Two-dimensional Projective Geometric Algebra (PGA).
This module implements the projective geometry of the Euclidean plane using the degenerate Clifford algebra:
Cl(2,0,1)with signature:
{1,1,0}and basis:
e1, e2, e0where e0 is the ideal (infinite) basis vector.
In PGA, geometric entities are represented homogeneously:
- Points are grade-2 bivectors
- Lines are grade-1 vectors
- Directions are ideal points
Finite points are represented as:
P = e12 + x*e20 + y*e01where the coefficient of e12 is the homogeneous scale factor.
Ideal points have no finite component:
P∞ = x*e20 + y*e01Examples
iex> p = Galixir.Algebras.PGA2.point(2, 3)
iex> Galixir.Algebras.PGA2.point_coordinates(p)
{2.0, 3.0}Summary
Functions
Adds two multivectors component-wise.
Checks whether a multivector is a blade.
Returns the mapping between blade names and storage indices.
Returns the canonical sign of a multivector.
Returns the coefficient of a basis blade.
Tests whether an object contains another object.
Returns the dimension of the algebra.
Extracts the Euclidean direction vector of a line.
Computes Euclidean distance between two finite points.
Computes the scalar product of two multivectors.
Computes the dual of a multivector.
Checks whether a PGA point is finite.
Computes the geometric product of two multivectors.
Extracts the grade-g component of a multivector.
Returns the grades present in a multivector.
Returns the grade if the multivector has a single grade.
Returns the ideal point representing the direction of a line.
Creates an ideal point representing a direction at infinity.
Checks whether a PGA point is ideal.
Tests whether two objects are incident.
Computes the inner product of two multivectors.
Computes the inverse of a multivector.
Computes the join of two objects.
Computes the left contraction of two multivectors.
Creates the line through two points.
Creates a line from coefficients.
Creates a line from a normal vector and a point.
Returns the normal vector of a line.
Returns the maximum absolute coefficient of a multivector.
Computes the meet of two objects.
Negates a multivector.
Returns the norm of a multivector.
Normalizes a multivector.
Returns the scalar identity element.
Returns the Euclidean origin point.
Creates a finite projective point.
Extracts Cartesian coordinates from a finite point.
Applies the reverse operation to a multivector.
Computes the right contraction of two multivectors.
Creates a rotation motor around the origin.
Checks whether a multivector contains only a scalar component.
Returns the scalar coefficient of a multivector.
Computes the scalar product of two multivectors.
Returns the metric signature of the algebra.
Returns the number of coefficients stored by the algebra.
Returns the squared norm of a multivector.
Subtracts two multivectors component-wise.
Returns the multiplication table for the algebra.
Formats a multivector using standard geometric algebra notation.
Applies a motor transformation.
Creates a translation motor from a vector.
Creates a translation motor.
Computes the inverse dual operation.
Returns the normalized direction vector of a line.
Returns the normalized line normal vector.
Creates a Euclidean direction vector.
Computes the outer product (wedge product) of two multivectors.
Returns the zero multivector.
Checks whether all coefficients of a multivector are zero.
Functions
Adds two multivectors component-wise.
Examples
iex> a = Galixir.Algebras.PGA2.new(scalar: 2)
iex> b = Galixir.Algebras.PGA2.new(scalar: 3)
iex> Galixir.Algebras.PGA2.add(a, b)
Galixir.Algebras.PGA2.new(scalar: 5)Checks whether a multivector is a blade.
A blade is a multivector containing components from at most one grade.
Scalars are considered blades.
Examples
iex> Galixir.Algebras.PGA2.blade?(Galixir.Algebras.PGA2.new(e1: 2))
true
iex> Galixir.Algebras.PGA2.blade?(Galixir.Algebras.PGA2.new(e12: 1))
true
iex> Galixir.Algebras.PGA2.blade?(Galixir.Algebras.PGA2.new(e1: 1, e2: 1))
true
iex> Galixir.Algebras.PGA2.blade?(Galixir.Algebras.PGA2.new(scalar: 2, e1: 2))
false
iex> Galixir.Algebras.PGA2.blade?(Galixir.Algebras.PGA2.new(e2: 2, e12: 2))
falseReturns the mapping between blade names and storage indices.
Blade coefficients are stored in a fixed-size tuple. This map translates canonical blade names into their corresponding tuple index.
Example
iex> Galixir.Algebras.PGA2.blade_indices()[:e1]
1Returns the canonical sign of a multivector.
The canonical sign is determined by the first non-zero coefficient in storage order.
Returns:
1if the first non-zero coefficient is positive-1if the first non-zero coefficient is negative0if all coefficients are zero
Examples
iex> Elixir.Galixir.Algebras.PGA2.canonical_sign(Elixir.Galixir.Algebras.PGA2.new(e1: 2))
1
iex> Elixir.Galixir.Algebras.PGA2.canonical_sign(Elixir.Galixir.Algebras.PGA2.new(e1: -2))
-1
iex> Elixir.Galixir.Algebras.PGA2.canonical_sign(Elixir.Galixir.Algebras.PGA2.new())
0Returns the coefficient of a basis blade.
The requested blade can be given in canonical form or as any registered blade alias. Aliases are automatically converted to the canonical blade and the appropriate sign is applied.
## Examples
iex> Elixir.Galixir.Algebras.PGA2.coefficient( ...> Elixir.Galixir.Algebras.PGA2.new(e1: 3), ...> :e1 ...> ) 3.0
Tests whether an object contains another object.
Returns the dimension of the algebra.
This is the number of basis vectors defined by the signature.
Extracts the Euclidean direction vector of a line.
Computes Euclidean distance between two finite points.
Computes the scalar product of two multivectors.
Computes the dual of a multivector.
The dual maps each basis blade to its complementary blade with the appropriate orientation sign. The complement is determined by the full pseudoscalar of the algebra.
The operation is linear and applies independently to every coefficient.
Examples
iex> Galixir.Algebras.PGA2.dual(Galixir.Algebras.PGA2.new(e1: 1)) |> inspect Galixir.Algebras.PGA2.new(e20: 1.0) |> inspect
Checks whether a PGA point is finite.
A finite point has a non-zero homogeneous e12 component.
Computes the geometric product of two multivectors.
The geometric product is the fundamental multiplication operation of geometric algebra. It combines the outer product and metric-dependent inner product into a single associative operation.
The result depends on the algebra's metric signature.
Examples
iex> Galixir.Algebras.PGA2.gp(
...> Galixir.Algebras.PGA2.new(e1: 1),
...> Galixir.Algebras.PGA2.new(e1: 1)
...> )
Galixir.Algebras.PGA2.new(scalar: 1)Extracts the grade-g component of a multivector.
All coefficients whose basis blades are not of grade g are set to zero.
Raises ArgumentError if g is outside the range 0..dimension().
Examples
iex> Galixir.Algebras.PGA2.grade(
...> Galixir.Algebras.PGA2.new(scalar: 1, e1: 2),
...> 1
...> )
Galixir.Algebras.PGA2.new(e1: 2)
iex> Galixir.Algebras.PGA2.grade(
...> Galixir.Algebras.PGA2.new(scalar: 1, e1: 2),
...> 0
...> )
Galixir.Algebras.PGA2.new(scalar: 1)Returns the grades present in a multivector.
The returned list contains every grade with at least one non-zero coefficient, ordered from lowest to highest.
Examples
iex> Galixir.Algebras.PGA2.grades(
...> Galixir.Algebras.PGA2.new(scalar: 1)
...> )
[0]
iex> Galixir.Algebras.PGA2.grades(
...> Galixir.Algebras.PGA2.new(e1: 2)
...> )
[1]
iex> Galixir.Algebras.PGA2.grades(
...> Galixir.Algebras.PGA2.new(scalar: 1, e1: 2)
...> )
[0, 1]
iex> Galixir.Algebras.PGA2.grades(
...> Galixir.Algebras.PGA2.new()
...> )
[]Returns the grade if the multivector has a single grade.
Returns nil for mixed-grade multivectors.
Returns the ideal point representing the direction of a line.
Creates an ideal point representing a direction at infinity.
Ideal points have no finite position and are used to represent directions and points at infinity.
Checks whether a PGA point is ideal.
An ideal point has zero homogeneous e12 component.
Tests whether two objects are incident.
Computes the inner product of two multivectors.
The operation is generated from the geometric product and retains only terms satisfying the grade selection rule.
Example
iex> Galixir.Algebras.PGA2.inner(
...> Galixir.Algebras.PGA2.new(e1: 2),
...> Galixir.Algebras.PGA2.new(e1: 3)
...> )Computes the inverse of a multivector.
The inverse is computed using the reverse:
inverse(a) = reverse(a) / scalar_part(a * reverse(a))
This formula is valid when a * reverse(a) is a non-zero scalar.
Raises ArgumentError if the multivector is not invertible by this formula.
## Examples
iex> Galixir.Algebras.PGA2.inverse(
...> Galixir.Algebras.PGA2.new(e1: 2)
...> )|> inspect
Galixir.Algebras.PGA2.new(e1: 0.5) |> inspectComputes the join of two objects.
In PGA this produces the smallest object containing both inputs.
Examples:
point ∨ point -> line
point ∨ line -> plane elementComputes the left contraction of two multivectors.
The operation is generated from the geometric product and retains only terms satisfying the grade selection rule.
Example
iex> Galixir.Algebras.PGA2.left_contraction(
...> Galixir.Algebras.PGA2.new(e1: 2),
...> Galixir.Algebras.PGA2.new(e1: 3)
...> )Creates the line through two points.
Creates a line from coefficients.
Represents the equation:
ax + by + c = 0as the PGA vector:
a*e1 + b*e2 + c*e0Creates a line from a normal vector and a point.
Returns the normal vector of a line.
The normal is:
a*e1 + b*e2Returns the maximum absolute coefficient of a multivector.
Accepts either a multivector struct or the internal coefficient tuple.
Example
iex> Elixir.Galixir.Algebras.PGA2.max_abs_component(Elixir.Galixir.Algebras.PGA2.new(e1: 2, scalar: 5))
5.0
iex> Elixir.Galixir.Algebras.PGA2.max_abs_component(Elixir.Galixir.Algebras.PGA2.new(e1: 5, scalar: 2))
5.0Computes the meet of two objects.
In PGA the meet is the outer product.
Examples:
line ∧ line -> pointNegates a multivector.
Returns the norm of a multivector.
The norm is the square root of the absolute squared norm.
Example
iex> a = Galixir.Algebras.PGA2.new(scalar: 3)
iex> Galixir.Algebras.PGA2.norm(a)
3.0Normalizes a multivector.
The result has unit norm while preserving the direction of the multivector.
Raises ArgumentError when attempting to normalize a null
multivector.
Example
iex> a = Galixir.Algebras.PGA2.new(scalar: 2)
iex> Galixir.Algebras.PGA2.norm(Galixir.Algebras.PGA2.normalize(a))
1.0Returns the scalar identity element.
Returns the Euclidean origin point.
Creates a finite projective point.
The point is represented homogeneously as:
P = e12 + x*e20 + y*e01Extracts Cartesian coordinates from a finite point.
Returns:
{x, y}Applies the reverse operation to a multivector.
Reverse (also called reversion) changes the sign of basis blades according to their grade:
grade 0: +
grade 1: +
grade 2: -
grade 3: -
grade 4: +
...For a blade with grade r, the sign is:
(-1)^(r(r-1)/2)Examples
iex> Galixir.Algebras.PGA2.reverse(Galixir.Algebras.PGA2.new(e1: 2)) |> inspect
Galixir.Algebras.PGA2.new(e1: 2)|> inspect
iex> Galixir.Algebras.PGA2.reverse(Galixir.Algebras.PGA2.new(e12: 2))|> inspect
Galixir.Algebras.PGA2.new(e12: -2)|> inspect
iex> Galixir.Algebras.PGA2.reverse(Galixir.Algebras.PGA2.new(scalar: 3))|> inspect
Galixir.Algebras.PGA2.new(scalar: 3)|> inspectComputes the right contraction of two multivectors.
The operation is generated from the geometric product and retains only terms satisfying the grade selection rule.
Example
iex> Galixir.Algebras.PGA2.right_contraction(
...> Galixir.Algebras.PGA2.new(e1: 2),
...> Galixir.Algebras.PGA2.new(e1: 3)
...> )Creates a rotation motor around the origin.
The angle is specified in radians.
Checks whether a multivector contains only a scalar component.
Components with an absolute value smaller than eps are considered
zero.
Examples
iex> Galixir.Algebras.PGA2.scalar?(Galixir.Algebras.PGA2.new(scalar: 3))
true
iex> Galixir.Algebras.PGA2.scalar?(Galixir.Algebras.PGA2.new(e1: 3))
false
iex> Galixir.Algebras.PGA2.scalar?(Galixir.Algebras.PGA2.new())
trueReturns the scalar coefficient of a multivector.
This is equivalent to retrieving the coefficient of the scalar blade.
Examples
iex> Elixir.Galixir.Algebras.PGA2.scalar_part(Elixir.Galixir.Algebras.PGA2.new(scalar: 5.0, e1: 2.0))
5.0Computes the scalar product of two multivectors.
The scalar product is the grade-0 component of the geometric product:
<a b>₀The result depends on the metric signature of the algebra. In particular, basis vectors with negative or null squares affect the result.
Examples
iex> Galixir.Algebras.PGA2.scalar_product(
...> Galixir.Algebras.PGA2.new(e1: 2),
...> Galixir.Algebras.PGA2.new(e1: 3)
...> )
6.0Returns the metric signature of the algebra.
Example:
{1, 1, 1, 0}represents a projective geometric algebra with three Euclidean basis vectors and one null basis vector.
Returns the number of coefficients stored by the algebra.
A dimension n algebra contains 2^n basis blades.
Returns the squared norm of a multivector.
The squared norm is computed as:
scalar_part(a * reverse(a))The result may be negative for algebras with indefinite metrics.
Example
iex> a = Galixir.Algebras.PGA2.new(scalar: 3)
iex> Galixir.Algebras.PGA2.squared_norm(a)
9.0Subtracts two multivectors component-wise.
Examples
iex> a = Galixir.Algebras.PGA2.new(scalar: 5)
iex> b = Galixir.Algebras.PGA2.new(scalar: 2)
iex> Galixir.Algebras.PGA2.sub(a, b)
Galixir.Algebras.PGA2.new(scalar: 3)Returns the multiplication table for the algebra.
The table contains precomputed geometric products between basis blades.
Each entry maps {left_blade, right_blade} to {coefficient, result_blade}.
The blades are represented internally as bitmasks.
Example
iex> Galixir.Algebras.PGA2.table() |> Map.has_key?({1, 1})
trueFormats a multivector using standard geometric algebra notation.
Zero coefficients are omitted. Coefficients of 1 and -1 are elided for
non-scalar basis blades.
Examples
iex> inspect(Galixir.Algebras.PGA2.new())
"0"
iex> inspect(Galixir.Algebras.PGA2.new(scalar: 2))
"2.0"
iex> inspect(Galixir.Algebras.PGA2.new(e1: 1))
"e1"
iex> inspect(Galixir.Algebras.PGA2.new(scalar: 1, e1: 2))
"1.0 + 2.0e1"Applies a motor transformation.
Uses the sandwich product:
M X M⁻¹Creates a translation motor from a vector.
Creates a translation motor.
Translates by the vector (x,y).
Computes the inverse dual operation.
undual/1 reverses the blade complement operation performed by dual/1.
For non-degenerate Euclidean algebras this corresponds to applying the dual operation twice with the appropriate pseudoscalar factor. In degenerate algebras the result depends on the implemented dual convention.
Examples
iex> Galixir.Algebras.PGA2.undual(Galixir.Algebras.PGA2.dual(Galixir.Algebras.PGA2.new(e1: 2)))
Galixir.Algebras.PGA2.new(e1: 2)Returns the normalized direction vector of a line.
Returns the normalized line normal vector.
Creates a Euclidean direction vector.
Computes the outer product (wedge product) of two multivectors.
The wedge product combines blades by joining their basis vectors. It is antisymmetric:
a ∧ b = -(b ∧ a)and vanishes when the operands share a basis vector.
Examples
iex> Galixir.Algebras.PGA2.wedge(
...> Galixir.Algebras.PGA2.new(e1: 1),
...> Galixir.Algebras.PGA2.new(e2: 1)
...> )
Galixir.Algebras.PGA2.new(e12: 1)
iex> Galixir.Algebras.PGA2.wedge(
...> Galixir.Algebras.PGA2.new(e2: 1),
...> Galixir.Algebras.PGA2.new(e1: 1)
...> )
Galixir.Algebras.PGA2.new(e12: -1)
iex> Galixir.Algebras.PGA2.wedge(
...> Galixir.Algebras.PGA2.new(e1: 1),
...> Galixir.Algebras.PGA2.new(e1: 1)
...> )
Galixir.Algebras.PGA2.new()Returns the zero multivector.
Checks whether all coefficients of a multivector are zero.
Examples
iex> Galixir.Algebras.PGA2.zero?(Galixir.Algebras.PGA2.new())
true
iex> Galixir.Algebras.PGA2.zero?(Galixir.Algebras.PGA2.new(e1: 1))
false