = d\) :

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 :

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

\(p_1 \vee p_2\)

\(a\mathbf e_1 + b\mathbf e_2 + c\mathbf e_3 - d\mathbf e_0\)

\(P_1 \wedge P_2\)

\(e^{\alpha L}\)

\(e^{\alpha L}\)

\(\cos^{-1}(P_1 \cdot P_2)\)

\( (L \cdot p) L\)

Algebra - in the bag.

Algebra - in the bag.

Algebra - in the bag.

Algebra - in the bag.

Algebra - in the bag.

point

direction

line

plane

position

rotation

Same as opening but now in PGA - implementation is shorter than function names !

no coordinates - no chirality - no gimbal lock - no conversions

Thanks !

Where to go from here ?

Remember that PGA is very, very, very young. Not a lot of material is available, and even less implementations and examples. Keep an eye out for the Siggraph courses on PGA, and in the mean time, here are some papers and links :

https://www.researchgate.net/project/Projective-geometric-algebra

Dr Gunn's page on PGA :

My ganja.js GA implementation :

https://github.com/enkimute/ganja.js

My collection of examples :

https://enkimute.github.io/ganja.js/examples/coffeeshop.html

questions ? need help ? \(\rightarrow\) steven at enki dot ws