emmy.calculus.manifold — org.mentat/emmy 0.31.0

This namespace defines a functional API for:

• differentiable manifolds (both manifold families like Rn and manifolds specialized to a concrete dimension)
• manifold points
• coordinate patches

As well as a whole bunch of defined manifolds and coordinate systems for exploration and fun!

alternate-angles clj/s

attach-coordinate-system clj/s

(attach-coordinate-system family
                          coordinate-system-name
                          patch-name
                          coordinate-system-ctor)

Returns a new manifold family generated by attaching the supplied coordinate system constructor to family, indexed by the supplied patch and coordinate system names.

attach-patch clj/s

(attach-patch family patch-name)

Takes a manifold family and attaches a patch template with the supplied patch-name. Returns a new manifold family.

All manifolds generated from the returned family will have this coordinate patch attached.

c:generate clj/s

(c:generate n orientation f)

Generates a coordinate structure of the supplied dimension n, and orientation using the supplied function f for entries. See the very similar emmy.structure/generate for more details.

NOTE from GJS: this is a kludge introduced only to allow a coordinate of dimension 1 to automatically unwrap itself.

chart clj/s

(chart coordinate-system)

Given an ICoordinateSystem, returns a function from a point on the coordinate system's manifold to the coordinate representation specified by the supplied coordinate-system.

constant-manifold-function clj/s

(constant-manifold-function c)

Takes some constant c and returns a manifold function that maps every input manifold point to c.

coordinate-prototype clj/s

(coordinate-prototype coordsys)

Returns the symbolic coordinate prototype associated with coordsys. This is a structure of the correct dimension for this coordinate system, with all symbolic entries.

Returns nil for non-valid inputs.

coordinate-system-at clj/s

(coordinate-system-at manifold coordinate-system-name patch-name)

Returns an ICoordinateSystem instance specialized to the patch named patch-name on manifold.

coordinate-system-names clj/s

(coordinate-system-names patch)

Returns a set of names of all coordinate system constructors registered in the supplied patch.

coordinate-system? clj/s

(coordinate-system? x)

Returns true if x implements ICoordinateSystem, false otherwise.

corresponding-velocities clj/s

(corresponding-velocities coords)

Takes a coordinate representation coords of a manifold point with all symbolic entries, and returns a structure of the same shape with v: prepended to all symbols.

This structure is appropriate for representing the velocities associated with each coordinate.

Euler-angles clj/s

get-coordinates clj/s

(get-coordinates manifold-point coordinate-system thunk)

Returns the representation of manifold-point in coordinate-system.

If an entry for the given coordinate-system is not found, thunk is called to produce the representation. The representation is cached in the point.

get-patch clj/s

(get-patch manifold patch-name)

Returns the patch named by patch-name within the supplied manifold if registered. Throws otherwise.

NOTE that the returned patch will keep a reference to the supplied manifold under a :manifold key.

A coordinate patch is a simply-connected open set around a point in the manifold. A manifold might have many patches. Coordinate systems are defined on patches; these allow the parameterization of any point on the patch in terms of a tuple of real numbers (the coordinates).

ICoordinateSystem clj/s protocol

check-coordinates clj/s

(check-coordinates this coords)

Returns true if the supplied coordinates coords can be converted into a point by this ICoordinateSystem, false otherwise.

check-point clj/s

(check-point this point)

Returns true if the supplied point can be converted into coordinates by this ICoordinateSystem, false otherwise.

coords->point clj/s

(coords->point this coords)

Returns the manifold point on this ICoordinateSystem's manifold corresponding to the supplied coords.

point->coords clj/s

(point->coords this point)

Returns a coordinate representation of the supplied manifold point point, as specified by this ICoordinateSystem.

uuid clj/s

(uuid this)

Returns a unique identifier for this instance of ICoordinateSystem.

(This is an internal implementation detail to allow us to attach coordinate prototypes and other items as metadata to an ICoordinateSystem without affecting equality.)

literal-manifold-function clj/s

(literal-manifold-function sym coordinate-system)

Given a symbolic name sym and an ICoordinateSystem, returns a literal function that maps coordinate-free manifold points to a scalar output.

Also aliased as literal-manifold-function.

literal-scalar-field clj/s

Alias for literal-manifold-function, present for scmutils codebase compatibility.

make-manifold clj/s

(make-manifold family n)

(make-manifold family n embedding-dimension)

Returns a concrete manifold generated by specializing the supplied manifold family into a concrete manifold of dimension n. n must be a positive integer.

Optionally takes an embedding-dimension; this must be >= the value of n. Use this in cases like an n-sphere embedded in a euclidean space of dimension n+1.

A manifold is a topological space that locally resembles Euclidean space near each point.

make-manifold-family clj/s

(make-manifold-family name-format)

(make-manifold-family name-format over)

Generates a manifold family (a template for building manifolds) from the supplied name-format.

Generated manifolds locally resemble Euclidean space (Rn) by default. You can optionally pass 'Complex or 'Quaternion to over to customize the field of the vector space that the manifold locally resembles at each point.

NOTE: only 'Real does anything as of 3.15.2021.

manifold clj/s

(manifold m)

If m is a manifold, acts as identity. Else, if given some structure associated with a manifold (like a coordinate system), returns the associated manifold.

manifold-family? clj/s

(manifold-family? m)

Returns true if m is a dictionary representing a manifold family, false otherwise.

manifold-point-representation clj/s

(manifold-point-representation point)

Returns the backing Euclidean space representation of the supplied manifold point.

manifold-point? clj/s

(manifold-point? p)

Returns true if p is a manifold point, false otherwise.

manifold-type clj/s

(manifold-type manifold)

The supplied manifold m locally resembles some vector space; this function returns the field over which that vector space was specified.

manifold? clj/s

(manifold? m)

Returns true if m is a dictionary representing a manifold, false otherwise.

one-manifold-function clj/s

(one-manifold-function point)

Manifold function that maps every input manifold point to the scalar value 1.

patch-names clj/s

(patch-names manifold)

Returns a set of patch names registered in the supplied manifold.

point clj/s

(point coordinate-system)

Given an ICoordinateSystem, returns a function from coordinates in coordinate-system's repesentation to the matching point on the manifold associated with coordinate-system.

point->manifold clj/s

(point->manifold point)

Return the manifold upon which this point was defined.

R1 clj/s

R1-rect clj/s

R2 clj/s

R2-polar clj/s

R2-rect clj/s

R3 clj/s

R3-cyl clj/s

R3-rect clj/s

R3-spherical clj/s

R4 clj/s

R4-cyl clj/s

R4-rect clj/s

Rn clj/s

S1 clj/s

S1-circular clj/s

S1-gnomonic clj/s

S1-slope clj/s

S1-tilted clj/s

S2 clj/s

S2-gnomonic clj/s

S2-Riemann clj/s

S2-spherical clj/s

S2-stereographic clj/s

S2-tilted clj/s

Similar to the S2-spherical coordinate system, with the north pole rotated 90 degrees and lying along the positive y axis.

S2-type clj/s

S2p clj/s

S2p-gnomonic clj/s

S2p-Riemann clj/s

S2p-spherical clj/s

S2p-stereographic clj/s

S2p-tilted clj/s

S3 clj/s

S3-gnomonic clj/s

S3-spherical clj/s

S3-stereographic clj/s

S3-tilted clj/s

Sn clj/s

SO3 clj/s

SO3-type clj/s

spacetime clj/s

spacetime-rect clj/s

spacetime-sphere clj/s

the-real-line clj/s

transfer-point clj/s

(transfer-point embedded embedding)

Returns a function that takes a single manifold point embedded in the manifold embedded and transfers the point to the supplied embedding manifold.

The embedding dimension must be the same for both manifolds.

NOTE that embedded and embedding can be either manifolds, or instances of ICoordinateSystem. In the latter case embedded and embedding will bind to the manifold associated with the supplied ICoordinateSystem.

typical-coords clj/s

(typical-coords coordinate-system)

Given an ICoordinateSystem, returns a structure that matches the coordinate-prototype of coordinate-system, with all unique, gensym-ed entries.

Use typical-coords if you require a unique symbolic coordinate representation compatible with coordinate-system.

See typical-point for a coordinate-free version of this function.

typical-point clj/s

(typical-point coordinate-system)

Given an ICoordinateSystem, returns a unique, symbolically-represented point on the manifold associated with coordinate-system.

See typical-coords for a coordinate-based version of this function.

with-coordinate-prototype clj/s

(with-coordinate-prototype coordsys prototype)

Returns an identical coordsys with the new coordinate-prototype installed.

zero-manifold-function clj/s

(zero-manifold-function point)

Manifold function that maps every input manifold point to the scalar value 0.