Geometric algebra

Geometric algebra is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

Geometric algebra is an other name for Clifford algebras and it's basically an algebra containing a vector space ${\mathcal {V}}$ and obeying the following axioms:

${\begin{array}{c}(ab)c=a(bc)\\a(b+c)=ab+ac\\(a+b)c=ac+bc\\\forall \mathbf {x} \in {\mathcal {V}},\,\mathbf {x} ^{2}\in \mathbb {R} \end{array}}$

The purpose of this task is to implement such an algebra with vectors of arbitrary size, or up to 32 dimensions if that's easier to implement in your language.

To demonstrate your solution, you will use it to implement quaternions:

• define a function $n\mapsto e(n)$ which allows for creation of an orthonormal basis

$(\mathbf {e} _{0},\mathbf {e} _{1},\mathbf {e} _{2},\ldots )$,

• define a scalar product function as the symmetric part of the geometric product $\mathbf {a} \cdot \mathbf {b} =(\mathbf {ab} +\mathbf {ba} )/2$
• verify the orthonormality $\mathbf {e} _{i}\cdot \mathbf {e} _{j}=\delta _{i,j}$ for i, j, k in $\{1,2,3\}$.
• define the following three constants:

${\begin{array}{c}i=\mathbf {e} _{1}\mathbf {e} _{2}\\j=\mathbf {e} _{2}\mathbf {e} _{3}\\k=\mathbf {e} _{1}\mathbf {e} _{3}\end{array}}$

• verify that $i^{2}=j^{2}=k^{2}=ijk=-1$

JavaScript

<lang javascript>var CGA = function () {

   function e(n) {

var result = []; result[1 << n] = 1; return result;

   }
   function neg(x) { return multiply([-1], x) }
   function bitCount(i) {

// Note that unsigned shifting (>>>) is not required. i = i - ((i >> 1) & 0x55555555); i = (i & 0x33333333) + ((i >> 2) & 0x33333333); i = (i + (i >> 4)) & 0x0F0F0F0F; i = i + (i >> 8); i = i + (i >> 16); return i & 0x0000003F;

   }
   function reorderingSign(a, b) {

a >>= 1; var sum = 0; while (a != 0) { sum += bitCount(a & b); a >>= 1; } return (sum & 1) == 0 ? 1 : -1;

   }
   function add(a, b) {

var result = []; for (var i = 0; i < 32; i++) { if (a[i] && b[i]) { var r = a[i] + b[i]; if (r !== 0) { result[i] = r; } } else if (a[i]) { result[i] = a[i]; } else if (b[i]) { result[i] = b[i]; } } return result;

   }
   function multiply(a, b)
   {

var result = []; for (var i = 0; i < 32; i++) { if (a[i]) { for (var j = 0; j < 32; j++) { if (b[j]) { var s = reorderingSign(i, j) * a[i] * b[j]; // if (i == 1 && j == 1) { s *= -1 } // e0*e0 == -1 var k = i ^ j; if (result[k]) { result[k] += s; } else { result[k] = s; } } } } } return result;

   }
   return {

e  : e, neg : neg, add : add, mul : multiply

   };

}(); </lang>

And then, from the console:

<lang javascript>var e = CGA.e; function cdot(a, b) { CGA.mul([0.5], CGA.add(CGA.mul(a, b), CGA.mul(b, a)) }

console.log(cdot(e(1), e(1))); // [1] console.log(cdot(e(2), e(2))); // [1] console.log(cdot(e(3), e(3))); // [1]

console.log(cdot(e(1), e(2))); // [] which means 0 console.log(cdot(e(1), e(3))); // [] console.log(cdot(e(2), e(3))); // []

var i = CGA.mul(e(1), e(2)); var j = CGA.mul(e(2), e(3)); var k = CGA.mul(e(1), e(3));

console.log(CGA.mul(i, i)); // [-1] console.log(CGA.mul(j, j)); // [-1] console.log(CGA.mul(k, k)); // [-1] console.log(CGA.mul(CGA.mul(i, j), k)); // [-1] </lang>

J

This example is incomplete. Please ensure that it meets all task requirements and remove this message.

Using the implementation from the Quaternion type task:

<lang J> e=: {&(#:2^3 A. i.4)

  dot=: +/ .*

  0 1 2 3 dot&e"0/0 1 2 3

1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1

  i=: (e 1) mul (e 2)
  j=: (e 2) mul (e 3)
  k=: (e 1) mul (e 3)

  i

0 1 0 0

  j

0 0 1 0

  k

0 0 0 1

  i mul j mul k

_1 0 0 0

  i mul i

_1 0 0 0

  j mul j

_1 0 0 0

  k mul k

_1 0 0 0</lang>

Note that the first element of this quaternion data structure is the "real" component.

Perl 6

<lang perl6>unit class MultiVector; has Real %.blades{UInt}; method clean { for %!blades { %!blades{.key} :delete unless .value; } } method narrow {

   for %!blades { return self if .key > 0 && .value !== 0; }
   return %!blades{0} // 0;

}

sub e(UInt $n?) returns MultiVector is export {

   $n.defined ?? MultiVector.new(:blades(my Real %{UInt} = (1 +< $n) => 1)) !! MultiVector.new

}

my sub grade(UInt $n) is cached { [+] $n.base(2).comb } my sub order(UInt:D $i is copy, UInt:D $j) is cached {

   my $n = 0;
   repeat {

$i +>= 1; $n += [+] ($i +& $j).base(2).comb;

   } until $i == 0;
   return $n +& 1 ?? -1 !! 1;

}

multi infix:<+>(MultiVector $A, MultiVector $B) returns MultiVector is export {

   my Real %blades{UInt} = $A.blades.clone;
   for $B.blades {

%blades{.key} += .value; %blades{.key} :delete unless %blades{.key};

   }
   return MultiVector.new: :%blades;

} multi infix:<+>(Real $s, MultiVector $A) returns MultiVector is export {

   my Real %blades{UInt} = $A.blades.clone;
   %blades{0} += $s;
   %blades{0} :delete unless %blades{0};
   return MultiVector.new: :%blades;

} multi infix:<+>(MultiVector $A, Real $s) returns MultiVector is export { $s + $A } multi infix:<*>(MultiVector $A, MultiVector $B) returns MultiVector is export {

   my Real %blades{UInt};
   for $A.blades -> $a {

for $B.blades -> $b { my $c = $a.key +^ $b.key; %blades{$c} += $a.value * $b.value * order($a.key, $b.key); %blades{$c} :delete unless %blades{$c}; }

   }
   return MultiVector.new: :%blades;

} multi infix:<**>(MultiVector $ , 0) returns MultiVector is export { MultiVector.new } multi infix:<**>(MultiVector $A, 1) returns MultiVector is export { $A } multi infix:<**>(MultiVector $A, 2) returns MultiVector is export { $A * $A } multi infix:<**>(MultiVector $A, UInt $n where $n %% 2) returns MultiVector is export { ($A ** ($n div 2)) ** 2 } multi infix:<**>(MultiVector $A, UInt $n) returns MultiVector is export { $A * ($A ** ($n div 2)) ** 2 }

multi infix:<*>(MultiVector $, 0) returns MultiVector is export { MultiVector.new } multi infix:<*>(MultiVector $A, 1) returns MultiVector is export { $A } multi infix:<*>(MultiVector $A, Real $s) returns MultiVector is export {

   return MultiVector.new: :blades(my Real %{UInt} = map { .key => $s * .value }, $A.blades);

} multi infix:<*>(Real $s, MultiVector $A) returns MultiVector is export { $A * $s } multi infix:</>(MultiVector $A, Real $s) returns MultiVector is export { $A * (1/$s) } multi prefix:<->(MultiVector $A) returns MultiVector is export { return -1 * $A } multi infix:<->(MultiVector $A, MultiVector $B) returns MultiVector is export { $A + -$B } multi infix:<->(MultiVector $A, Real $s) returns MultiVector is export { $A + -$s } multi infix:<->(Real $s, MultiVector $A) returns MultiVector is export { $s + -$A }

multi infix:<==>(MultiVector $A, MultiVector $B) returns Bool is export { $A - $B == 0 } multi infix:<==>(Real $x, MultiVector $A) returns Bool is export { $A == $x } multi infix:<==>(MultiVector $A, Real $x) returns Bool is export {

   my $narrowed = $A.narrow;
   $narrowed ~~ Real and $narrowed == $x;

}</lang>

And here is the code implementing and verifying quaternions:

<lang perl6>use MultiVector; use Test;

plan 7;

sub infix:<cdot>($a, $b) { ($a*$b + $b*$a) / 2 }

ok e(1) cdot e(1) == 1, "e1.e1 = 1"; ok e(1) cdot e(2) == 0, "e1.e2 = 0"; ok e(1) cdot e(3) == 0, "e1.e3 = 0"; ok e(2) cdot e(2) == 1, "e2.e2 = 1"; ok e(2) cdot e(3) == 0, "e2.e3 = 0"; ok e(3) cdot e(3) == 1, "e3.e3 = 1";

my constant i = e(1)*e(2); my constant j = e(2)*e(3); my constant k = e(1)*e(3);

ok i**2 == j**2 == k**2 == i*j*k == -1, "i² = j² = k² = ijk = -1";</lang>

Output:

1..7
ok 1 - e1.e1 = 1
ok 2 - e1.e2 = 0
ok 3 - e1.e3 = 0
ok 4 - e2.e2 = 1
ok 5 - e2.e3 = 0
ok 6 - e3.e3 = 1
ok 7 - i² = j² = k² = ijk = -1