# 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:

RT: CPAN's request tracker (report bugs here)

http://rt.cpan.org/NoAuth/Bugs.html?Dist=Math-Geometry-Multidimensional

AnnoCPAN: Annotated CPAN documentation

CPAN Ratings

http://cpanratings.perl.org/d/Math-Geometry-Multidimensional

Search CPAN

# 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

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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.

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