# Overview

While the halfedge mesh encodes the connectivity of a surface, this section covers the classes which sit atop a halfedge mesh to define its geometry.

The first section below explains the class structure used to expose geometric logic, and the second section explains the system of automatically-cached quantities.

## Geometry hierarchy

TL;DR

Construct a VertexPositionGeometry object using vertex positions; it offers all the geometric routines you would expect, and can be passed to any method that demands geometry.

Geometry central is intentionally designed to allow flexibility in defining the geometry of a surface. Traditional code might assume a 3D position for every vertex, but many algorithms actually need only the intrinsic geometry of a surface, aka the edge lengths. More generally, specifying algorithms to only use the geometric data they really need allows us to seamlessly leverage powerful techniques.

We (sparingly) make use of polymorphism via inheritance in C++ to encode a hierarchy of geometric quantities that one might compute for a surface.

• Interfaces define which quantities can be computed from the geometry; for instance, an EmbeddedGeometryInterface can compute face normals, and it can also compute face areas because it extends the more basic IntrinsicGeometryInterface. Interfaces are abstract, and cannot be instantiated by themselves.
• Realizations are concrete classes that allow the user instantiate a geometry object from data; for instance, a VertexPositionGeometry can be constructed from vertex positions, and implements the EmbeddedGeometryInterface giving access to a wide range of intrinsic and extrinsic geometric quantities.

The following diagram outlines the interfaces and realizations currently available.

## Quantity management

### Immediate computation

In the most basic usage, realizations can compute simple quantities for a given element directly from input data. For instance, double VertexPositionGeometry::faceArea(Face f) will compute the area of a face. However, this is not the typical intended method for working with geometric quantities in geometry central.

### Managed quantities

A common pattern in geometry code is to maintain precomputed arrays of values that are used repeatedly (e.g. vertex normals). However, naive use of this pattern requires the programmer to coordinate these arrays throughout their codebase, or risk computing and storing the same array many times in distant subroutines. Geometry central embraces this pattern, and provides automatic support for proper use of it.

All geometry objects automatically maintain of system of caches for geometric quantities; the user can simply call (for instance) geometry.requireFaceAreas() at the beginning of a subroutine to ensure that the face area buffer is populated, then access geometry.faceAreas[f] in any subsequent code. This strategy keep storage and computation to a minimum by sharing repeated values across any use of the geometry object.

The following example demonstrates immediate computation vs cached quantities.

VertexPositionGeometry& geometry = /* ... */;

for(Face f : mesh->faces()) {
Vector3 normal = geometry.faceNormal(f);
}

// good: automatic caching and storage
geometry.requireFaceNormals();
for(Face f : mesh->faces()) {
Vector3 normal = geometry.faceNormals[f];
}


In fact, the inheritance in this design allows geometry central to leverage alternate ways of computing a quantity depending on the underlying data available. For instance, face areas are computed for an EdgeLengthGeometry using Heron’s rule, but for a VertexPositionGeometry a faster and more stable edge cross-product is used. Of course, this is all happening under the hood– the user just needs to call myGeometry.requireFaceAreas().

#### Dependencies

In addition, dependencies between these quantities are managed internally; for instance, if vertex normals are requested, face normals will be internally populated and used to compute vertex normals. However, these dependencies are internal and subject to change; the programmer should always explicitly call geometry.requireFaceNormals() if they intend to access face normals.

#### Updating

If the underlying geometric data changes (e.g., vertices are moved or the mesh is mutated), invoking geometry.refreshQuantities() will recompute all required values.

#### Minimizing storage usage

To minimize memory usage, invoke geometry.unrequireFaceNormals() at the conclusion of a subroutine to indicate that the quantity is no longer needed, decrementing an internal counter. The quantity is not instantly deleted after being un-required, but invoking geometry.purgeQuantities() will delete any quantities that are not currently required, reducing memory usage. Most users find that un-requiring and purging quantities is not necessary, and one can simply allow them to accumulate and eventually be deleted with the geometry object.

#### Quantity API

#include "geometrycentral/surface/geometry.h" to get all geometry interfaces.

All quantities offer methods and storage following the same naming pattern. For a quantity named YYYs (e.g. faceAreas), which is defined in an interface GeometryInterface (e.g. IntrinsicGeometry) the pattern is given below.

An exauhstive list is given in quantities.

MeshData<> GeometryInterface::YYYs

The member variable array for quantity YYY. Initially empty, but can be populated with requireYYYs() below.

For instance, for the quantity face areas, there is a member FaceData<double> IntrinsicGeometry::faceAreas.

void GeometryInterface::requireYYYs()

Request that the buffer for quantity YYYs be populated. If it is already populated the method will return right away, otherwise the quantity will be computed.

For instance, for the quantity face areas, one would call IntrinsicGeometry::requireFaceAreas().

void GeometryInterface::unrequireYYYs()

Indicate that the buffer for quantity YYYs no longer needs to be populated. Internally decrements a counter; once there is a unrequireYYYs() for every requireYYYs() call, the quantity can be purged by GeometryInterface::purgeQuantities().

For instance, for the quantity face areas, one would call IntrinsicGeometry::unrequireFaceAreas().

Note: most users find that un-requiring and purging quantities is not necessary, and one can simply allow them to accumulate and eventually be deleted with the geometry object. This functionality can be used only if reducing memory usage is very important.

T GeometryRealization::computeYYY(Element e)

Immediate computation: rather than using the caching system described above, directly compute the value from the input data.

Only available for quantities which can be easily computed in O(1) from inputs (e.g. face normals), but not for quantities with significant dependencies (e.g. vertex normals, which depend on all incident face normals).

For instance, face areas can be immediately computed with double VertexPositionGeometry::faceArea(Face f).

Note: immediate computation is generally only preferred if you are frequently changing the geometry; managed quantities are the primary method for computing geometric values.

In addition, the caching system provides two methods.

void GeometryInterface::refreshQuantities()

Recompute all required quantities from the input geometric data.

Should be called, for instance if vertices are moved or the underlying mesh is mutated.

void GeometryInterface::purgeQuantities()

Delete all cached quantities which are not currently require()‘d, reducing memory usage.

Note: most users find that un-requiring and purging quantities is not necessary, and one can simply allow them to accumulate and eventually be deleted with the geometry object. This functionality can be used only if reducing memory usage is very important.

## Interfaces

Interfaces are abstract classes which define which quantities are available for a given geometry, and compute/manage caches of these quantities.

For the full list of the managed quantities that can be computed by these interfaces, see the quantities section.

Base Geometry

#### Base Geometry

Class: BaseGeometryInterface

This is a simple base class which is serves as a parent of all geometry interfaces. It does not actually correspond to any geometric data, and the only quantities it manages are convenience element indices.

#include "geometrycentral/surface/base_geometry_interface.h"

Intrinsic Geometry

#### Intrinsic Geometry

Class: IntrinsicGeometryInterface

Extends BaseGeometryInterface. These quantites depend only on the notion of lengths and angles on the surface, but not how it might sit in space.

#include "geometrycentral/surface/intrinsic_geometry_interface.h"

An intrinsic geometry can be instantiated via any of the realizations which inherit from it:

• EdgeLengthGeometry
• VertexPositionGeometry (additionally inherits from the child EmbeddedGeometryInterface)
Extrinsic Geometry

#### Extrinsic Geometry

Class: ExtrinsicGeometryInterface

Extends IntrinsicGeometryInterface. These quantites depend on extrinsic angles (like dihedral angles at edges), but are rotation-invariant.

#include "geometrycentral/surface/extrinsic_geometry_interface.h"

An extrinsic geometry can be instantiated via any of the realizations which inherit from it:

• VertexPositionGeometry (additionally inherits from the child EmbeddedGeometryInterface)

Note that there is not currently any realization which instantiates an ExtrinsicGeometryInterface which is not also an EmbeddedGeometryInterface, but such a realization might one day exist.

Embedded Geometry

#### Embedded Geometry

Class: EmbeddedGeometryInterface

Extends ExtrinsicGeometryInterface. These quantites depend explicitly on how the surface sits in 3D space.

#include "geometrycentral/surface/embedded_geometry_interface.h"

An embedded geometry can be instantiated via any of the realizations which inherit from it:

• VertexPositionGeometry

## Realizations

Realizations construct a geometry object from input data. There are two widely used realizations in geometry central: VertexPositionGeometry and EdgeLengthGeometry. Other realizations of geometry can be created as needed to make use of other kinds of data, ranging from small modifications to e.g. make use of predefined tangent spaces, to completely new geometries defined from alternate input data.

In addition to the quantities listed in their own section, the realizations each offer a few immediate computations and some utility functions.

### Vertex Position Geometry

The usual notion of geometry for a mesh, with a position in 3D for each vertex. These positions are stored in the member VertexPositionGeometry::vertexPositions (which is inherited from the EmbeddedGeometryInterface).

This class inherits from all of the geometry interfaces mentioned above, so all quantities will be available.

#include "geometrycentral/surface/vertex_position_geometry.h"

void VertexPositionGeometry::VertexPositionGeometry(VertexData<Vector3> positions)

Construct a new geometry from vertex positions.

The positions input is copied, and stored in the member VertexPositionGeometry::vertexPositions.

void VertexPositionGeometry::VertexPositionGeometry(SurfaceMesh& mesh)

Construct a new geometry for the mesh, with all positions set to the origin Vector3{0., 0., 0.,}.

void VertexPositionGeometry::VertexPositionGeometry(SurfaceMesh& mesh, const VertexData<Vector3>& positions)

Construct a new geometry for a mesh from known vertex positions.

The positions input is copied, and stored in the member VertexPositionGeometry::vertexPositions.

void VertexPositionGeometry::VertexPositionGeometry(SurfaceMesh& mesh, const Eigen::MatrixBase<T>& positions)

Construct a new geometry for a mesh from known vertex positions, where the positions are stored as an Eigen matrix.

The input matrix should be a Vx3 matrix of floating point values. The Eigen::MatrixBase type is just a generic type which accepts most Eigen matrices as input, including the geometry-central DenseMatrix<> type.

std::unique_ptr<VertexPositionGeometry> VertexPositionGeometry::copy()

Copy the geometry, creating a new identical geometry on the same mesh. Any require() counts or already-computed quantities are not transferred, the new geometry is a blank slate.

std::unique_ptr<VertexPositionGeometry> VertexPositionGeometry::reinterpretTo(SurfaceMesh& targetMesh)

Copy the geometry, creating a new identical geometry on targetMesh. The target mesh must be in vertex-correspondence with the input mesh, in the sense that both meshes have the same number of vertices and iterating through the vertex sets yields matching vertices.

Any require() counts or already-computed quantities are not transferred, the new geometry is a blank slate.

Immediate computations. These routines directly compute geometry quantities from the input data, without touching the caching system. For the full list of available managed quantities, see the quantities section.

double VertexPositionGeometry::edgeLength(Edge e)

Compute the length of a single edge.

double VertexPositionGeometry::faceArea(Face f)

Compute the area of a single face.

double VertexPositionGeometry::cornerAngle(Corner c)

Compute the angle (in radians) formed by the two edges incident on a corner.

double VertexPositionGeometry::halfedgeCotanWeight(Halfedge he)

Compute the cotangent weight of a hafedge.

double VertexPositionGeometry::edgeCotanWeight(Edge e)

Compute the cotangent weight of an edge.

Vector3 VertexPositionGeometry::faceNormal(Face f)

Compute the normal of a face.

### Edge Length Geometry

A weaker notion of geometry where one knows only edge lengths. This data turns out to be sufficient to implement many algorithms in geometry processing, and offers valuable flexibility in defining the geometry.

This class inherits from the IntrinsicGeometryInterface, so only intrinsic quantities will be available.

#include "geometrycentral/surface/edge_length_geometry.h"

void EdgeLengthGeometry::EdgeLengthGeometry(SurfaceMesh& mesh, EdgeData<double> edgeLengths)

Construct a new geometry for a mesh from known edge lengths.

The edgeLengths input is copied, and stored in the member EdgeLengthGeometry::edgeLengths (inherited from IntrinsicGeometryInterface).

std::unique_ptr<EdgeLengthGeometry> EdgeLengthGeometry::copy()

Copy the geometry, creating a new identical geometry on the same mesh. Any require() counts or already-computed quantities are not transferred, the new geometry is a blank slate.

std::unique_ptr<EdgeLengthGeometry> reinterpretTo(SurfaceMesh& targetMesh)

Copy the geometry, creating a new identical geometry on targetMesh. The target mesh must be in edge-correspondence with the input mesh, in the sense that both meshes have the same number of edges and iterating through the edge sets yields matching edges.

Any require() counts or already-computed quantities are not transferred, the new geometry is a blank slate.

Immediate computations. These routines directly compute geometry quantities from the input data, without touching the caching system. For the full list of available managed quantities, see the quantities section.

double EdgeLengthGeometry::faceArea(Face f)

Compute the area of a single face.

double EdgeLengthGeometry::cornerAngle(Corner c)

Compute the angle (in radians) formed by the two edges incident on a corner.

double EdgeLengthGeometry::halfedgeCotanWeight(Halfedge he)

Compute the cotangent weight of a hafedge.

double EdgeLengthGeometry::edgeCotanWeight(Edge e)

Compute the cotangent weight of an edge.