Discrete Connections

A connection is an object from differential geometry that maps between tangent planes. On a manifold, tangent planes are the local planar approximation of the surface. When comparing objects living in two adjacent planes, one must first align the plane and their corresponding bases. This can be done by parallel transporting objects from one plane to the other.

On a surface mesh, this process is discrete and not differential. The parallel transport is then simply the angle formed by the two bases of the two planes.

Interesting blog post on the topic: http://wordpress.discretization.de/geometryprocessingandapplicationsws19/connections-and-parallel-transport/

SurfaceConnectionVertices

Bases: SurfaceConnection

Local bases and parallel transport defined between tangent planes of the mesh at the vertices

Reference

Globally Optimal Direction Fields, Knöppel et al. (2013)

Parameters:

Name	Type	Description	Default
`mesh`	`SurfaceMesh`	the supporting mesh	required
`feat`	`FeatureEdgeDetector`	feature edges of the mesh. If not provided, will be initialized with a FeatureEdgeDetector object that flags only the boundary. Defaults to None.	`None`

Other Parameters:

Name	Type	Description
`vnormals`	`Attribute`	An attribute representing the vertex normals. If not provided, will be computed at initialization
`angles`	`Attribute)`	An attribute representing angles at corners of faces. If not provided, will be computed at initialization

`base(i)`

Local basis of element i

Parameters:

Name	Type	Description	Default
`i`	`int`	index	required

Returns:

Type	Description
`(Vec, Vec)`	the X vector and Y vector of the tangent basis.

Note

cross(X,Y) is the normal of element i

`transport(iA, iB)`

summary

Parameters:

Name	Type	Description	Default
`iA`	`int`	first vertex index	required
`iB`	`int`	second vertex index	required

Returns:

Name	Type	Description
`float`	`float`	angle needed to change from basis of iA to basis of iB

SurfaceConnectionFaces

Bases: SurfaceConnection

Local bases and parallel transport defined between tangent planes of the mesh at the vertices

Reference

Globally Optimal Direction Fields, Knöppel et al. (2013)

Parameters:

Name	Type	Description	Default
`mesh`	`SurfaceMesh`	the supporting mesh	required
`feat`	`FeatureEdgeDetector`	feature edges of the mesh. If not provided, will be initialized with a FeatureEdgeDetector object that flags only the boundary. Defaults to None.	`None`

`base(i)`

Local basis of element i

Parameters:

Name	Type	Description	Default
`i`	`int`	index	required

Returns:

Type	Description
`(Vec, Vec)`	the X vector and Y vector of the tangent basis.

Note

cross(X,Y) is the normal of element i

`transport(iA, iB)`

summary

Parameters:

Name	Type	Description	Default
`iA`	`int`	first vertex index	required
`iB`	`int`	second vertex index	required

Returns:

Name	Type	Description
`float`	`float`	angle needed to change from basis of iA to basis of iB

DiscreteExponentialMap

https://en.wikipedia.org/wiki/Exponential_map_(Riemannian_geometry)

Usage

conn = SurfaceConnectionVertices(mesh)
expm = DiscreteExponentialMap(mesh, conn, rad)
expm.run({0, 2, 42}) # computes map for vertices 0, 2 and 42

expm.run() # computes map for all vertices

u,v = expm.map(0, 3) # coordinates of vertex 3 in exp map of vertex 0
u,v = expm.map(3, 0) # coordinates of vertex 0 in exp map of vertex 3. Exp map of 3 is computed on the go if necessary

Bases: Worker

References

[1] Interactive decal compositing with discrete exponential maps, Schmidt et al., 2006

Example

https://github.com/GCoiffier/mouette/blob/main/examples/expmap.py

Parameters:

Name	Type	Description	Default
`mesh`	`SurfaceMesh`	input surface mesh	required
`conn`	`SurfaceConnectionVertices`	connection over vertices for the mesh.	required
`radius`	`int`	Radius in number of edges of the exponential map of each vertex. Defaults to 2.	`2`
`verbose`	`bool`	verbose options. Defaults to False.	`False`

`_compute_map(u)`

Computes map originating from vertex u for all vertices v at most at distance 'self.d' from u in terms of number of edges

`export_map_as_mesh(v_source)`

Exports the map of vertex v_source as a flat surface mesh for visualization purposes

Parameters:

Name	Type	Description	Default
`v_source`	`int`	source vertex index	required

Returns:

Name	Type	Description
`SurfaceMesh`	`SurfaceMesh`	exponential map of v_source as a surface mesh

`map(v_source, v_target)`

Returns the uv-coordinates of v_target in the tangent plane of v_source. If the map of v_source was not computed, it is computed on the go.

Parameters:

Name	Type	Description	Default
`v_source`	`int`	target vertex index, center of the exponential map	required
`v_target`	`int`	query vertex index	required

Returns:

Name	Type	Description
`Vec`		uv-coordinates of v_target as projected in the tangent plane of v_source. Returns None if v_target is not in the map of v_source (vertex is invalid or too far)