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Math-Geometry-Multidimensional-0.02
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/ Math::Geometry::Multidimensional

NAME

Math::Geometry::Multidimensional - geometrical functions in n-dimensions

VERSION

Version 0.02

SYNOPSIS

This module has a bunch of functions that work in mulitiple dimensions, e.g. distance of a point from a line in n-dimensions.

    use Math::Geometry::Multidimensional qw/distanceToLineN/;
    # parametric:
    my $distance = distanceToLineN($point, $gradients, $intersect);
    # symmetric:
    my $distance = distanceToLineP($point, $denominators, $constants);

EXPORT

distanceToLineN
diagonalComponentsN
diagonalDistancesFromOriginN

SUBROUTINES/METHODS

distanceToLineN

We have a line with symmetric form:

(x+a)/m = (y+b)/n = (z+c)/p ...

@M is the list of denominators and @C is the list of constants.

For a point $P,

        distanceToLineN($P,\@M,\@C)

returns the distance to the closest point on the line... in n-dimensions.

lineFromTwoPoints

diagonalDistanceFromOriginN

This is the distance along the y=z=x=... line from any point to the origin. First we find the closest point on y=z=x=... from our point, which happens to be the average of the coordinates, e.g. if the point is (10,8) then the closest point on y=z is 9,9. If the point is (9,8,4) then the closest point on z=y=x is (7,7,7). If the point is (2,3,4,7) then the closest point on z=y=x=w is (4,4,4,4). Why?

For P(u,v,w) and L: (x+a)/k = (y+b)/l = (z+c)/m = t

we know that x=kt-a ; y=lt-b ; z=my-c

so k(kt-a) + l(lt-b) + m(mt-c) = kkt-ka + llt-lb + mmt-mc = ku+lv+mw OR t(kk+ll+mm) = k(u+a)+l(v+b)+m(w+c) so t = (k(u+a)+l(v+b)+m(w+c)) / (kk+ll+mm)

BUT, if a=b=c=0 and k=l=m=1, then:

t = (x+y+z)/(3)

in general, t is the average of the coordinates.

then, x' = kt-a, and if k is 1 and a is 0, then x' is t.

P' is (t,t,t) so the distance to P' from the origin is sqrt(3 t^2) or sqrt( 3 * ((x+y+z)/3)^2) or sqrt( 3 * (x+y+z)^2 / 9 ) or sqrt( (x+y+z)^2 / 3) or (x+y+z)/sqrt(3) or SUM(coords)/sqrt(n)

Does that make sense?

diagonalDistancesFromOriginN

Acts on columns rather than an individual point... give it column number, row number and list of columns.

        my $arrayref = diagonalDistancesFromOriginN ($k,$n,@cols)

diagonalComponentsN

Here, we are basically rotating all the data so that the "y-axis" or whatever you want to call the left-most co-ordinate now lies diagonally through the data.

distanceFromDiagonalN

As above, we know that the point P' on the diagonal closest to our point P has the average coordinates of point P. And the distance PP' (x-x', y-y', z-z') is the root of the sum of the squares. So

so, if x' = t, which is (x+y+z)/3 ...

PP' = sqrt( (x - x/3 - y/3 - z/3)^2 + (y - x/3 - y/3 - z/3)^2 + (z + x/3 + y/3 + z/3)^2 )

= sqrt( x^2 (2/3) + y^2 (2/3) + z^2 (2/3) + 2xy + 2xz + 2yz )

this is not implemented yet.

AUTHOR

Jimi Wills, <jimi at webu.co.uk>

BUGS

Please report any bugs or feature requests to bug-math-geometry-multidimensional at rt.cpan.org, or through the web interface at http://rt.cpan.org/NoAuth/ReportBug.html?Queue=Math-Geometry-Multidimensional. I will be notified, and then you'll automatically be notified of progress on your bug as I make changes.

SUPPORT

You can find documentation for this module with the perldoc command.

    perldoc Math::Geometry::Multidimensional

You can also look for information at:

ACKNOWLEDGEMENTS

LICENSE AND COPYRIGHT

Copyright 2013 Jimi Wills.

This program is free software; you can redistribute it and/or modify it under the terms of the the Artistic License (2.0). You may obtain a copy of the full license at:

http://www.perlfoundation.org/artistic_license_2_0

Any use, modification, and distribution of the Standard or Modified Versions is governed by this Artistic License. By using, modifying or distributing the Package, you accept this license. Do not use, modify, or distribute the Package, if you do not accept this license.

If your Modified Version has been derived from a Modified Version made by someone other than you, you are nevertheless required to ensure that your Modified Version complies with the requirements of this license.

This license does not grant you the right to use any trademark, service mark, tradename, or logo of the Copyright Holder.

This license includes the non-exclusive, worldwide, free-of-charge patent license to make, have made, use, offer to sell, sell, import and otherwise transfer the Package with respect to any patent claims licensable by the Copyright Holder that are necessarily infringed by the Package. If you institute patent litigation (including a cross-claim or counterclaim) against any party alleging that the Package constitutes direct or contributory patent infringement, then this Artistic License to you shall terminate on the date that such litigation is filed.

Disclaimer of Warranty: THE PACKAGE IS PROVIDED BY THE COPYRIGHT HOLDER AND CONTRIBUTORS "AS IS' AND WITHOUT ANY EXPRESS OR IMPLIED WARRANTIES. THE IMPLIED WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE, OR NON-INFRINGEMENT ARE DISCLAIMED TO THE EXTENT PERMITTED BY YOUR LOCAL LAW. UNLESS REQUIRED BY LAW, NO COPYRIGHT HOLDER OR CONTRIBUTOR WILL BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, OR CONSEQUENTIAL DAMAGES ARISING IN ANY WAY OUT OF THE USE OF THE PACKAGE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.