NAME¶
Math::Vector::Real - Real vector arithmetic in Perl
SYNOPSIS¶
use Math::Vector::Real;
my $v = V(1.1, 2.0, 3.1, -4.0, -12.0);
my $u = V(2.0, 0.0, 0.0, 1.0, 0.3);
printf "abs(%s) = %d\n", $v, abs($b);
my $dot = $u * $v;
my $sub = $u - $v;
# etc...
DESCRIPTION¶
A simple pure perl module to manipulate vectors of any dimension.
The function "V", always exported by the module, allows one to create
new vectors:
my $v = V(0, 1, 3, -1);
Vectors are represented as blessed array references. It is allowed to manipulate
the arrays directly as far as only real numbers are inserted (well, actually,
integers are also allowed because from a mathematical point of view, integers
are a subset of the real numbers).
Example:
my $v = V(0.0, 1.0);
# extending the 2D vector to 3D:
push @$v, 0.0;
# setting some component value:
$v->[0] = 23;
Vectors can be used in mathematical expressions:
my $u = V(3, 3, 0);
$p = $u * $v; # dot product
$f = 1.4 * $u + $v; # scalar product and vector addition
$c = $u x $v; # cross product, only defined for 3D vectors
# etc.
The currently supported operations are:
+ * /
- (both unary and binary)
x (cross product for 3D vectors)
+= -= *= /= x=
== !=
"" (stringfication)
abs (returns the norm)
atan2 (returns the angle between two vectors)
That, AFAIK, are all the operations that can be applied to vectors.
When an array reference is used in an operation involving a vector, it is
automatically upgraded to a vector. For instance:
my $v = V(1, 2);
$v += [0, 2];
Besides the common mathematical operations described above, the following
methods are available from the package.
Note that all these methods are non destructive returning new objects with the
result.
- $v = Math::Vector::Real->new(@components)
- Equivalent to "V(@components)".
- $zero = Math::Vector::Real->zero($dim)
- Returns the zero vector of the given dimension.
- $v = Math::Vector::Real->cube($dim, $size)
- Returns a vector of the given dimension with all its components set to
$size.
- $u = Math::Vector::Real->axis_versor($dim, $ix)
- Returns a unitary vector of the given dimension parallel to the axis with
index $ix (0-based).
For instance:
Math::Vector::Real->axis_versor(5, 3); # V(0, 0, 0, 1, 0)
Math::Vector::Real->axis_versor(2, 0); # V(1, 0)
- @b = Math::Vector::Real->canonical_base($dim)
- Returns the canonical base for the vector space of the given
dimension.
- $u = $v->versor
- Returns the versor for the given vector.
It is equivalent to:
$u = $v / abs($v);
- $wrapped = $w->wrap($v)
- Returns the result of wrapping the given vector in the box (hyper-cube)
defined by $w.
Long description:
Given the vector "W" and the canonical base "U1, U2,
...Un" such that "W = w1*U1 + w2*U2 +...+ wn*Un". For every
component "wi" we can consider the infinite set of affine
hyperplanes perpendicular to "Ui" such that they contain the
point "j * wi * Ui" being "j" an integer number.
The combination of all the hyperplanes defined by every component define a
grid that divides the space into an infinite set of affine hypercubes.
Every hypercube can be identified by its lower corner indexes "j1,
j2, ..., jN" or its lower corner point "j1*w1*U1 + j2*w2*U2
+...+ jn*wn*Un".
Given the vector "V", wrapping it by "W" is equivalent
to finding where it lays relative to the lower corner point of the
hypercube inside the grid containing it:
Wrapped = V - (j1*w1*U1 + j2*w2*U2 +...+ jn*wn*Un)
such that ji*wi <= vi < (ji+1)*wi
- $max = $v->max_component
- Returns the maximum of the absolute values of the vector components.
- $min = $v->min_component
- Returns the minimum of the absolute values of the vector components.
- $d2 = $b->norm2
- Returns the norm of the vector squared.
- $d = $v->dist($u)
- Returns the distance between the two vectors.
- $d = $v->dist2($u)
- Returns the distance between the two vectors squared.
- ($bottom, $top) = Math::Vector::Real->box($v0, $v1, $v2, ...)
- Returns the two corners of the axis-aligned minimum bounding box
<http://en.wikipedia.org/wiki/Minimum_bounding_box#Axis-aligned_minimum_bounding_box>
(or hyperrectangle <http://en.wikipedia.org/wiki/Hyperrectangle>)
for the given vectors.
In scalar context returns the difference between the two corners (the box
diagonal vector).
- $p = $v->nearest_in_box($w0, $w1, ...)
- Returns the vector nearest to $v from the axis-aligned minimum box
bounding the given set of vectors.
For instance, given a point $v and an axis-aligned rectangle defined by two
opposite corners ($c0 and $c1), this method can be used to find the point
nearest to $v from inside the rectangle:
my $n = $v->nearest_in_box($c0, $c1);
Note that if $v lays inside the box, the nearest point is $v itself.
Otherwise it will be a point from the box hyper-surface.
- $d2 = $v->dist2_to_box($w0, $w1, ...)
- Calculates the square of the minimal distance between the vector $v and
the minimal axis-aligned box containing all the vectors "($w0, $w1,
...)".
- $d2 = $v->max_dist2_to_box($w0, $w1, ...)
- Calculates the square of the maximum distance between the vector $v and
the minimal axis-aligned box containing all the vectors "($w0, $w1,
...)".
- $d2 = Math::Vector::Real->max_dist2_between_boxes($a0, $a1, $b0,
$b1)
- Returns the square of the maximum distance between any two points
belonging respectively to the boxes defined by "($a0, $a1)" and
"($b0, $b1)".
- $v->set($u)
- Equivalent to "$v = $u" but without allocating a new object.
Note that this method is destructive.
- $d = $v->max_component_index
- Returns the index of the vector component with the maximum size.
- $r = $v->first_orthant_reflection
- Given the set of vectors formed by $v and all its reflections around the
axis-aligned hyperplanes, this method returns the one lying on the first
orthant.
See also
[http://en.wikipedia.org/wiki/Reflection_%28mathematics%29|reflection] and
[http://en.wikipedia.org/wiki/Orthant|orthant].
- ($p, $n) = $v->decompose($u)
- Decompose the given vector $u in two vectors: one parallel to $v and
another normal.
In scalar context returns the normal vector.
- $v = Math::Vector::Real->sum(@v)
- Returns the sum of all the given vectors.
- @b = Math::Vector::Real->complementary_base(@v)
- Returns a base for the subspace complementary to the one defined by the
base @v.
The vectors on @v must be linearly independent. Otherwise a division by zero
error may pop up or probably due to rounding errors, just a wrong result
may be generated.
- @b = $v->normal_base
- Returns a set of vectors forming an orthonormal base for the hyperplane
normal to $v.
In scalar context returns just some unitary vector normal to $v.
Note that this two expressions are equivalent:
@b = $v->normal_base;
@b = Math::Vector::Real->complementary_base($v);
- ($i, $j, $k) = $v->rotation_base_3d
- Given a 3D vector, returns a list of 3 vectors forming an orthonormal base
where $i has the same direction as the given vector $v and "$k = $i x
$j".
- @r = $v->rotate_3d($angle, @s)
- Returns the vectors @u rotated around the vector $v an angle $angle in
radians in anticlockwise direction.
See
<http://en.wikipedia.org/wiki/Rotation_operator_(vector_space)>.
- @s = $center->select_in_ball($radius, $v1, $v2, $v3, ...)
- Selects from the list of given vectors those that lay inside the n-ball
determined by the given radius and center ($radius and $center
respectively).
Zero vector handling¶
Passing the zero vector to some methods (i.e. "versor",
"decompose", "normal_base", etc.) is not acceptable. In
those cases, the module will croak with an "Illegal division by
zero" error.
"atan2" is an exceptional case that will return 0 when any of its
arguments is the zero vector (for consistency with the "atan2"
builtin operating over real numbers).
In any case note that, in practice, rounding errors frequently cause the check
for the zero vector to fail resulting in numerical instabilities.
The correct way to handle this problem is to introduce in your code checks of
this kind:
if ($v->norm2 < $epsilon2) {
croak "$v is too small";
}
Or even better, reorder the operations to minimize the chance of instabilities
if the algorithm allows it.
Math::Vector::Real::XS¶
The module Math::Vector::Real::XS reimplements most of the methods available
from this module in XS. When it is installed, "Math::Vector::Real"
when automatically load and use it.
SEE ALSO¶
Math::Vector::Real::Random extends this module with random vector generation
methods.
Math::GSL::Vector, PDL.
There are other vector manipulation packages in CPAN (Math::Vec,
Math::VectorReal, Math::Vector), but they can only handle 3 dimensional
vectors.
SUPPORT¶
In order to report bugs you can send me and email to the address that appears
below or use the CPAN RT bug-tracking system available at
<
http://rt.cpan.org>.
The source for the development version of the module is hosted at GitHub:
<
https://github.com/salva/p5-Math-Vector-Real>.
My wishlist¶
If you like this module and you're feeling generous, take a look at my wishlist:
<
http://amzn.com/w/1WU1P6IR5QZ42>
COPYRIGHT AND LICENSE¶
Copyright (C) 2009-2012, 2014 by Salvador Fandin~o (sfandino@yahoo.com)
This library is free software; you can redistribute it and/or modify it under
the same terms as Perl itself, either Perl version 5.10.0 or, at your option,
any later version of Perl 5 you may have available.