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\author{J. Andreas B{\ae}rentzen}

\title{CGLA usage}

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\begin{document}

\maketitle

\section{Introduction}

CGLA is a set of numerical C++ vector and matrix classes and class

templates designed with computer graphics in mind. CGLA stands for

``Computer Graphics Linear Algebra''. It is a part of the larger

package called GEL but deserves its own document since it is an  important part.

Let us get right down to the obvious question: Why create another

linear algebra package?

Well, CGLA evolved from a few matrix and vector classes because I

didn't have anything better. Also, I created CGLA to experiment with

some template programming techniques. This led to the most important

feature of CGLA, namely the fact that all vector types are derived

from the same template. 

This makes it easy to ensure identical semantics: Since all vectors

have inherited, say, the * operator from a common ancestor, it works

the same for all of them.

It is important to note that CGLA was designed for Computer Graphics 

(not numerical computations) and this had a number of

implications. Since, in computer graphics we mainly need small vectors

of dimension 2,3, or 4 CGLA was designed for vectors of low

dimensionality. Moreover, the amount of memory allocated for a vector

is decided by its type at compile time. CGLA does not use dynamic

memory. CGLA also does not use virtual functions, and most functions

are inline. These features all help making CGLA relatively fast. 

Of course, other libraries of vector templates for computer graphics

exist, but to my knowledge none where the fundamental templates are

parametrized w.r.t. dimension as well as type. In other words, we have

a template (\texttt{ArithVec})that gets both type

(e.g. \texttt{float}) and dimension 

(e.g. 3) as arguments. the intended use of this template is as

ancestor of concrete types such as \texttt{Vec3f} - a 3D floating

point type. 

The use of just one template as basis is very important, I believe,

since it makes it extremely simple to add new types of

vectors. Another very generic template is \texttt{ArithMat} which is a

template for matrix classes. (and not necessarily NxN matrices). 

From a users perspective CGLA contains a number of vector and matrix

classes, a quaternion and some utility classes. In summary, the most

important features are

\begin{itemize}

\item A number of 2, 3 and 4 d vector classes.

\item A number of Matrix classes.

\item A Quaternion class.

\item Some test programs.

\item Works well with OpenGL.

\end{itemize}

\subsection{Naming conventions}

Vectors in CGLA are named \texttt{VecDT} where \texttt{D} stands for

dimension at  \texttt{T} 

for type. For instance a 3D floating point vector is named

\texttt{Vec3f}. Other types are d (double), s (short), i (int), uc

(unsigned char), and usi (unsigned shourt int).

Matrices are similiarly named \texttt{MatDxDT}. For instance a 4D

\texttt{double} matrix is called \texttt{Mat4x4d}.

\subsection{How to use CGLA}

If you need a given CGLA class you can find the header file that

contains it in this document. Simply include the header file and use

the class. Remember also that all CGLA functions and classes live in

the CGLA namespace! Lastly, look at the example programs that came

with the code.

An important point is that you should never use the Arith... classes

directly. Classes whose names begin with Arith are templates used for

deriving concrete types. It is simpler, cleaner, and the intended

thing to do to only use the derived types.

In some cases, you may find that you need a vector or matrix class

that I haven't defined. If so, it is fortunate that CGLA is easy to

extend. Just look at, say, \texttt{Vec4f} if you need a \texttt{Vec5d}

class.

For some more specific help look at the next section where some of the

commonly used operations are shown.

\subsection{How to get CGLA}

If you have this document but not the code, look at\\

\href{http://www2.imm.dtu.dk/projects/GEL/}{http://www2.imm.dtu.dk/projects/GEL/}

\subsection{Bugs etc.}

CGLA was written (mostly) by Andreas B{\ae}entzen (jab{@}imm.dtu.dk), and

any bug fixes, contributions, or questions should be addressed to me.

\section{CGLA: Examples of Usage}

The examples below are by no means complete. Many things are

possible but not covered below. However, most of the common usage is

shown, so this should be enough to get you started. Note that in the

following we assume that you are \texttt{using namespace CGLA} and

hence don't prefix with \texttt{CGLA::}.

In short, to compile the examples below you would need the following

in the top of your file

\begin{lstlisting}{}

#include <iostream> // For input output

#include "CGLA/Vec3f.h"

#include "CGLA/Quaternion.h"

#include "CGLA/Mat4x4f.h"

using namespace std; // For input output

using namespace CGLA;

\end{lstlisting}

To begin with let us create 3 3D vectors. This is done as follows:

\begin{lstlisting}{}

  Vec3f p0(10,10,10);

  Vec3f p1(20,10,10);

  Vec3f p2(10,20,10);

\end{lstlisting}

if we had left out the arguments the three vectors would have been

uninitialized, at least in release mode. If we compile in debug mode

they would have been initialized to ``Not a Number'' to aid

debugging. However, the $[10 10 10]^T$ vector could also have been

created differently, using

\begin{lstlisting}{}

  Vec3f p0(10);

\end{lstlisting}

A very common operation is to compute the normal of a triangle from

the position of its vertices. Assuming the three vectors represent the

vertices of a triangle, we can compute the normal by finding the

vector from the first vertex to the two other vertices, taking the

cross product and normalizing the result. This is a one-liner:

\begin{lstlisting}{}

  Vec3f n = normalize(cross(p1-p0,p2-p0));

\end{lstlisting}

Quite a lot goes on in the snippet above. Observe that the - operator

also works for vectors. In fact almost all the arithmetic operators

work for vectors. You can also use assignment operators (i.e

\texttt{+=}) which is often faster. Then there is the function

\texttt{cross} which simply computes the cross product of its

arguments. Another frequently used function is \texttt{dot} which

takes the dot product. Finally the vector is normalized using the

function normalize.

Of course, we can print all or at least most CGLA entities. For

example 

\begin{lstlisting}{}

  cout << n << endl;

\end{lstlisting}

will print the normal vector just computed. We can also treat a vector

as an array as shown below

\begin{lstlisting}{}

  float x = n[0];

\end{lstlisting}

here, of course, we just extracted the first coordinate of the

vector. 

CGLA contains a number of features that are not used very frequently,

but which are used frequently enough to warrent inclusion. A good

example is assigning to a vector using spherical coordinates:

\begin{lstlisting}{}

  Vec3f p;

  p.set_spherical(0.955317, 3.1415926f/4.0f , 1);

\end{lstlisting}

CGLA also includes a quaternion class. Here it is used to construct a

quaternion which will rotate the x axis into the y axis.

\begin{lstlisting}{}

  Quaternion q;

  q.make_rot(Vec3f(1,0,0), Vec3f(0,1,0));

\end{lstlisting}

Next, we construct a $4\times 4$ matrix \texttt{m} and assign a translation

matrix to the newly constructed matrix. After that we ask the

quaternion to return a $4\times 4$ matrix corresponding to its

rotation. This rotation matrix is then multiplied onto \texttt{m}.

\begin{lstlisting}{} 

  Mat4x4f m = translation_Mat4x4f(Vec3f(1,2,3));

  m *= q.get_mat4x4f();

\end{lstlisting}

Just like for vectors, the subscript operator works on

matrices. However, in this case there are two indices. Just using one

index will return the ith row as a vector as shown on the first line

below. On the second line we see that using two indices will get us an

element in the matrix. 

\begin{lstlisting}{} 

  Vec4f v4 = m[0];

  float c = m[0][3];

\end{lstlisting}

There is a number of constructors for vectors. The default constructor

will create a null vector as we have already seen. We can also specify

all the coordinates. Finally, we can pass just a single number

$a$. This will create the $[a\:a\:a]^T$ vector. For instance, below we

create the $[1\: 1\: 1]^T$ vector. Subsequently, this vector is

multiplied onto \texttt{m}. 

\begin{lstlisting}{}

  Vec3f p(1);

  Vec3f p2 = m.mul_3D_point(p);

\end{lstlisting}

Note though that \texttt{m} is a $4\times

4$ matrix so ... how is that possible? Well, we use the function

\texttt{mul\_3D\_point} which, effectively, adds a $w=1$ coordinate to p

making it a 4D vector. This $w$ coordinate is stripped afterwards. In

practice, this means that the translation part of the matrix is also

applied. There is a similar function \texttt{mul\_3D\_vector} if we want

to transform vectors without having the translation. This function,

effectively, sets $w=0$.

Finally, CGLA is often used together with OpenGL although there is no

explicit tie to the OpenGL library. However, we can call the

\texttt{get} function of most CGLA entities to get a pointer to the

contents. E.g. \texttt{p.get()} will return a pointer to the first

float in the 3D vector \texttt{p}. This can be used with OpenGL's

``v'' functions as shown below. 

\begin{lstlisting}{}

  glVertex3fv(p.get());

\end{lstlisting}

\end{document}