6.1 Manifolds

9.1.0.11

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6.1 Manifolds🔗

package: rktsicm

This module provides objects to construct manifolds. Some important manifolds are predefined but it is possible to construct more.

Although the OO-like implementation is procedure/message based, for the purpose of this document the racket-class syntax has been used.

But there are (currently) no manifold% or other classes actualy defined, and there is no good way to check if something is-a? manifold%.

Methods, like ((manifold-spec%__’new-patch) rst ...) should be called as ((manifold-spec% 'new-patch) rst ...).

6.1.1 (Fake) classes🔗

class
manifold-spec% : class?

superclass: procedure?

constructor
(specify-manifold spec [type]) → manifold-spec%
  spec : symbol?
  type : symbol? = Real
The constructor function for a new manifold-spec%. Predefined manifold-spec%s include R^n for n-dimensional space, S^n for surface of an n-dimensional sphere. an "^n" within the spec will be used to attach the dimension to a manifold% generated by this manifold-spec%
method
((manifold-spec%__’new-patch patch-name
generator
setup) → void?
  patch-name : symbol?
  generator : any
  setup : any
Attaches a new patch to the manifold. Don’t use directly but use attach-patch.
method
((manifold-spec%__’patch-setup) patch-name) → any
  patch-name : symbol?
Get the patch-generator for a patch linked to the manifold-spec%. It should not be necessary to call this directly.
method
((manifold-spec%__’generator dimension
[ embedding-dimension]) → manifold%
  dimension : positive-integer?
  embedding-dimension : positive-integer? = dimension
Set up a new manifold%. Don’t use directly but use make-manifold.

class
manifold% : class?

superclass: procedure?

A manifold is ...???.
field
(manifold%__’name) → symbol?
field
(manifold%__’the-manifold) → patch%
Returns itself
field
(manifold%__’type) → symbol?
field
(manifold%__’dimension) → positive-integer?
field
(manifold%__’manifold-dimension) → positive-integer?
field
(manifold%__’embedding-dimension) → positive-integer?
field
(manifold%__’patch-names) → (listof symbol?)
method
((manifold%__’get-patch) name) → patch%
  name : symbol?
Get the patch from a manifold. Don’t use directly but use patch
field
(manifold%__’distinguished-points) → (listof ???)
¿leftovers?
method
((manifold%__’add-distinguished-point! sspec
m) → void?
  sspec : symbol?
  m : ???
¿leftovers?

class
patch% : class?

superclass: procedure?

A patch is ...???. ... linked to a manifold ... and one or more coordinate systems. Messages not handled by the patch will be passed to the manifold. A new patch should be added via attach-patch.
field
(patch%__’name) → symbol?
The name of the patch.
field
(patch%__’the-patch) → patch%
Returns itself
field
(patch%__’coordinate-system-names) → (listof symbol?)
All the defined coordinate systems for a patch.
method
((patch%__’get-coordinate-system) name) → coord-sys%
name : symbol?
Get the coordinate system on a patch. Don’t use directly but use coordinate-system-at

class
coord-sys% : class?

superclass: procedure?

A coordinate-system is ...???. ... linked to a manifold and a patch. Messages not handled by the coordinate-system will be passed to the manifold. A new coordinate-system should be added via attach-coordinate-system.
field
(coord-sys%__’name) → symbol?
The name of the coordinate-system.
field
(coord-sys%__’patch) → patch%
The patch the coordinate-system belongs to.
method
((coord-sys%__’->point) coord) → manifold-point%
  coord : structure?
method
((coord-sys%__’check-coords) coord) → boolean?
  coord : structure?
method
((coord-sys%__’->coords) point) → manifold-point%
  point : manifold-point%
method
((coord-sys%__’check-point) point) → boolean?
  point : manifold-point%

class
coord-transformer% : class?

superclass: procedure?

6.1.2 Functions🔗

class
manifold-point% : class?

superclass: procedure?

procedure
(temp) → number?

procedure
(patch name manifold) → patch%
name : symbol?
manifold : manifold%

Get the patch from a manifold.

procedure
(attach-patch patch-name manifold-spec) → void?
patch-name : symbol?
manifold-spec : manifold-spec%

Attaches a new patch to the manifold. For example (attach-patch 'north-pole S^n) or (attach-patch 'origin R^n)

procedure
(make-manifold manifold-spec
dimension
[ embedding-dimension]) → manifold%
  manifold-spec : manifold-spec%
  dimension : positive-integer?
  embedding-dimension : positive-integer? = dimension

Create a new manifold% within manifold-spec%.

procedure
(coordinate-system name patch) → coord-sys%
name : symbol?
patch : patch%

Get the coordinate system on a patch.

procedure
(coordinate-system-at coord-name
patch-name
manifold) → coord-sys%
  coord-name : symbol?
  patch-name : symbol?
  manifold : manifold%

procedure
(attach-coordinate-system name
patch-name
spec
transformations
[ prototype]) → void?
  name : symbol?
  patch-name : symbol?
  spec : manifold-spec%
  transformations : (-> manifold% coord-transformer%)
  prototype : structure? = (up 'x0 'x1 ...)

Sets up and a coord-sys% to a patch% on a manifold-spec% so it can be used on it’s manifolds. If a prototype is supplied only manifolds with the same dimension as the prototype can be used with this coordinate system.

procedure
(transfer-point embedded embedding)
→ (-> manifold-point% manifold-point%)
embedded : manifold%
embedding : manifold%

Creates a function that will copy points from one manifold% to another. For this to work the (embedded) dimensions of the manifold%s must be the same. See also the examples at the end of this section.

6.1.3 Predefined elements🔗

value
R^n : manifold-spec%

Pre-installed patch%es:
'origin
Pre-installed coordinate-system generators:
'rectangular
'polar/cylindrical
'spherical/cylindrical
'spacetime-spherical
With names for folowing manifold%s and coord-sys%tems:
value
R1 : manifold%
value
R1-rect : coord-sys%
value
the-real-line : coord-sys%
value
R2 : manifold%
value
R2-rect : coord-sys%
value
R2-polar : coord-sys%
value
R3 : manifold%
value
R3-rect : coord-sys%
value
R3-cyl : coord-sys%
value
R3-spherical : coord-sys%
value
R4 : manifold%
value
R4-rect : coord-sys%
value
R4-cyl : coord-sys%
value
spacetime : manifold%
value
spacetime-rect : coord-sys%
value
spacetime-sphere : coord-sys%

value
S^n : manifold-spec%

Pre-installed patch%es:
'north-pole
'tilted
'south-pole
Pre-installed coordinate-system generators:
'spherical
'stereographic
'gnomic
With names for folowing manifold%s and coord-sys%tems:
value
S1 : manifold%
value
S1-circular : (coord-sys% 'spherical 'north-pole)
value
S1-tilted : (coord-sys% 'spherical 'tilted)
value
S1-sloped : (coord-sys% 'stereographic 'north-pole)
value
S1-gnomic : (coord-sys% 'gnomic 'north-pole)
value
S2p : manifold%
value
S2p-spherical : (coord-sys% 'spherical 'north-pole)
value
S2p-tilted : (coord-sys% 'spherical 'tilted)
value
S2p-sloped : (coord-sys% 'stereographic 'north-pole)
value
S2p-Riemann : (coord-sys% 'stereographic 'north-pole)
value
S2p-gnomic : (coord-sys% 'gnomic 'north-pole)
value
S3 : manifold%
value
S3-spherical : (coord-sys% 'spherical 'north-pole)
value
S3-tilted : (coord-sys% 'spherical 'tilted)
value
S3-gnomic : (coord-sys% 'gnomic 'north-pole)
value
S3-stereographic : (coord-sys% 'stereographic 'south-pole)

value
S^2 : manifold-spec%

Pre-installed patch%es:
'north-pole
'tilted
'south-pole
Pre-installed coordinate-system generators:
'spherical
'stereographic
'gnomic
With names for folowing manifold%s and coord-sys%tems:
value
S2 : manifold%
value
S2-spherical : (coord-sys% 'spherical 'north-pole)
value
S2-tilted : (coord-sys% 'spherical 'tilted)
value
S2-stereographic : (coord-sys% 'stereographic 'north-pole)
value
S2-Riemann : (coord-sys% 'stereographic 'north-pole)
value
S2-gnomic : (coord-sys% 'gnomic 'north-pole)

value
SO3-type : manifold-spec%

Pre-installed patch%es:
'Euler-patch
'alternate-patch
Pre-installed coordinate-system generators:
'Euler
'alternate
With names for folowing manifold%s and coord-sys%tems:
value
SO3 : manifold%
value
Euler-angles : (coord-sys% 'Euler 'Euler-patch)
value
alternate-angles : (coord-sys% 'alternate 'alternate-patch)

6.1.4 Examples🔗

With all this in place you can define a point using spherical coordinates, and extracts its rectangular coordinates:

Examples:

> (define P ((R3-spherical '->point) (up 'r 'θ 'ϕ)))
> (expression ((R3-rect '->coords) P))
'(up (* r (cos ϕ) (sin θ)) (* r (sin θ) (sin ϕ)) (* r (cos θ)))

For this to work points need to be defined on the same manifold. If the embedding dimension is the same, you can also transfer points from one manifold to another:

Examples:

> (define P ((R3-rect '->point) (up 'x 'y 'z)))
> (expression
((S2-spherical '->coords)
((transfer-point R3-rect S2-spherical) P)))
'(up (acos z) (atan y x))

contents ← prev up next →

1	Introduction
2	General
3	Kernel
4	Display
5	Polynomials
6	Calculus

6.1.1	(Fake) classes
6.1.2	Functions
6.1.3	Predefined elements
6.1.4	Examples