Simple Vector Library        (1.5)

The Simple Vector Library (SVL) provides vector and matrix classes, as well as a number of functions for performing vector arithmetic with them. Equation-like syntax is supported via class operators, for example:

`    #include "svl/SVL.h"    Vec3    v(1.0, 2.0, 3.0);    Mat3    m(1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0);    v =  2 * v + m * v;    v *= (m / 3.0) * norm(v);        cout << v << endl;`
Both generic (arbitrarily-sized), and fixed-size (2, 3 and 4 element) vectors and matrices are supported. The latter are provided for the efficient manipulation of vectors or points in 2D or 3D space, and make heavy use of inlining for frequently-used operations. (One of the design goals of SVL was to ensure that it was as fast as the C-language, macro-based libraries it was written to replace.)

SVL is a subset of VL, a more extensive vector library, which in addition contains classes for sparse vector/matrices, sub-vector/matrices, and implementations of some iterative solvers. Whereas with SVL the component type of vectors and matrices is defined with a compile-time switch, with VL you specify the type explicitly (Vec2f, Vec2d) and can, for example, mix matrices of doubles with vectors of floats.

SVL requires C++. It is known to compile under Irix CC, g++, and MSVC. The latest version can be retrieved from http://www.cs.cmu.edu/~ajw/public/dist/. This documentation can be found online at http://www.cs.cmu.edu/~ajw/doc/svl.html.

SVL contains the following types and classes:

`    Fixed-size:        Real            component type (float or double)        Vec2            2-vector        Vec3            3-vector        Vec4            4-vector        Mat2            2 x 2 matrix        Mat3            3 x 3 matrix        Mat4            4 x 4 matrix    Generic:        Vec             n-vector        Mat             n x m matrix`

The elements of a vector or matrix are accessed with standard C array notation:

`       v = 4.0;         // set element 2 of the vector    m = 5.0       // set row 3, column 4 of the matrix    m = v;           // set row 2 of the matrix`
For the resizeable vector types, the current size can be obtained from the Elts() method for vectors, and the Rows() and Cols() methods for matrices. To iterate over all members of these types, you can use code of the form:
`    for (i = 0; i < v.Elts(); i++)        v[i] = i;`
`    for (i = 0; i < m.Rows(); i++)        for (j = 0; j < m.Cols(); j++)            m[i][j] = i + j;`
Though it seems slightly unintuitive, if you have a pointer to a vector or matrix, you must dereference it first before indexing it:
`    (*vPtr) = 3.0;`
If you need a pointer to the data belonging to a vector or matrix, use the Ref() method. (Matrices are stored by row.)
`    Real *vecDataPtr = v.Ref(), *matDataPtr = m.Ref();`
Warning: Any pointer to a generic matrix or vector will become invalid as soon as it is resized.

Note: If you compile with the symbol VL_DEBUG, index range checks will be performed on all element accesses.

### Arithmetic Operators

The following binary operators are defined for all vector and matrix classes, as long as both operands are of the same type.
`    Basic arithmetic:         + - * /    Accumulation arithmetic:  += -= *= /=    Comparison:               ==, !=`
Vector multiplication and division is pairwise: (a * b)[i] = a[i] * b[i]. (See below for how to form the dot product of two vectors with dot().) Matrix multiplication is defined as usual, and matrix division is undefined.

For both matrices and vectors, multiplication and division by a scalar is also allowed. Matrices can be multiplied either on the left or the right by a vector. In the expression m * v, v is treated as a column vector; in the expression v * m, it is treated as a row vector.

### Vector Functions

The following is a list of the various vector functions, together with a short description of what they return.
`       Real dot(const VecN &a, const VecN &b);     // inner product of a and b    Real len(const VecN &v);                    // length of v: || v ||    Real sqrlen(const VecN &v);                 // length of v, squared    VecN norm(const VecN &v);                   // v / || v ||    Vec2 cross(const Vec2 &a);                  // vector orthogonal to a     Vec3 cross(const Vec3 &a, const Vec3 &b);   // vector orthogonal to a and b    Vec4 cross(const Vec4 &a, const Vec4 &b, const Vec4 &c);                                                // vector orthogonal to a, b and c    Vec2 proj(const Vec3 &v);                   // homog. projection: v[0..1] / v    Vec3 proj(const Vec4 &v);                   // homog. projection: v[0..2] / v`
In the above, VecN is either a Vec or a Vec, and Real is a Float or a Double, depending on the argument type. For more on the use of the proj() operator, see Transformations.

### Matrix Functions

The following functions can be used with matrices.
`    MatN trans(const MatN &m);                  // Transpose of m               Real trace(const MatN &m);                  // Trace of m    MatN adj(const MatN &m);                    // Adjoint of m    Real det(const MatN &m);                    // Determinant of m    MatN inv(const MatN &m);                    // Inverse of m, if it exists.`
In the above, MatN is any matrix type, a, though the det() and adj() functions are only defined for Mat.

There are a number of 'magic' constants in SVL that can be used to initialise vectors or matrices with simple assignment statements. For example:

`    Vec3 v; Mat3 m; Vec v8(8);    v = vl_0            [0, 0, 0]    v = vl_y            [0, 1, 0]    v = vl_1            [1, 1, 1]        m = vl_0;           3 x 3 matrix, all elts. set to zero.    m = vl_1;           3 x 3 identity matrix    m = vl_B;           3 x 3 matrix, all elts. set to one.        v8 = vl_axis(6);    [0, 0, 0, 0, 0, 0, 1, 0]`
Below is a summary of the constants defined by SVL.
`    vl_one/vl_1/vl_I              vector of all 1s, or identity matrix    vl_zero/vl_0/vl_Z             vector or matrix of all 0s    vl_B                          matrix of all 1s    vl_x, vl_y, vl_z, vl_w        x, y, z and w axis vectors    vl_axis(n)                    zero vector with element n set to 1    vl_pi                         pi!    vl_halfPi                     pi/2`

In general, a vector or matrix constructor should be given either one of the initialiser constants listed above, or a list of values for its elements. If neither of these is supplied, the variable will be uninitialised. The first arguments to the constructor of a generic vector or matrix should always be the required size. Thus matrices and vectors are declared as follows:

`    Vec    v([initialisation_constant | element_list]);    Vec         v([elements, [initialisation_constant | element_list]]);    Mat    m([initialisation_constant | element_list]);    Mat         m([rows, columns, [initialisation_constant | element_list]]);`
If generic vectors or matrices are not given a size when first created, they are regarded as empty, with no associated storage. This state persists until they are assigned a matrix/vector or the result of some computation, at which point they take on the dimensions of that result.

Examples:

`    Vec3    v(vl_1);    Vec3    v(1.0, 2.0, 3.0);    Vec     v(6, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0);     Vec     v(20, vl_axis(10));    Mat2    m(1.0, 2.0, 3.0, 4.0);    Mat     m(10, 20, vl_I);`
Warning: When initialising a generic vector or matrix with a list of elements, you must always ensure there is no possibility of the element being mistaken for an integer. (This is due to limitations of the stdarg package.) Make sure that each element value has either an exponent or a decimal point, i.e., use '2.0' rather than just '2'.

Finally, to set the size of a empty matrix or vector explicitly, or resize an existing matrix or vector, use the SetSize method:

`       v.SetSize(23);    m.SetSize(10, 20);`

All of the vector and matrix types in SVL can be used in iostream-type expressions. For example:

`    #include <iostream.h>    Vec3 v(vl_1);    cout << v << 2 * v << endl;    cin >> v;`
will output
`    [1 1 1][2 2 2]`
and then prompt for input. Vectors and matrices are parsed in the same format that they are output: vectors are delimited by square brackets, elements separated by white space, and matrices consist of a series of row vectors, again delimited by square brackets.

The following are the transformations supported by SVL.

`    Mat2    Rot2(Real theta)            // rotate a 2d vector CCW by theta    Mat2    Scale2(const Vec2 &s)            // scale by s around the origin        Mat3    HRot3(Real theta)            // rotate a homogeneous 2d vector CCW by theta    Mat3    HScale3(const Vec2 &s)            // scale by s around the origin, in homogeneous 2d coords.    Mat3    HTrans3(const Vec2 &t)            // translate a homogeneous 2d vector by t    Mat3    Rot3(const Vec3 &axis, Real theta)            // rotate a 3d vector CCW around axis by theta    Mat3    Rot3(const Vec4 &q)            // rotate a 3d vector by the quaternion q    Mat3    Scale3(const Vec3 &s)            // scale by s around the origin    Mat4    HRot4(const Vec3 &axis, Real theta)            // rotate a homogeneous 3d vector CCW around axis by theta    Mat4    HRot4(const Vec4 &q)            // rotate a homogeneous 3d vector by the quaternion q    Mat4    HScale4(const Vec3 &s)            // scale by s around the origin, in homogeneous 3d coords    Mat4    HTrans4(const Vec3 &t)            // translate a homogeneous 3d vector by t`
Transformations with a prefix of 'H' operate in the homogeneous coordinate system, which allows translation and shear transformations, as well as the usual rotation and scale. In this coordinate system an n-vector is embedded in a (n+1)-dimensional space, e.g., a homogeneous point in 2d is represented by a 3-vector.

To convert from non-homogeneous to homogeneous vectors, make the extra coordinate (usually 1) the second argument in a constructor of/cast to the next-higher dimension vector. To project from a homogeneous vector down to a non-homogeneous one (doing a homogeneous divide in the process), use the proj() function. This process can be simplified by the use of the xform() function, which applies a transform to a vector, doing homogeneous/nonhomogeneous conversions if necessary. For example:

`    Vec3 x,y;    // apply homogeneous transformations to a 3-vector     x = proj(Scale4(...) * Rot4(...) * Trans4(...) * Vec4(y, 1.0));    // do the same thing with xform()    x = xform(Scale4(...) * Rot4(...) * Trans4(...), y);`
By default, SVL assumes that transformations should operate on column vectors (v = T * v), though it can be compiled to assume row vectors instead (v = v * T).

For basic use, the only header file needed is svl/SVL.h.

For your final build, link with -lsvl (libsvl.a). To use the debugging version of SVL, which has assertions and range checking turned on, use -lsvl.dbg (libsvl.dbg.a), and compile with the VL_DEBUG symbol. This debugging version includes checks for correct matrix and vector sizes during arithmetic operations.

### Compile options

SVL uses the following compile-time options:
`    VL_FLOAT        - use floats instead of doubles    VL_DEBUG        - turn on index checking and assertions    VL_ROW_ORIENT   - transformations operate on row vectors instead of column vectors`

SVL comes with a header file, SVLgl.h, which makes using SVL vectors with OpenGL more convenient. For example:

`    #include "svl/SVLgl.h"        Vec3 x(24, 0, 100), y(40, 20, 10);        glBegin(GL_LINES);    glVertex(x);     glVertex(y);    glEnd();`