#include "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 CC/DCC (Irix), g++, and Metrowerks C++.
The latest version can be retrieved from
http://www.cs.cmu.edu/~ajw/public/dist/svl.tar.gz.
Classes
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
v[2] = 4.0; // set element 2 of the vector
m[3][4] = 5.0 // set row 3, column 4 of the matrix
m[2] = v; // set row 2 of the matrix
Though it seems slightly unintuitive, if you have a pointer to a vector or matrix,
you must dereference it first before indexing it:
(*vPtr)[20] = 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 -DVL_CHECKING, index range checks will be performed on
all element accesses.
Arithmetic Operators and Functions
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.
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[2]
Vec3 proj(const Vec4 &v); // homog. projection: v[0..2] / v[3]
For more on the use of the proj() operator, see Transformations.
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.
The det() and adj() functions are only defined for Mat[234].
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
Vec[234] v([initialisation_constant | element_list]);
Vec v([elements, [initialisation_constant | element_list]]);
Mat[234] 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);
#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.
Mat2 Rot2(Real theta)
// rotate a 2d vector CCW by theta
Mat2 Scale2(const Mat2 &s)
// scale by s around the origin
Mat3 HRot3(Real theta)
// rotate a homogeneous 2d vector CCW by theta
Mat3 HScale3(const Mat2 &s)
// scale by s around the origin, in homogeneous 2d coords.
Mat3 HTrans3(const Mat2 &t)
// translate a homogeneous 2d vector by t
Mat3 Rot3(const Mat3 &axis, Real theta)
// rotate a 3d vector CCW around axis by theta
Mat3 Rot3(const Mat4 &q)
// rotate a 3d vector by the quaternion q
Mat3 Scale3(const Vec3 &s)
// scale by s around the origin
Mat4 HRot4(const Mat3 &axis, Real theta)
// rotate a homogeneous 3d vector CCW around axis by theta
Mat4 HRot4(const Mat4 &q)
// rotate a homogeneous 3d vector by the quaternion q
Mat4 HScale4(const Mat3 &s)
// scale by s around the origin, in homogeneous 3d coords
Mat4 HTrans4(const Mat3 &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(Scale4d(...) * Rot4d(...) * Trans4d(...) * Vec4(y, 1.0));
// do the same thing with xform()
x = xform(Scale4d(...) * Rot4d(...) * Trans4d(...), 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).
VL_FLOAT - use floats instead of doubles
VL_CHECKING - turn on index checking and assertions
VL_ROW_ORIENT - transformations operate on row vectors instead of column vectors
#include "SVLgl.h"
Vec3 x(24, 0, 100), y(40, 20, 10);
glBegin(GL_LINES);
glVertex(x);
glVertex(y);
glEnd();
Andrew Willmott (ajw+vl@cs.cmu.edu), Graphics Group, SCS, CMU.