Adds two multivectors component-wise.

Examples

iex> a = new(scalar: 2)
iex> b = new(scalar: 3)
iex> add(a, b)
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> blade?(new(e1: 2))
true

iex> blade?(new(e12: 1))
true

iex> blade?(new(e1: 1, e2: 1))
true

iex> blade?(new(scalar: 2, e1: 2))
false

iex> blade?(new(e2: 2, e12: 2))
false

Returns the number of coefficients stored by the algebra.

A dimension n algebra contains 2^n basis blades.

Returns the mapping between blade names and storage indices.

Blade coefficients are stored in a fixed-blade_count tuple. This map translates canonical blade names into their corresponding tuple index.

Example

iex> blade_indices()[:e1]
1

Returns the canonical sign of a multivector.

The canonical sign is determined by the first non-zero coefficient in storage order.

Returns:

• 1 if the first non-zero coefficient is positive
• -1 if the first non-zero coefficient is negative
• 0 if all coefficients are zero

Examples

iex> canonical_sign(new(e1: 2))
1

iex> canonical_sign(new(e1: -2))
-1

iex> canonical_sign(new())
0

Returns 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> coefficient(
...>  new(e1: 3),
...>  :e1
...> )
3.0

Computes the commutator product of two multivectors.

The commutator is defined as half the difference between the geometric products in opposite orders:

[a, b] = (a * b - b * a) / 2

This measures the non-commutativity of the geometric product.

Examples

iex> commutator(new(e1: 1), new(e2: 1))
new(e12: 1)

iex> commutator(new(e2: 1), new(e1: 1))
new(e12: -1)

iex> commutator(new(e1: 1), new(e1: 1))
new()

iex> commutator(new(e1: 1), new(scalar: 2))
new()

Returns the dimension of the algebra.

This is the number of basis vectors defined by the metric.

Returns the ideal point representing the direction of a line.

The result is a point at infinity, not a Euclidean vector.

Extracts the Euclidean direction vector of a line.

Examples

iex> direction_vector(line(point(0,1), point(2,0)))
new(e1: 2, e2: -1)

Computes Euclidean distance between two finite points.

iex> distance(point(1,2), point(1,2))
0.0

iex> distance(point(-1,-1), point(2,3))
5.0

Checks whether a PGA point is finite.

A finite point has a non-zero homogeneous e12 component.

Examples

iex> finite_point?(point(1,2))
true


iex> finite_point?(ideal_point(1,2))
false

iex> finite_point?(vector(1,2))
false

Formats this multivector as geometric algebra notation.

This is the same representation used by Inspect.

Examples

iex> inspect(new())
"~G[0.0]"

iex> inspect(new(), custom_options: [all_blades: true])
"~G[0.0 + 0.0e1 + 0.0e2 + 0.0e12 + 0.0e0 + 0.0e10 + 0.0e20 + 0.0e120]"

iex> inspect(new(scalar: -2))
"~G[-2.0]"

iex> inspect(new(scalar: 2))
"~G[2.0]"

iex> inspect(new(scalar: -2))
"~G[-2.0]"

iex> inspect(new(e1: 1))
"~G[1.0e1]"

iex> inspect(new(scalar: 1, e1: 2))
"~G[1.0 + 2.0e1]"

iex> inspect(new(scalar: 1, e1: -2))
"~G[1.0 - 2.0e1]"

iex> inspect(new(scalar: -1, e1: -2))
"~G[-1.0 - 2.0e1]"

iex> inspect(new(scalar: -1, e1: 2))
"~G[-1.0 + 2.0e1]"

iex> inspect(new(e1: 2))
"~G[2.0e1]"

iex> inspect(new(e1: -2))
"~G[-2.0e1]"

iex> inspect(~G[ε + εe1])
"~G[ε + εe1]"
iex> inspect(~G[-ε - εe1])
"~G[-ε - εe1]"

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.

Examples

iex> gp(
...>   new(e1: 1),
...>   new(e1: 1)
...> )
new(scalar: 1)

Returns the grade of a homogeneous multivector.

The zero multivector is considered grade 0.

Returns nil for mixed-grade multivectors.

Examples

iex> grade(
...>   new(scalar: 1)
...> )
0

iex> grade(
...>   new(e1: 2)
...> )
1

iex> grade(
...>   new(scalar: 1, e1: 2)
...> )
nil

iex> grade(
...>   new()
...> )
nil

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> grade(
...>   new(scalar: 1, e1: 2),
...>   1
...> )
new(e1: 2)

iex> grade(
...>   new(scalar: 1, e1: 2),
...>   0
...> )
new(scalar: 1)

Checks whether a multivector contains components of the given grade only.

A multivector is considered to have a grade if all non-zero components belong to that grade. The zero multivector is considered to have grade 0.

Examples

iex> grade?(new(e1: 1), 1)
true

iex> grade?(new(scalar: 1, e1: 2), 2)
false

iex> grade?(new(scalar: 1), 0)
true

iex> grade?(new(), 0)
true

See is_grade/2

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> grades(
...>   new(scalar: 1)
...> )
[0]

iex> grades(
...>   new(e1: 2)
...> )
[1]

iex> grades(
...>   new(scalar: 1, e1: 2)
...> )
[0, 1]

iex> grades(
...>   new()
...> )
[]

Computes the Hodge dual of a multivector.

The Hodge dual maps each basis blade to its complementary blade with the appropriate orientation sign. The complementary blade is determined by the pseudoscalar of the algebra.

The operation is linear and is applied independently to each basis blade and its coefficient.

Unlike the pseudoscalar dual, the Hodge dual does not require the pseudoscalar to be invertible and is therefore also defined for degenerate algebras.

The operations are inverses:

hodge_undual(hodge_dual(a)) == a
hodge_dual(hodge_undual(a)) == a

See hodge_undual/1 for the inverse operation.

Examples

iex> hodge_dual(new(e1: 1)) |> inspect
new(e20: 1.0) |> inspect

Computes the inverse Hodge dual of a multivector.

hodge_undual/1 reverses the blade-complement operation performed by hodge_dual/1, mapping each basis blade back to its complementary blade with the appropriate orientation sign.

The operation is linear and is applied independently to each basis blade and its coefficient.

Unlike the pseudoscalar dual, the Hodge dual and its inverse are defined for degenerate algebras as well.

The operations are inverses:

hodge_undual(hodge_dual(a)) == a
hodge_dual(hodge_undual(a)) == a

See hodge_dual/1 for the inverse operation.

Examples

iex> hodge_undual(hodge_dual(new(e1: 2)))
new(e1: 2)

Creates an ideal point representing a direction.

Ideal points are points at infinity and have no finite location.

Note that Euclidean vectors are not ideal points. Use direction_point/1 to convert a line direction into an ideal point.

Checks whether a PGA point is ideal.

An ideal point has zero homogeneous e12 component.

Examples

iex> ideal_point?(ideal_point(1,0))
true

iex> ideal_point?(point(1,0))
false

iex> ideal_point?(vector(1,0))
false

Tests whether two objects are incident.

In PGA, two objects are incident when their join vanishes.

Note that identical objects are incident with themselves. Parallel distinct lines are not incident.

incident?/2 is not the same as geometric intersection. Parallel coincident lines are incident.

This includes cases such as:

• a point lying on a line
• an ideal point representing the direction of a line
• identical geometric objects

Examples

iex> incident?(point(0,0), line(0,1,0))
true

iex> p = point(1, 2)
iex> l = line(point(1, 2), point(3, 4))
iex> incident?(p, l)
true

iex> p = point(1, 2)
iex> l = line(point(0, 0), point(1, 0))
iex> incident?(p, l)
false

iex> l1 = line(point(0, 0), point(1, 0))
iex> l2 = line(point(2, 0), point(3, 0))
iex> incident?(l1, l2)
true

iex> incident?(point(0, 0), line(point(0, 0), point(1, 0)))
true

iex> incident?(point(0, 1), line(point(0, 0), point(1, 0)))
false

iex> incident?(ideal_point(1, 0), line(point(0, 0), point(1, 0)))
true

iex> p = ideal_point(1, 0)
iex> l = line(point(0, 0), point(1, 0))
iex> incident?(p, l)
true

iex> p = ideal_point(0, 1)
iex> l = line(point(0, 0), point(1, 0))
iex> incident?(p, l)
false

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> inner(
...>   new(e1: 2),
...>   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.

Returns either {:ok, result} if inverse can be compute or :error otherwise

Examples

iex> inverse(
...>   new(e1: 2)
...> )
{:ok, new(e1: 0.5) }

iex> inverse(
...>   new(e1: 2)
...> )
{:ok, new(e1: 0.5)}

iex> inverse(
...>   new(scalar: 0.0)
...> )
:error

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> inverse!(
...>   new(e1: 2)
...> )
new(e1: 0.5)

iex> inverse!(
...>   new(e1: 2)
...> )
new(e1: 0.5)

iex> assert_raise ArgumentError, fn ->
...>   inverse!(new(scalar: 0))
...> end

A guard that checks if a floating point number is below the epsilon threshold.

Can be used in function guards:

def foo(x) when is_near_zero(x), do: x

Checks whether a multivector contains components of the given grade only.

A multivector is considered to have a grade if all non-zero components belong to that grade. The zero multivector is considered to have grade 0.

The is_grade(vector, grade) guard can be used in function guards:

def foo(mv) when is_grade(mv, 1) do
  mv
end

Examples

iex> grade?(new(e1: 1), 1)
true

iex> grade?(new(scalar: 1, e1: 2), 2)
false

iex> grade?(new(scalar: 1), 0)
true

iex> grade?(new(), 0)
true

See grade?/2

A guard that matches multi vectors that are all zero.

Can be used in function guards:

def foo(x) when is_zero(x), do: x

Computes the join of two geometric objects.

The join returns the smallest blade containing both geometric objects.

It is implemented using the dual outer product:

a ∨ b = dual(dual(a) ∧ dual(b))

In PGA, common cases are:

point ∨ point -> line
point ∨ line -> top-grade element (pseudoscalar, ie. trivector)

In this dual representation, incidence is tested by the vanishing of the join.

Examples

iex> p1 = point(0, 0)
iex> p2 = point(1, 0)
iex> incident?(p1, line(p1, p2))
true

Computes 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> left_contraction(
...>   new(e1: 2),
...>   new(e1: 3)
...> )

Creates the line through two points.

Equivalent to:

join(point1, point2)

Examples

iex> l = line(point(23, 42), point(3,4))
iex> incident?(point(23,42), l)
true

Creates a line from coefficients.

Represents the equation:

ax + by + c = 0

as the PGA vector:

a*e1 + b*e2 + c*e0

The line is represented up to scale.

iex> normalize!(line(1,2,3)) == normalize!(line(2,4,6))
true

Creates a line from a normal vector and a point.

Returns the normal vector of a line.

The normal is:

a*e1 + b*e2

Returns the maximum absolute coefficient of a multivector.

Accepts either a multivector struct or the internal coefficient tuple.

Example

iex> max_abs_component(new(e1: 2, scalar: 5))
5.0

iex> max_abs_component(new(e1: 5, scalar: 2))
5.0

Computes the meet of two geometric objects.

The meet returns the intersection of geometric objects. In this PGA representation it is computed using the outer product:

a ∧ b

Common case:

line ∧ line -> point

For non-intersecting objects, the result may be an ideal object or zero, depending on the geometric configuration.

Examples

iex> l1 = line(point(0, 0), point(1, 0))
iex> l2 = line(point(0, 0), point(0, 1))
iex> p = meet(l1, l2)
iex> point_coordinates(p)
{0.0, 0.0}

Returns the metric 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 norm of a multivector.

The norm is the square root of the absolute squared norm.

Example

iex> a = new(scalar: 3)
iex> norm(a)
3.0

Normalizes 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 = new(scalar: 2)
iex> norm(normalize!(a))
1.0

iex> normalize!(new())
** (ArgumentError) cannot normalize given null multivector ...

iex> normalize!(new(scalar: epsilon() / 2.0))
** (ArgumentError) cannot normalize given null multivector ...

Returns the normalized direction vector of a line.

Returns the normalized line normal vector.

Returns the scalar identity element.

This is the multiplicative identity.

Examples

iex> one()
~G"1.0"

Returns the Euclidean origin point.

Parses a string representation into a multivector.

The string may contain scalar coefficients and basis blades combined using + and - operators.

Examples:

iex> parse("1 + e1")
new(scalar: 1, e1: 1)

iex> parse("foo")
** (ArgumentError) expected number ...

Creates a finite projective point.

The point is represented homogeneously as:

P = e12 + x*e20 + y*e01

The optional w argument controls the homogeneous scale. Normally points should be created using the default value 1. The homogeneous scale w must not be zero.

Examples

iex> point_coordinates(point(1, 2))
{1.0, 2.0}

iex> point_coordinates(point(1, 2, 2))
{1.0, 2.0}

iex> point_coordinates(point(1, 2, 0.5))
{1.0, 2.0}

Extracts Cartesian coordinates from a finite point.

Returns: {x, y}

Examples

iex> point_coordinates(origin())
{0.0, 0.0}


iex> point_coordinates(point(2, 3))
{2.0, 3.0}

Returns the pseudoscalar:

e1 ∧ e2 ∧ e0

iex> grades(pseudoscalar())
[3]

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> reverse(new(e1: 2))
new(e1: 2)

iex> reverse(new(e12: 2))
new(e12: -2)

iex> reverse(new(scalar: 3))
new(scalar: 3)

Computes 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> right_contraction(
...>   new(e1: 2),
...>   new(e1: 3)
...> )

Creates a rotation motor around the origin.

The angle is specified in radians.

Examples

iex> p = point(1,0)
iex> {x, y} = point_coordinates(transform(p, rotor(:math.pi / 2)))
iex> {Float.round(x, 10), Float.round(y, 10)}
{0.0, 1.0}

iex> p = point(3,4)
iex> {x, y} = p
...>  |> transform(rotor(:math.pi() / 2))
...>  |> transform(rotor(-:math.pi() / 2))
...>  |> point_coordinates()
iex> {Float.round(x,10), Float.round(y,10)}
{3.0,4.0}

Computes the rotor that maps one frame to another.

The function constructs blades from the source and target frames and computes the transformation rotor:

R = normalize(1 + T * S⁻¹)

where S is the source frame blade and T is the target frame blade.

The resulting rotor can be applied to multivectors to rotate the source frame into the target frame.

Turns a Multivector with only a scalar part into an elixir number.

iex> ~G[3.0] |> scalar()
{:ok, 3.0}

iex> ~G[3.0 + e1] |> scalar()
:error

Checks whether a multivector contains only a scalar component.

Components with an absolute value smaller than eps are considered zero.

Examples

iex> scalar?(new(scalar: 3))
true

iex> scalar?(new(e1: 3))
false

iex> scalar?(new())
true

Returns the scalar coefficient of a multivector.

This is equivalent to retrieving the coefficient of the scalar blade.

Examples

iex> scalar_part(new(scalar: 5.0, e1: 2.0))
5.0

Computes the scalar product of two multivectors.

The result depends on the metric of the algebra.

In PGA2, points are represented as bivectors:

P = e12 + x*e20 + y*e01

They are not null elements as conformal points are in conformal geometric algebra. The metric of PGA2 gives:

scalar_product(P, P) = -1

for normalized finite points.

Examples

iex> scalar_product(vector(1, 2), vector(3, 4))
11.0

iex> scalar_product(new(e1: 1), new(e1: 2))
2.0

iex> scalar_product(point(1, 2), point(1, 2))
-1.0

Creates a multivector from a string representation.

The ~G sigil provides a convenient syntax for constructing multivectors using basis blades and coefficients.

Examples:

iex> ~G"e1 + e2"
new(e1: 1, e2: 1)

iex> ~G"2e1 - 0.5e12"
new(e1: 2, e12: -0.5)

iex> ~G"e12"
new(e12: 1)

iex> ~G"3"
new(scalar: 3)

iex> ~G[ε + εe1]
new(scalar: epsilon(), e1: epsilon())

iex> ~G[-ε - εe1]
new(scalar: -epsilon(), e1: -epsilon())

The parsed expression is converted into the same representation accepted by new/1, so blade ordering and signs are handled by the algebra implementation.

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.

This is also called the Spinor Magnitude.

Example

iex> a = new(scalar: 3)
iex> squared_norm(a)
9.0

Subtracts two multivectors component-wise.

Examples

iex> a = new(scalar: 5)
iex> b = new(scalar: 2)
iex> sub(a, b)
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> table() |> Map.has_key?({1, 1})
true

Turns a Multivector with only a scalar part into an elixir number.

Raises if the given Multivector is not grade 0.

iex> ~G[3.0] |> to_scalar()
3.0

iex> ~G[3.0 + e1] |> to_scalar()
** (ArgumentError) cannot convert multivector to scalar: only grade-0 multivectors can be converted to a scalar, got ...

Applies a motor transformation.

Uses the sandwich product:

M X M⁻¹

Creates a translation motor from a vector.

Examples

iex> m = translator(vector(3, 4))
iex> point_coordinates(transform(origin(), m))
{3.0, 4.0}

Creates a translation motor.

Translates by the vector (x,y).

Examples

iex> m = translator(3, 4)
iex> point_coordinates(transform(origin(), m))
{3.0, 4.0}


iex> point_coordinates(transform(origin(), translator(3, 4)))
{3.0, 4.0}

iex> point_coordinates(transform(point(1, 2), translator(-2, 5)))
{-1.0, 7.0}

iex> p = point(1, 2)
iex> point_coordinates(transform(p, translator(0, 0)))
{1.0, 2.0}

iex> m = translator(3,4)
iex> point_coordinates(transform(transform(point(5,6), m), inverse!(m)))
{5.0,6.0}

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> wedge(
...>   new(e1: 1),
...>   new(e2: 1)
...> )
new(e12: 1)

iex> wedge(
...>   new(e2: 1),
...>   new(e1: 1)
...> )
new(e12: -1)

iex> wedge(
...>   new(e1: 1),
...>   new(e1: 1)
...> )
new()

Computes the outer product of a list of multivectors.

The vectors are combined from left to right using the wedge product.

The result is a blade representing the subspace spanned by all input multivectors.

Examples

iex> wedge_all([])
new(scalar: 1)

iex> wedge_all([new(e1: 1)])
new(e1: 1)

iex> wedge_all([new(e1: 1), new(e1: 1)])
new()

iex> wedge_all([new(e1: 1), new(e2: 1)])
new(e12: 1)

iex> wedge_all([new(e2: 1), new(e1: 1)])
new(e12: -1)

Returns the zero multivector.

Examples

iex> zero()
~G"0"

Checks whether all coefficients of a multivector are zero.

Examples

iex> zero?(new())
true

iex> zero?(new(e1: 1))
false