Name

Math::Vectors2 - Vectors in two dimensions

Synopsis

use Math::Vectors2;

my ($zero, $x, $y) = Math::Vectors2::zeroAndUnits;

ok near deg2rad(-60),  $x + $y * sqrt(3)    <    $x;
ok near deg2rad(+30), ($x + $y * sqrt(3))->angle($y);

Description

Vectors in two dimensions

Version 20200402.

The following sections describe the methods in each functional area of this module. For an alphabetic listing of all methods by name see Index.

Methods

Vector methods.

new($x, $y)

Create new vector from components.

   Parameter  Description
1  $x         X component
2  $y         Y component

Example:

my ($zero, $x, $y) = zeroAndUnits;
ok near $y->angle(𝗻𝗲𝘄(+1, -1)), deg2rad(-135);
ok near $y->angle(𝗻𝗲𝘄(+1,  0)), deg2rad(-90);
ok near $y->angle(𝗻𝗲𝘄(+1, +1)), deg2rad(-45);
ok near $y->angle(𝗻𝗲𝘄( 0, +1)), deg2rad(+0);
ok near $y->angle(𝗻𝗲𝘄(-1, +1)), deg2rad(+45);
ok near $y->angle(𝗻𝗲𝘄(-1,  0)), deg2rad(+90);
ok near $y->angle(𝗻𝗲𝘄(-1, -1)), deg2rad(+135);

ok near 𝗻𝗲𝘄(1,1) < 𝗻𝗲𝘄( 0, -1), deg2rad(-135);
ok near 𝗻𝗲𝘄(1,1) < 𝗻𝗲𝘄( 1, -1), deg2rad(-90);
ok near 𝗻𝗲𝘄(1,1) < 𝗻𝗲𝘄( 1,  0), deg2rad(-45);
ok near 𝗻𝗲𝘄(1,1) < 𝗻𝗲𝘄( 1,  1), deg2rad(0);
ok near 𝗻𝗲𝘄(1,1) < 𝗻𝗲𝘄( 0,  1), deg2rad(+45);
ok near 𝗻𝗲𝘄(1,1) < 𝗻𝗲𝘄(-1,  1), deg2rad(+90);
ok near 𝗻𝗲𝘄(1,1) < 𝗻𝗲𝘄(-1,  0), deg2rad(+135);

ok near $x + $y * sqrt(3) < $x, deg2rad(-60);
ok near $x + $y * sqrt(3) < $y, deg2rad(+30);

for my $i(-179..179)
 {ok near $x < 𝗻𝗲𝘄(cos(deg2rad($i)), sin(deg2rad($i))), deg2rad($i);
 }

This is a static method and so should either be imported or invoked as:

Math::Vectors2::new

zeroAndUnits()

Create the useful vectors: o=(0,0), x=(1,0), y=(0,1)

Example:

my ($z, $x, $y) = 𝘇𝗲𝗿𝗼𝗔𝗻𝗱𝗨𝗻𝗶𝘁𝘀;
ok $x + $y + $z == $x->plus($y);
ok $x - $y == $x->minus($y);
ok $x * 3  == $x->multiply(3);
ok $y / 2  == $y->divide(2);
ok $x + $y eq '(1,1)';
ok $x - $y eq '(1,-1)';
ok $x * 3  eq '(3,0)';
ok $y / 2  eq '(0,0.5)';
ok (($x * 2 + $y * 3)-> print eq '(2,3)');

This is a static method and so should either be imported or invoked as:

Math::Vectors2::zeroAndUnits

eq($o, $p)

Whether two vectors are equal to within the accuracy of floating point arithmetic

   Parameter  Description
1  $o         First vector
2  $p         Second vector

Example:

my ($z, $x, $y) = zeroAndUnits;
ok $x + $y + $z == $x->plus($y);
ok $x - $y == $x->minus($y);
ok $x * 3  == $x->multiply(3);
ok $y / 2  == $y->divide(2);
ok $x + $y 𝗲𝗾 '(1,1)';
ok $x - $y 𝗲𝗾 '(1,-1)';
ok $x * 3  𝗲𝗾 '(3,0)';
ok $y / 2  𝗲𝗾 '(0,0.5)';
ok (($x * 2 + $y * 3)-> print 𝗲𝗾 '(2,3)');

zero($o)

Whether a vector is equal to zero within the accuracy of floating point arithmetic

   Parameter  Description
1  $o         Vector

Example:

my ($𝘇𝗲𝗿𝗼, $x, $y) = zeroAndUnits;
ok $𝘇𝗲𝗿𝗼->𝘇𝗲𝗿𝗼;
ok !$x->𝘇𝗲𝗿𝗼;
ok !$y->𝘇𝗲𝗿𝗼;

print($p, @p)

Print one or more vectors.

   Parameter  Description
1  $p         Vector to print
2  @p         More vectors to print

Example:

my ($z, $x, $y) = zeroAndUnits;
ok $x + $y + $z == $x->plus($y);
ok $x - $y == $x->minus($y);
ok $x * 3  == $x->multiply(3);
ok $y / 2  == $y->divide(2);
ok $x + $y eq '(1,1)';
ok $x - $y eq '(1,-1)';
ok $x * 3  eq '(3,0)';
ok $y / 2  eq '(0,0.5)';
ok (($x * 2 + $y * 3)-> 𝗽𝗿𝗶𝗻𝘁 eq '(2,3)');

clone($o)

Clone a vector.

   Parameter  Description
1  $o         Vector to clone

Example:

my ($z, $x, $y) = zeroAndUnits;
ok $x->swap == $y;
ok $x->𝗰𝗹𝗼𝗻𝗲 == $x;

Plus($o, @p)

Add zero or more other vectors to the first vector and return the result.

   Parameter  Description
1  $o         First vector
2  @p         Other vectors

Example:

my ($zero, $x, $y) = zeroAndUnits;
$x->𝗣𝗹𝘂𝘀(new(1,1));
ok $x eq '(2,1)';
$y += new(1,1);
ok $y eq '(1,2)';

plus($o, @p)

Add zero or more other vectors to a copy of the first vector and return the result.

   Parameter  Description
1  $o         First vector
2  @p         Other vectors

Example:

my ($z, $x, $y) = zeroAndUnits;
ok $x + $y + $z == $x->𝗽𝗹𝘂𝘀($y);
ok $x - $y == $x->minus($y);
ok $x * 3  == $x->multiply(3);
ok $y / 2  == $y->divide(2);
ok $x + $y eq '(1,1)';
ok $x - $y eq '(1,-1)';
ok $x * 3  eq '(3,0)';
ok $y / 2  eq '(0,0.5)';
ok (($x * 2 + $y * 3)-> print eq '(2,3)');

Minus($o, @p)

Subtract zero or more vectors from the first vector and return the result.

   Parameter  Description
1  $o         First vector
2  @p         Other vectors

Example:

my ($zero, $x, $y) = zeroAndUnits;
$x->𝗠𝗶𝗻𝘂𝘀(new(0, 1));
ok $x eq '(1,-1)';
$y -= new(1,1);
ok $y eq '(-1,0)';

minus($o, @p)

Subtract zero or more vectors from a copy of the first vector and return the result.

   Parameter  Description
1  $o         First vector
2  @p         Other vectors

Example:

my ($z, $x, $y) = zeroAndUnits;
ok $x + $y + $z == $x->plus($y);
ok $x - $y == $x->𝗺𝗶𝗻𝘂𝘀($y);
ok $x * 3  == $x->multiply(3);
ok $y / 2  == $y->divide(2);
ok $x + $y eq '(1,1)';
ok $x - $y eq '(1,-1)';
ok $x * 3  eq '(3,0)';
ok $y / 2  eq '(0,0.5)';
ok (($x * 2 + $y * 3)-> print eq '(2,3)');

Multiply($o, $m)

Multiply a vector by a scalar and return the result.

   Parameter  Description
1  $o         Vector
2  $m         Scalar to multiply by

Example:

my ($zero, $x, $y) = zeroAndUnits;
$x->𝗠𝘂𝗹𝘁𝗶𝗽𝗹𝘆(2);
ok $x eq '(2,0)';
$y *= 2;
ok $y eq '(0,2)';

multiply($o, $m)

Multiply a copy of a vector by a scalar and return the result.

   Parameter  Description
1  $o         Vector
2  $m         Scalar to multiply by

Example:

my ($z, $x, $y) = zeroAndUnits;
ok $x + $y + $z == $x->plus($y);
ok $x - $y == $x->minus($y);
ok $x * 3  == $x->𝗺𝘂𝗹𝘁𝗶𝗽𝗹𝘆(3);
ok $y / 2  == $y->divide(2);
ok $x + $y eq '(1,1)';
ok $x - $y eq '(1,-1)';
ok $x * 3  eq '(3,0)';
ok $y / 2  eq '(0,0.5)';
ok (($x * 2 + $y * 3)-> print eq '(2,3)');

Divide($o, $d)

Divide a vector by a scalar and return the result.

   Parameter  Description
1  $o         Vector
2  $d         Scalar to multiply by

Example:

my ($zero, $x, $y) = zeroAndUnits;
$x->𝗗𝗶𝘃𝗶𝗱𝗲(1/2);
ok $x eq '(2,0)';
$y /= 1/2;
ok $y eq '(0,2)';

divide($o, $d)

Divide a copy of a vector by a scalar and return the result.

   Parameter  Description
1  $o         Vector
2  $d         Scalar to divide by

Example:

my ($z, $x, $y) = zeroAndUnits;
ok $x + $y + $z == $x->plus($y);
ok $x - $y == $x->minus($y);
ok $x * 3  == $x->multiply(3);
ok $y / 2  == $y->𝗱𝗶𝘃𝗶𝗱𝗲(2);
ok $x + $y eq '(1,1)';
ok $x - $y eq '(1,-1)';
ok $x * 3  eq '(3,0)';
ok $y / 2  eq '(0,0.5)';
ok (($x * 2 + $y * 3)-> print eq '(2,3)');

l($o)

Length of a vector.

   Parameter  Description
1  $o         Vector

Example:

my ($z, $x, $y) = zeroAndUnits;

ok  5 == ($x * 3 + $y * 4)->𝗹;
ok 25 == ($x * 3 + $y * 4)->l2;

ok 2 * ($x + $y)->𝗹  == ($x + $y)->d (-$x - $y);
ok 4 * ($x + $y)->l2 == ($x + $y)->d2(-$x - $y);

l2($o)

Length squared of a vector.

   Parameter  Description
1  $o         Vector

Example:

my ($z, $x, $y) = zeroAndUnits;

ok  5 == ($x * 3 + $y * 4)->l;
ok 25 == ($x * 3 + $y * 4)->𝗹𝟮;

ok 2 * ($x + $y)->l  == ($x + $y)->d (-$x - $y);
ok 4 * ($x + $y)->𝗹𝟮 == ($x + $y)->d2(-$x - $y);

d($o, $p)

Distance between the points identified by two vectors when placed on the same point.

   Parameter  Description
1  $o         Vector 1
2  $p         Vector 2

Example:

my ($z, $x, $y) = zeroAndUnits;

ok  5 == ($x * 3 + $y * 4)->l;
ok 25 == ($x * 3 + $y * 4)->l2;

ok 2 * ($x + $y)->l  == ($x + $y)->𝗱 (-$x - $y);
ok 4 * ($x + $y)->l2 == ($x + $y)->d2(-$x - $y);

d2($o, $p)

Distance squared between the points identified by two vectors when placed on the same point.

   Parameter  Description
1  $o         Vector 1
2  $p         Vector 2

Example:

my ($z, $x, $y) = zeroAndUnits;

ok  5 == ($x * 3 + $y * 4)->l;
ok 25 == ($x * 3 + $y * 4)->l2;

ok 2 * ($x + $y)->l  == ($x + $y)->d (-$x - $y);
ok 4 * ($x + $y)->l2 == ($x + $y)->𝗱𝟮(-$x - $y);

n($o)

Return a normalized a copy of a vector.

   Parameter  Description
1  $o         Vector

Example:

my ($z, $x, $y) = zeroAndUnits;
ok (($x * 3 + $y * 4)->𝗻 == $x * 3/5 + $y * 4/5);

ok 0 == $x . $y;
ok 1 == $x . $x;
ok 1 == $y . $y;
ok 8 == ($x * 1 + $y * 2) .($x * 2 + $y * 3);

dot($o, $p)

Dot product of two vectors.

   Parameter  Description
1  $o         Vector 1
2  $p         Vector 2

Example:

my ($z, $x, $y) = zeroAndUnits;
ok (($x * 3 + $y * 4)->n == $x * 3/5 + $y * 4/5);

ok 0 == $x . $y;
ok 1 == $x . $x;
ok 1 == $y . $y;
ok 8 == ($x * 1 + $y * 2) .($x * 2 + $y * 3);

area($o, $p)

Signed area of the parallelogram defined by the two vectors. The area is negative if the second vector appears to the right of the first if they are both placed at the origin and the observer stands against the z-axis in a left handed coordinate system.

   Parameter  Description
1  $o         Vector 1
2  $p         Vector 2

Example:

my ($z, $x, $y) = zeroAndUnits;
ok +1 == $x->cosine($x);
ok +1 == $y->cosine($y);
ok  0 == $x->cosine($y);
ok  0 == $y->cosine($x);

ok  0 == $x->sine($x);
ok  0 == $y->sine($y);
ok +1 == $x->sine($y);
ok -1 == $y->sine($x);

ok near -sqrt(1/2), ($x + $y)->sine($x);
ok near +sqrt(1/2), ($x + $y)->sine($y);
ok near -2,         ($x + $y)->𝗮𝗿𝗲𝗮($x * 2);
ok near +2,         ($x + $y)->𝗮𝗿𝗲𝗮($y * 2);

cosine($o, $p)

cos(angle between two vectors)

   Parameter  Description
1  $o         Vector 1
2  $p         Vector 2

Example:

my ($z, $x, $y) = zeroAndUnits;
ok +1 == $x->𝗰𝗼𝘀𝗶𝗻𝗲($x);
ok +1 == $y->𝗰𝗼𝘀𝗶𝗻𝗲($y);
ok  0 == $x->𝗰𝗼𝘀𝗶𝗻𝗲($y);
ok  0 == $y->𝗰𝗼𝘀𝗶𝗻𝗲($x);

ok  0 == $x->sine($x);
ok  0 == $y->sine($y);
ok +1 == $x->sine($y);
ok -1 == $y->sine($x);

ok near -sqrt(1/2), ($x + $y)->sine($x);
ok near +sqrt(1/2), ($x + $y)->sine($y);
ok near -2,         ($x + $y)->area($x * 2);
ok near +2,         ($x + $y)->area($y * 2);

sine($o, $p)

sin(angle between two vectors)

   Parameter  Description
1  $o         Vector 1
2  $p         Vector 2

Example:

my ($z, $x, $y) = zeroAndUnits;
ok +1 == $x->cosine($x);
ok +1 == $y->cosine($y);
ok  0 == $x->cosine($y);
ok  0 == $y->cosine($x);

ok  0 == $x->𝘀𝗶𝗻𝗲($x);
ok  0 == $y->𝘀𝗶𝗻𝗲($y);
ok +1 == $x->𝘀𝗶𝗻𝗲($y);
ok -1 == $y->𝘀𝗶𝗻𝗲($x);

ok near -sqrt(1/2), ($x + $y)->𝘀𝗶𝗻𝗲($x);
ok near +sqrt(1/2), ($x + $y)->𝘀𝗶𝗻𝗲($y);
ok near -2,         ($x + $y)->area($x * 2);
ok near +2,         ($x + $y)->area($y * 2);

angle($o, $p)

Angle in radians anticlockwise that the first vector must be rotated to point along the second vector normalized to the range: -pi to +pi.

   Parameter  Description
1  $o         Vector 1
2  $p         Vector 2

Example:

my ($zero, $x, $y) = zeroAndUnits;
ok near $y->𝗮𝗻𝗴𝗹𝗲(new(+1, -1)), deg2rad(-135);
ok near $y->𝗮𝗻𝗴𝗹𝗲(new(+1,  0)), deg2rad(-90);
ok near $y->𝗮𝗻𝗴𝗹𝗲(new(+1, +1)), deg2rad(-45);
ok near $y->𝗮𝗻𝗴𝗹𝗲(new( 0, +1)), deg2rad(+0);
ok near $y->𝗮𝗻𝗴𝗹𝗲(new(-1, +1)), deg2rad(+45);
ok near $y->𝗮𝗻𝗴𝗹𝗲(new(-1,  0)), deg2rad(+90);
ok near $y->𝗮𝗻𝗴𝗹𝗲(new(-1, -1)), deg2rad(+135);

ok near new(1,1) < new( 0, -1), deg2rad(-135);
ok near new(1,1) < new( 1, -1), deg2rad(-90);
ok near new(1,1) < new( 1,  0), deg2rad(-45);
ok near new(1,1) < new( 1,  1), deg2rad(0);
ok near new(1,1) < new( 0,  1), deg2rad(+45);
ok near new(1,1) < new(-1,  1), deg2rad(+90);
ok near new(1,1) < new(-1,  0), deg2rad(+135);

ok near $x + $y * sqrt(3) < $x, deg2rad(-60);
ok near $x + $y * sqrt(3) < $y, deg2rad(+30);

for my $i(-179..179)
 {ok near $x < new(cos(deg2rad($i)), sin(deg2rad($i))), deg2rad($i);
 }

r90($o)

Rotate a vector by 90 degrees anticlockwise.

   Parameter  Description
1  $o         Vector to rotate

Example:

my ($z, $x, $y) = zeroAndUnits;
ok $x->𝗿𝟵𝟬           ==  $y;
ok $y->𝗿𝟵𝟬           == -$x;
ok $x->𝗿𝟵𝟬->𝗿𝟵𝟬      == -$x;
ok $y->𝗿𝟵𝟬->𝗿𝟵𝟬      == -$y;
ok $x->𝗿𝟵𝟬->𝗿𝟵𝟬->𝗿𝟵𝟬 == -$y;
ok $y->𝗿𝟵𝟬->𝗿𝟵𝟬->𝗿𝟵𝟬 ==  $x;

r180($o)

Rotate a vector by 180 degrees.

   Parameter  Description
1  $o         Vector to rotate

Example:

my ($z, $x, $y) = zeroAndUnits;
ok $x->r90           ==  $y;
ok $y->r90           == -$x;
ok $x->r90->r90      == -$x;
ok $y->r90->r90      == -$y;
ok $x->r90->r90->r90 == -$y;
ok $y->r90->r90->r90 ==  $x;

r270($o)

Rotate a vector by 270 degrees anticlockwise.

   Parameter  Description
1  $o         Vector to rotate

Example:

my ($z, $x, $y) = zeroAndUnits;
ok $x->r90           ==  $y;
ok $y->r90           == -$x;
ok $x->r90->r90      == -$x;
ok $y->r90->r90      == -$y;
ok $x->r90->r90->r90 == -$y;
ok $y->r90->r90->r90 ==  $x;

swap($o)

Swap the components of a vector

   Parameter  Description
1  $o         Vector

Example:

my ($z, $x, $y) = zeroAndUnits;
ok $x->𝘀𝘄𝗮𝗽 == $y;
ok $x->clone == $x;

Math::Vectors2 Definition

Attributes of a vector

Output fields

x - X coordinate

y - Y coordinate

Index

1 angle - Angle in radians anticlockwise that the first vector must be rotated to point along the second vector normalized to the range: -pi to +pi.

2 area - Signed area of the parallelogram defined by the two vectors.

3 clone - Clone a vector.

4 cosine - cos(angle between two vectors)

5 d - Distance between the points identified by two vectors when placed on the same point.

6 d2 - Distance squared between the points identified by two vectors when placed on the same point.

7 divide - Divide a copy of a vector by a scalar and return the result.

8 Divide - Divide a vector by a scalar and return the result.

9 dot - Dot product of two vectors.

10 eq - Whether two vectors are equal to within the accuracy of floating point arithmetic

11 l - Length of a vector.

12 l2 - Length squared of a vector.

13 Minus - Subtract zero or more vectors from the first vector and return the result.

14 minus - Subtract zero or more vectors from a copy of the first vector and return the result.

15 Multiply - Multiply a vector by a scalar and return the result.

16 multiply - Multiply a copy of a vector by a scalar and return the result.

17 n - Return a normalized a copy of a vector.

18 new - Create new vector from components.

19 Plus - Add zero or more other vectors to the first vector and return the result.

20 plus - Add zero or more other vectors to a copy of the first vector and return the result.

21 print - Print one or more vectors.

22 r180 - Rotate a vector by 180 degrees.

23 r270 - Rotate a vector by 270 degrees anticlockwise.

24 r90 - Rotate a vector by 90 degrees anticlockwise.

25 sine - sin(angle between two vectors)

26 swap - Swap the components of a vector

27 zero - Whether a vector is equal to zero within the accuracy of floating point arithmetic

28 zeroAndUnits - Create the useful vectors: o=(0,0), x=(1,0), y=(0,1)

Installation

This module is written in 100% Pure Perl and, thus, it is easy to read, comprehend, use, modify and install via cpan:

sudo cpan install Math::Vectors2

Author

philiprbrenan@gmail.com

http://www.appaapps.com

Copyright

This module is free software. It may be used, redistributed and/or modified under the same terms as Perl itself.

To install Math::Vectors2, copy and paste the appropriate command in to your terminal.

cpanm

cpanm Math::Vectors2

CPAN shell

perl -MCPAN -e shell
install Math::Vectors2

For more information on module installation, please visit the detailed CPAN module installation guide.

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Name

Synopsis

Description

Methods

new($x, $y)

zeroAndUnits()

eq($o, $p)

zero($o)

print($p, @p)

clone($o)

Plus($o, @p)

plus($o, @p)

Minus($o, @p)

minus($o, @p)

Multiply($o, $m)

multiply($o, $m)

Divide($o, $d)

divide($o, $d)

l($o)

l2($o)

d($o, $p)

d2($o, $p)

n($o)

dot($o, $p)

area($o, $p)

cosine($o, $p)

sine($o, $p)

angle($o, $p)

r90($o)

r180($o)

r270($o)

swap($o)

Math::Vectors2 Definition

Output fields

Index

Installation

Author

Copyright

Module Install Instructions