Adds two multivectors component-wise.

Examples

iex> a = new(scalar: 2)
iex> b = new(scalar: 3)
iex> add(a, b)
new(scalar: 5)

This function checks that the object has bivector grade and contains finite point-pair structure. Use split/1 to extract the actual points.

This function only checks the grade. Use split/1 to determine whether the bivector represents a valid point pair.

Examples

iex> pp = meet(
...>   circle(point(0, 0), 2),
...>   line(point(-2, 0), point(2, 0))
...> )
iex> bivector_candidate?(pp)
true

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 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> 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

Creates a circle from a center point and radius.

The returned multivector represents the conformal circle object.

Examples

iex> circle(1, 2, 3)
~G"1.5e12p + 2.5e12m + 2.0e1pm - e2pm"

iex> circle?(circle(1, 2, 3))
true

iex> circle_parameters(circle(1, 2, 3))
{:circle, {{1.0, 2.0}, 3.0}}

iex> c = circle(point(1, 2), 3)
iex> contains?(c, point(4, 2))
true
iex> contains?(c, point(5, 2))
false

Creates a circle from Euclidean coordinates and radius.

Returns whether a multivector represents an OPNS circle.

Lines are also grade-3 blades in CGA2, so they are excluded.

Examples

iex> circle?(circle(point(0, 0), 2))
true

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

Returns whether a multivector is an OPNS circle or line.

In CGA2 both circles and lines are grade-3 objects. Lines are exactly those trivectors containing e_inf.

Use line?/1 to distinguish lines.

Examples

iex> circle_or_line?(circle(point(0, 0), 2))
true

Extracts the Euclidean parameters of an OPNS circle or line.

Returns either

• {:circle, {{x, y}, radius}}
• {:line, {a, b, c}}

where the line satisfies ax + by + c = 0.

Examples

iex> circle_parameters(circle(point(1, 2), 3))
{:circle, {{1.0, 2.0}, 3.0}}

Classifies a geometric object and extracts its Euclidean parameters.

Returns one of

• {:point, {x, y}}
• {:line, {a, b, c}}
• {:circle, {{x, y}, radius}}
• {:point_pair, kind, {p1, p2}}
• {:unknown, multivector}

Examples

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

iex> classify(point(-5, 4))
{:point, {-5.0, 4.0}}

iex> {:line, {a, b, c}} = classify(line(point(0, 0), point(0, 1)))
iex> abs(a) == 1.0 and b == 0.0 and c == 0.0
true

iex> {:line, {a, b, c}} = classify(line(point(0, 0), point(1, 0)))
iex> a == 0.0 and abs(b) == 1.0 and c == 0.0
true

iex> classify(circle(point(0, 0), 2))
{:circle, {{0.0, 0.0}, 2.0}}

iex> classify(circle(point(3, -2), 5))
{:circle, {{3.0, -2.0}, 5.0}}

iex> pp =
...>   meet(
...>     circle(point(0, 0), 2),
...>     line(point(-3, 0), point(3, 0))
...>   )
iex> {:point_pair, :real, points} = classify(pp)
iex> Enum.sort(Tuple.to_list(points))
[{-2.0, 0.0}, {2.0, 0.0}]

iex> pp =
...>   meet(
...>     circle(point(0, 0), 1),
...>     line(point(-2, 0), point(2, 0))
...>   )
iex> match?({:point_pair, :real, _}, classify(pp))
true

iex> pp =
...>   meet(
...>     circle(point(1, 2), 1),
...>     circle(point(3, 2), 1)
...>   )
iex> classify(pp)
{:point_pair, :tangent, {2.0, 2.0}}

iex> pp =
...>   meet(
...>     circle(point(1, -1), 1),
...>     circle(point(1, -3), 1)
...>   )
iex> classify(pp)
{:point_pair, :tangent, {1.0, -2.0}}

iex> classify(new(e1: 1))
{:unknown, new(e1: 1)}

iex> {:point_pair, :real, _} =  classify(join(point(0,0), point(1,0)))

Removes coefficients whose absolute value is below eps.

This is useful for cleaning up floating-point round-off errors after geometric computations.

Examples

iex> new(e1: 1.0, e2: 1.0e-12)
...> |> cleanup()
...> |> coefficient(:e2)
0.0

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

Tests whether a point lies on a conformal object.

Returns true when the incidence meet operation produces the zero multivector.

iex> l = line(point(0,0), point(1,0)) iex> contains?(l, point(2,0)) true iex> contains?(l, point(0,1)) false

Returns the dimension of the algebra.

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

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> dual(new(e1: 1)) |> inspect
new(e2pm: 1.0) |> inspect

Returns the conformal infinity vector.

The infinity vector represents the point at infinity in conformal space:

e_inf = e_m + e_p

Examples

iex> scalar_product(e_inf(), e_inf())
0.0

Returns the conformal origin vector.

The origin is defined as:

e_o = (e_m - e_p) / 2

Examples

iex> scalar_product(e_o(), e_inf())
-1.0

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> 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.

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 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.

Raises ArgumentError if the multivector is not invertible by this formula.

## Examples

    iex> inverse(
    ...>   new(e1: 2)
    ...> )|> inspect
    new(e1: 0.5) |> inspect

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 outer product join of two CGA objects.

The join is the wedge product:

join(a, b) = a ∧ b

This operates on the actual multivector representations. Objects represented in IPNS form should be converted to OPNS form before using the join.

For OPNS objects, the result is the smallest blade containing both.

Examples

iex> l =
...>   join(
...>     point(0, 0),
...>     point(1, 0)
...>   )
iex> grades(l)
[2]

iex> c =
...>   join(
...>     join(
...>       point(1, 0),
...>       point(0, 1)
...>     ),
...>     point(-1, 0)
...>   )
iex> grades(c)
[3]


iex> c =
...>   join(
...>     point(1,0),
...>     point(0,1),
...>     point(-1,0)
...>   )
iex> circle_parameters(c)
{:circle, {{0.0, 0.0}, 1.0}}

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)
...> )

Lifts an affine point representative back into the conformal point embedding.

Examples

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

iex> lift_point(point(2,3))
add(add(e_o(), add(new(e1: 2), new(e2: 3))), scale(6.5, e_inf()))

iex> point_coordinates(lift_point(point(4,5)))
{4.0,5.0}

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

Creates a conformal line through two points.

The line is represented using the outer product:

L = a ∧ b ∧ e_inf

Examples

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

iex> contains?(line(point(0,0), point(1,0)), point(0.5,0))
true

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

iex> contains?(line(point(0, 0), point(1, 0)), point(2, 0))
true

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

iex> l = line(point(2, -1), point(2, 3))
iex> contains?(l, point(2, 10))
true
iex> contains?(l, point(3, 0))
false

Returns whether a multivector is an OPNS line.

Examples

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

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

Creates an OPNS circle or degenerate line through three conformal points.

The circle is represented by the outer product:

C = a ∧ b ∧ c

For collinear points this degenerates to a line.

Examples

iex> c = ...> line_or_circle_from_points( ...> point(1, 0), ...> point(0, 1), ...> point(-1, 0) ...> ) iex> circle?(c) true iex> circle_parameters(c) {:circle, {{0.0, 0.0}, 1.0}}

iex> c = ...> line_or_circle_from_points( ...> point(0, 0), ...> point(1, 0), ...> point(0, 1) ...> ) iex> contains?(c, point(0, 0)) true

iex> c = ...> line_or_circle_from_points( ...> point(0, 0), ...> point(1, 0), ...> point(2, 0) ...> ) iex> line?(c) true

Returns the Euclidean coefficients of an OPNS line.

The returned tuple {a, b, c} satisfies

ax + by + c = 0

Examples

iex> {a, b, c} = line_parameters(line(point(0, 0), point(0, 1)))
iex> {abs(a), b, c}
{1.0, 0.0, 0.0}

iex> line_parameters(line(point(0, 0), point(1, 2)))
{-2.0, 1.0, 0.0}

iex> line_parameters(line(point(0,0), point(1,0)))
{0.0, 1.0, 0.0}

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 (incidence) operation between two objects.

The meet is implemented using duality:

meet(a,b) = dual(dual(a) ∧ dual(b))

Examples

iex> l = line(point(-2,0), point(2,0))
iex> c = circle(point(0,0),1)
iex> bivector_candidate?(meet(c,l))
true

iex> c1 = circle(point(1, 2), 1)
iex> c2 = circle(point(3, 2), 1)
iex> {:tangent, p} = split(meet(c1, c2))
iex> p |> point_coordinates()
{2.0, 2.0}

iex> c1 = circle(point(1, -1), 1)
iex> c2 = circle(point(1, -3), 1)
iex> {:tangent, p} = split(meet(c1, c2))
iex> p |> point_coordinates()
{1.0, -2.0}


iex> pp =
...>   meet(
...>     circle(point(0,0),1),
...>     circle(point(5,0),1)
...>   )
iex> match?({:point_pair, :imag, _}, classify(pp))
true

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

Normalizes a conformal point so that its conformal weight is -1.

The weight is defined as:

w = p · e_o

and this module uses the convention that normalized points satisfy:

p · e_o = -1

Raises ArgumentError if the point has zero weight.

Examples

iex> p = point(2, 3) |> scale(5)
iex> normalized = normalize_point(p)
iex> scalar_product(normalized, e_o())
-1.0

iex> {x, y} = normalize_point(scale(point(8, 2), 100)) |> point_coordinates()
iex> abs(x - 8.0) < 1.0e-10 and abs(y - 2.0) < 1.0e-10
true

Returns the scalar identity element.

This is the multiplicative identity.

Examples

iex> one()
~G"1.0"

Embeds a Euclidean point into conformal space.

Uses the standard CGA point representation:

P = e_o + x*e1 + y*e2 + 1/2(x²+y²)e_inf

Examples

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

Returns whether a multivector is a finite conformal point.

A finite point is a grade-1 null vector with non-zero weight.

Examples

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

iex> point?(new(e1: 1, e2: 1))
false

Extracts Euclidean coordinates from a conformal point.

Returns a tuple:

{x, y}

Examples

iex> point(2, 3)
...> |> scale(7)
...> |> point_coordinates()
{2.0, 3.0}

Returns the CGA pseudoscalar:

e1 ∧ e2 ∧ ep ∧ em

iex> grades(pseudoscalar())
[4]

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 Euclidean rotation rotor.

Rotates by angle radians around the origin.

The rotor is normalized before being returned.

iex> {x, y} = point(1, 0) ...> |> transform(rotor(:math.pi() / 2)) ...> |> cleanup() ...> |> point_coordinates() iex> abs(x) < 1.0e-10 true iex> abs(y - 1.0) < 1.0e-10 true

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.

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.

This is the scalar part of the geometric product. It uses the CGA metric defined by the module signature.

iex> scalar_product(new(e1: 1), new(e1: 1)) 1.0

iex> scalar_product(point(1,0), new(e1: 1)) 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)

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 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.

Splits a point pair into its two conformal points.

Returns

• {:real, p1, p2} for two real points,
• {:imag, p1, p2} for an imaginary point pair, or
• :invalid if the bivector is not a valid point pair.

Examples

iex> l = line(point(-2, 0), point(2, 0))
iex> c = circle(point(0, 0), 1)
iex> {:real, p1, p2} = split(meet(c, l))
iex> Enum.sort([point_coordinates(p1), point_coordinates(p2)])
[{-1.0, 0.0}, {1.0, 0.0}]

iex> c1 = circle(point(-0.5, 0), 1)
iex> c2 = circle(point(0.5, 0), 1)
iex> {:real, p1, p2} = split(meet(c1, c2))
iex> [{x1, y1}, {x2, y2}] = Enum.sort([point_coordinates(p1), point_coordinates(p2)])
iex> abs(x1) < 1.0e-10 and abs(x2) < 1.0e-10
true
iex> (y1 < 0) != (y2 < 0)
true
iex> abs(abs(y1) - :math.sqrt(0.75)) < 1.0e-10
true
iex> abs(abs(y2) - :math.sqrt(0.75)) < 1.0e-10
true

iex> split(meet(circle(point(0,0),1), circle(point(0,0),1)))
:invalid

iex> c1 = circle(point(0,0),1)
iex> c2 = circle(point(5,0),1)
iex> {:imag, _, _} = split(meet(c1,c2))

iex> l = line(point(-1,1), point(1,1))
iex> c = circle(point(0,0),1)
iex> {:tangent, p} = split(meet(c,l))
iex> point_coordinates(p)
{0.0,1.0}

 iex> pp = join(point(0.0,0.0), point(1,0))
 iex> {:real, a, b} = split(pp)
 iex> point_coordinates(a)
 {0.0, 0.0}
 iex> point_coordinates(b)
 {1.0, 0.0}

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 = 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

Formats 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(new())
"0"

iex> inspect(new(scalar: 2))
"2.0"

iex> inspect(new(e1: 1))
"e1"

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

Applies a motor transformation to a CGA object.

Performs the sandwich product:

M * X * reverse(M)

Examples

iex> point(1, 2) ...> |> transform(translator(3, -1)) ...> |> point_coordinates() {4.0, 1.0}

Creates a translator motor for translating by (x, y).

The returned motor can be applied with transform/2.

iex> p = point(1, 2) |> transform(translator(3, -1)) iex> point_coordinates(p) {4.0, 1.0}

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> undual(dual(new(e1: 2)))
new(e1: 2)

Creates a Euclidean vector embedded in CGA.

The vector is represented only by its Euclidean components:

x*e1 + y*e2

It is not a conformal point. Use point/2 to embed a point.

Examples

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

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

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.

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