Homepage (versor.mat.ucsb.edu)
Bibtex (for referencing this work in your paper)
Versor is a C++ Library for Geometric Algebra, sometimes called Clifford Algebra, a system for encoding geometric concepts numerically.
The headers generate optimized code at compiletime through template metaprogramming. The core of the library is under 150kb, and supports arbitrary dimensions and metrics (limited by your compiler…).
The library can be used as a mathonly, or as an application with builtin graphics. Both OpenGL and OpeGLES draw routines are supported.
Lead Developer: Pablo Colapinto
gmail: wolftype
Some major revisions have been enacted that change the API.
vsr::nga::
is the namespace for generic ndimensional functionsvsr::cga::
is the namespace for 3D Conformal Geometric Algebra
vsr::nga::Round::
calls the generic implementation of nD round elements whereas vsr::cga::Round::
calls the 3D CGA specificationRound::
, Flat::
, Tangent::
) are written in full (as opposed to Ro::
, Fl::
and Ta::
). We note these are Capitalized, since earlier versions of the devel branch used lowercase namespaces.Construct::
is now used to construct useful objects within a particular namespace
Construct::point(0,1,0)
constructs a point at coordinate (0,1,0)
/detail /space /form /draw
and /util
organize the various filesMailing List  AlloSphere Research Group
See also my page at Academia.edu
My disseration, Articulating Space, introduces geometric algebra and explores applications to parametric design. My Master’s Thesis before it, serves as an introduction as well.
At C++ Now in Aspen I gave a Presentation on Metaprogramming Implementation Details
As long as algebra and geometry have been separated, their progress have been slow and their uses limited; but when these two sciences have been united, they have lent each mutual forces, and have marched together towards perfection.
Joseph Louis Lagrange
No attention should be paid to the fact that algebra and geometry are different in appearance.
Omar Khayyám
L’algèbre n’est qu’une géométrie écrite; la géométrie n’est qu’une algèbre figurée.
Sophie Germain
If you want to see, learn how to act
Heinz von Foerster
For this version you need C++11 support (gcc 4.7 or higher or clang 3.2 or higher) and for graphics support you’ll want glew.
See the Troubleshooting section below for instructions on installing glew.
git clone git://github.com/wolftype/versor.git
You’ll need to initialize the submodules to build any graphics examples:
cd versor
git submodule init
git submodule update
To build all examples
./buildAll.sh
To build library only
./build.sh
#include "vsr.h"
using Vec = vsr::euclidean_vec<3,float>; //< A 3 dimensional euclidean vector defined over floats
using Biv = vsr::euclidean_bivector<3,float>; //< A 3 dimensional bivector or "directed area element"
int main(){
auto v = Vec(1,2,3); //< A 3D vector at coordinates 1,2,3;
v.rotate( Biv::xy * .25 ).print(); //< Rotate the vector in the xy plane and print result
return 0;
};
While fully enabling arbitrary metric spaces, Versor has a lot of builtin functionality for specifically working with Conformal Geometric Algebra of 3D space, which is THE way to model all Euclidean transformations:
Point p = Round::point(0,0,0);
auto tp = p.translate(x, y, z);
auto rp = p.rotate( Biv::xy * theta)
TROUBLESHOOTING —
You may need to install Glew on newer macs. Best way to do this is with brew:
brew install glew
You’ll need C++11 support on your compiler (See makefile notes below). For C++11 you’ll want clang 3.2 (mac) or above or gcc 4.7 or above (linux).
Alternatively an earlier version of Versor is available at github.com/wolftype/versor_1.0.git This older version runs just as fast, but is strictly 3D CGA (i.e. R4,1 metric) since I generated headers ahead of time.
Versor provides operations and draw routines for Euclidean and Conformal Geometric Algebras, a relatively new spatial computing model used by physicists, engineers, and artists. Versor is designed to make graphical experimentation of geometric algebra within a C++ environment easier. You can use this library to draw geometrical things, explore spherical and hyperbolic spaces, transformations, design robots, etc. I am using it for my PhD on bioinspired engineering.
I first developed Versor while reading “Geometric Algebra for Computer Science” by Leo Dorst, Daniel Fontijne, and Stephen Mann. It’s a fantastic book and if you’re reading this you should also consider reading that.
Built to aid in my modelling of organic forms, the initial development was partially funded by the Olivia Long Converse Fellowship for Botanic research, courtesy of the Graduate Division at the University of California in Santa Barbara. Currently it is supported by the Robert W. Deutsch Foundation, and under a UC Regents General Public License. Feel free to use and distribute for noncommercial use as long as copyrights and credits are maintained. For commercial use, a licensing agreement may be necessary.
One quick word: clifford algebras and the spatial relationships they embody can often feel abstract and daunting. But it’s a twisty, boosty ride, full of weird discoveries. You’re bound to make some, so have fun!
The homogenous 5D CGA model used here was initially proposed by David Hestenes, Hongbo Li, and Alan Rockwood in 2001, and given full form and weight through the excellent and careful work of Leo Dorst, Joan and Anthony Lasenby, and Eduardo BayroCorrochano, and others. These researchers’ writings have helped me quite a bit. CGA is particular breed of Clifford Algebras (also known as Geometric Algebras), which operate upon combinatoric hypercomplex vector spaces that emerged from William Clifford’s attempt to fuse Hamilton’s quaternions with Grassmans’ extension algebras. Thus transformations were married with a system of abstraction. For more information, take a look at the links to the sites at the bottom of this page. For instance, for practical applications in robotics and “Geometric Cybernetics”, see Eduardo BayroCorrochano’s work. For some very helpful algorithms in rigid body dynamics and gravitational physics see the variety of publications by Joan and Anthony Lasenby. To get at the beginning of it all, read David Hestenes’ New Foundations for Classical Mechanics.
Typical matrix operation libraries have templated inlined functions for Vector and Matrix multiplication. Versor is similar, but on steroids, where vectors and sparse matrices of various sizes are all just called multivectors and represent geometric elements beyond just xyz directions and transformation matrices. Circles, lines, spheres, planes, points are all algebraic elements, as are operators that spin, twist, dilate, and bend those variables. Both these elements and operators are multivectors which multiply together in many many many different ways.
Versor compiles much faster than before, and without any silly predetermined list of allowable operations or types. Most notably, arbitrary metrics are now possible. For example, the xRoots.cpp example calculates all the Euclidean 4D reflections of a couple of point groups (F4 and D4, namely). So you can hypercube and polychoron away (8D cubes no problem!). Number of dimensions allowed are somewhat limited by your compiler infrastructure – let me know if you have a need that is not being met!
As for CGA, all the Pnt, Vec, Dll notation remains as before, but i’ve started adding utility functions since it helps people out.
auto pa = Round::point( 1,0,0 );
auto pb = Round::point( 0,1,0 );
auto pc = Round::point(1,0,0 );
auto circle = pa ^ pb ^ pc;
Draw(c);
If you like functional template metaprogramming, take a look at the code
and please let me know what you think. If you don’t, then I wouldn’t …
But if you have ideas or questions please do not hesitate to contact me.
GA combines many other maths (matrix, tensor, vector, and lie algebras). It is holistic. CGA uses a particular mapping (a conformal one) of 3D Euclidean space to a 4D sphere. Operations on that hypersphere are then projected back down to 3D. That how it works in a nutshell.
A fuller treatment of this question (er, the question of why we do this) can be found in my Master’s thesis on the subject. But basically, Geometic Algebra offers a particular richness of spatial expression. Imagine needing glasses and not knowing you needed glasses. Then, when you do get glasses, the world changes unexpectedly. GA is like glasses for the inside of your brain. Conformal Geometric Algebra, especially the 5D variety enlisted here, are like xray glasses. One point of clarification that occurs are disambiguations of previously collapsed concepts.
For instance, the main disambiguation, is that between a Point in space and a Vector in space.
A Point has no magnitude, but a Vector does. A Point has no direction, but a Vector does. Points are null Vectors. We can make them
by writing
Vec( 1,0,0 ).null();
More on that last point later … there are various binary operators defined (mainly three). We can introduce one right now, which is the dot or inner product.
In mathematics, the inner product of two points pa
and pb
is written \(p_{a} \rfloor p_{b}\). In Versor we use the <=
operator:
Point pa = Vec(1,0,0).null();
Point pb = Vec(1,0,0).null();
Scalar squaredDist = ( pa <= pb ) * 2;
which in this case would return a Scalar value of 4
. The 2
is there since the inner product really returns half the negative squared distance.
We can extract the Scalar into a c++ double like so:
double squaredDist = ( pa <= pb )[0] * 2;
Points thought of as Spheres (really, Dual Spheres, more on Duality later): they are Spheres of zero radius. As such they are a type of Round element. We can also build points this way:
Round::null( 1,0,0 );
or you can pass in another element
Round::null( Vec(1,0,0) );
or use the builtin method
Point pa = Vec(1,0,0).null();
Points can also be made with the macro PT
Point pa = PT(1,0,0);
which is just “syntactic sugar” for Vec(1,0,0).null()
Speaking of Spheres, we can also make spheres with a radius this way:
DualSphere dls = Round::dls( Vec( 1,0,0 ).null(), 1 );
or
DualSphere dls = Round::dls( Vec( 1,0,0 ), 1 );
or, specifying the radius first and then the coordinate:
DualSphere dls = Round::dls( 1 /* <radius */ , 1,0,0 )
all of which give a dual sphere of radius 1 at coordinate 1,0,0;
Versor is named after the one of the basic category of elements of geometric algebra.
A versor is a type of multivector which can be used to compose geometric transformations,
namely reflections, translations, rotations, twists, dilations, and transversions (special conformal transformations).
More on all of those transformations later.
In Versor, a Vector
(or Vec
) is a typical Euclidean 3D element. It can be built in the normal way:
Vec v(1,2,3);
Some builtin Vectors exist:
Vec::x x; //< X Direction Unit Vector Vec(1,0,0)
Vec::y y; //< Y Direction Unit Vector Vec(0,1,0)
Vec::z z; //< Z Direction Unit Vector Vec(0,0,1)
A Vector
can be spun around using a Rotor
, which is exactly like a quaternion. However, whereas quaternions are often built by specifying an axis
and an angle, rotors are built by specifying the plane of rotation. Eventually this will make much more sense to you: in general planes are what we
will be using to transform things. For instance, a reflection is a reflection in a plane. As we will see, planes can become hyperplanes which will allow for more extraordinary transformations.
The first completely new element to introduce is the Bivector
, which is the plane we will use to generate our Rotor
. Bivectors represent directed areas and are dual
to the cross product: the cross product of two vectors in typical vector algebra returns a vector normal to the plane they define. So it is not completely new,
but just sort of new.
Bivectors are also just three elements long, and are built the same way Vectors are.
Biv b(1,2,3);
Some builtin Bivectors exist:
Biv::xy xy; //< XY Counterclockwise Unit Area Biv(1,0,0)
Biv::xz xy; //< XZ Counterclockwise Unit Area Biv(0,1,0)
Biv::yz xy; //< YZ Counterclockwise Unit Area Biv(0,0,1)
While it is perfectly valid to write Vector
, Bivector
and Rotor
, you’ll notice I’ve truncated them to their three letter nicknames, Vec
and Rot
.
That’s up to you: Both longname and nickname versions are valid in libvsr (they are typedef’ed to each other).
Biv b = Biv::xy;
double theta = PIOVERTWO;
Vec v1 = Vec::x.rot( b * theta )
You can also generate rotors using Gen::rot( <some bivector> )
In fact, all transformations can be generated this way, and then later applied to arbitrary elements.
For instance, Motors
can be generated which translate and rotate an element at the same time. This is also called a twist.
Motor m = Gen::mot(<some dual line>); //< Makes A Twisting Motor around Some Dual Line
Point p = Vec(0,0,0).null().sp(m); //< Applies above motor to a Point
You’ll notice there are dual versions of elements: as in a DualLine
(or Dll
for short). That’s because in the real world of abstract geometry, there are usually
two ways of defining an element. For instance, we can build a direct Line
on the Yaxis by wedging two points together, along with infinity:
Line lin = Vec(0,0,0).null() ^ Vec(0,1,0).null() ^ Inf(1);
Or we can define a line by the bivector plane that it is normal to, and a support vector that determines how far away the line is from the origin. To convert the above line into its dual representation, we just call the dual() method:
Dll dll = lin.dual();
For those who are interested, this dual representation is isomorphic to the Plücker coordinates, which are used in screw theory to twist things around. Here, too, we can use dual lines to generate transformations which twist things around them.
The examples/*.cpp files include bindings to the GLV framework for windowing and user interface controls. This provides the minumum necessary glue to get started quickly building your own GA based graphics applications.
The interface has a built in gui, mouse info, and keyboard info stored.
//... a member of your App
Circle circle;
//...in App::setup()
objectController.attach(&circle);
Putting the above code inside your application will enable you to click and modify geometric elements by hitting the “T”, “R” and “S” keys (for translate, rotate, and scale) Hit any other key to deselect all elements.
[BUILTIN INTERFACE]
Key  Response 

~ 
Toggle full screen. 
SHIFT + Mouse or Arrow Keys 
Translate the camera in x and z directions. 
CTRL + Mouse or Arrow Keys 
Rotate the camera 
ALT +Arrow Keys 
Rotate the model view around. 
T 
Translate an Element 
R 
Rotate an Element 
S 
Scale an Element 
Tab 
Switch from navigation mode (default) to object manipulation mode 
Any other key  Release all Elements 
The elements of the algebra are geometric entities (circles, planes, spheres, etc) and operators (rotations, translations, twists, etc) which act on the elements of the algebra. All are known as multivectors since they are more than just your typical vectors.
Multivector elements are most often combined using three overloaded binary operators:
The Geometric Product of elements A
and B
:
A * B
multiplies two multivector elements together. This is most useful when multiplying one by the inverse of another (see !
operator, below).
The Outer Product of elements A
and B
:
A ^ B
“wedges” two multivectors together. Its from Grassman’s algebra of extensions, and can be thought of as a way of creating higher dimensions from smaller ones.
For instance, wedging two Vectors
(directed magnitudes) together returns a Bivector
(a directed Area). Wedging two Points
together returns a PointPair
.
Wedging three Points
together returns a Circle
.
The Inner Product of elements A
and B
:
A<=B
There is also a Commutator product (differential)
A%B
And a few overloaded operations, including,
The Inverse:
!A
returns \(A^{1}\)
The Reverse:
~A
returns \(\tilde{A}\)
And finally, since I ran out of overloadable operators, some basic methods
A.conj()
which returns \(\bar{A}\)
A.inv()
which returns \(\hat{A}\)
In summary:
Versor  Math  Description  

A * B 
\(AB\)  Multiplies two elements together (and, in the case of A * !B finds ratios between elements).  
A ^ B 
\(A \wedge B\)  Wedges two elements together (builds up higher dimensional elements).  
A <= B 
\(A \rfloor B\) or \(\boldsymbol{a} \cdot B\)  Contracts A out of B (returns the part of B “least like A”, sort of).  
A % B 
\(A \times B\)  Commutator, equal to \(\frac{1}{2}(ABBA)\)  
!A 
\(A^{1}\)  The Inverse of A.  
~A 
\(\tilde{A}\)  The Reverse of A.  
A.conj() 
\(\bar{A}\)  Conjugation.  
A.inv() 
\(\hat{A}\)  Involution. 
To make the process of writing code faster, all elements of the algebra are represented by types 3 letters long. Alternatively, you can also use the longform name.
Type  Long Form  Descrription 

Euclidean  
Sca 
Scalar 
A real number 
Vec 
Vector 
A Directed Magnitude, or 3D Vector, typical cartesian stuff 
Biv 
Bivector 
A Directed Area. Use them to make Rotors: Gen::Rot( Biv b ) 
Tri 
Trivector 
A Directed Volume Element 
Round  
Pnt 
Point 
A Null Vector: Pnt a = Vec(1,0,0).null() 
Par 
PointPair 
A 0Sphere (Sphere on a Line): Par par = Pnt a ^ Pnt b 
Cir 
Circle 
A 1Sphere: Cir cir = Pnt a ^ Pnt b ^ Pnt c 
Sph 
Sphere 
A 2Sphere: Sph sph = Pnt a ^ Pnt b ^ Pnt c ^ Pnt d 
Dls 
DualSphere 
Typedef’ed as a point: typedef Pnt Dls 
Flat  
Lin 
Line 
A Direct Line: e.g. Lin lin = Par par ^ Inf(1) 
Dll 
DualLine 
A Dual Line: e.g. Dll dll = lin.dual() 
Pln 
Plane 
A Direct Plane: e.g. Pln pln = Cir cir ^ Inf(1) 
Dlp 
DualPlane 
A Dual Plane: e.g. Dlp dlp = 
Flp 
FlatPoint 

Versors  
Rot 
Rotor 
Spins an Element (as a Quaternion would) 
Trs 
Translator 
Translates an Element 
Dil 
Dilator 
Dilates an Element 
Mot 
Motor 
Twists an Element along an axis 
Trv 
Transversor 
Bends an Element about the Origin 
Bst 
Booster 
Bends an Element around an “Orbit” 
Abstract  
Mnk 
MinkowskiPlane 

Pss 
Pseudoscalar 

Inf 
Infinity 
There are others as well (for instance, affine planes, lines, and points) but the above are more than sufficient to start with. There are also built in macros, for instance
EP
 Sphere At the Origin.
EM
 Imaginary Sphere at the Origin.
PT(x,y,z)
 A null Point at x,y,z
EP
and EM
can be invoked instead of Inf
to work in nonEuclidean metrics ( Spherical and Hyperbolic, respectively)
Many Euclidean elements can be drawn by invoking Draw::Render(
All elements can be dualized by invoking their dual()
method
All elements can be reflected over spinors with the sp(<spinor>)
method
All elements can be reflected over versors with the re(<versor>)
method
The versors are constructed by the geometric entities, typically by using the Gen::
routines. Operators can also be acted on by operators – you can rotate a translation, or twist a boost.
vsr_generic_op.h
and vsr_cga3D_op.h
contain the bulk of the functions for generating elements from other elements. Some guidelines:
Gen::
methods generate or otherwise operate on versorsroond::
methods create or otherwise operate on Round elements (Points, Point Pairs, Circles, Spheres)Flat::
methods create or otherwise operate on Flat elements (Lines, Dual Lines, Planes, Dual Planes, or Flat Points)Tangent::
methods create or otherwise operate on Tangent elements (Tangent Vectors, Tangent Bivectors, Tangent Trivectors)Returns  Function  Description 

Rot  Gen::rot( const Biv& b );  //<– Generate a Rotor from a Bivector 
Trs  Gen::trs( const Drv& v);  //<– Generate a Translator from a Direction Vector 
Mot  Gen::mot( const Dll& d);  //<– Generate a Motor from a Dual Line 
Dil  Gen::dil( const Pnt& p, double amt );  //<– Generate a Dilator from a Point and an amount 
Trv  Gen::trv( cont Tnv& v);  //<– Generate a Transveror from a Tangent Vector 
Bst  Gen::bst( const Par& p);  //<– Generate a Booster from a Point Pair 
In addition to the above “even” spinors, we can also reflect. Reflections (in a sphere, circle, or point pair, or over a line or plane ) can be calculated by writing
Pnt p = PT(1,0,0);
Pnt r = p.re( CXY(1) ); //Reflection of a point in a circle
r = r / r[3]; //Renormalization of a point
The re() method calculates v.re(C)
as C*v.inv()*~C
where inv() is an involution. With a versor C
and an element v
you might also try C * v * !C
. Inversion in a circle or a sphere may change the
weight of the element (for a Point at x, it will change it by x^2)