Geometry Optimization

Detailed Description

Provides an abstraction for reasoning about geometry in optimization problems, and using optimization problems to solve geometry problems.

Relationship to other components in Drake.

SceneGraph handles many types of geometry (see Shape). It is specialized to 3D and puts a strong emphasis on efficient implementation of a particular subset of geometry queries, like collision detection and signed-distance queries. SceneGraph also provides a lot of valuable tools for content management, including parsing geometries from file (via multibody::Parser) and Role.

MathematicalProgram has many relevant solvers::Cost / solvers::Constraint for reasoning about geometry (e.g., the LorentzCone or even LinearConstraint). The class and methods in this group add a level of modeling power above these individual constraints (there are many different types of constraints one would write given various optimization on these sets).

The geometry::optimization tools support:

• The ability to model sets in arbitrary dimensions, not just 3D.
• Many advanced queries (intersections, Minkowski sum, set containment, etc).
• The ability to use sets in mathematical programs
• The ability to use mathematical programs to perform operations on sets.
• The ability to include the geometry from a SceneGraph into a MathematicalProgram.
• The ability to use SceneGraph queries for some of the geometry operations (signed distance, collision), when the set is in 3D.
• The ability to use SceneGraph for rendering the geometries when they are in 3D.

Modules
	Geodesic Convexity
	A robot that has revolute joints without any limits has an inherently non-Euclidean configuration space, but one can still consider "geodesically-convex" sets, akin to convex sets in Euclidean space.

Classes
class	AffineBall
	Implements an ellipsoidal convex set represented as an affine scaling of the unit ball {Bu + center | |u|₂ ≤ 1}. More...
class	AffineSubspace
	An affine subspace (also known as a "flat", a "linear variety", or a "linear manifold") is a vector subspace of some Euclidean space, potentially translated so as to not pass through the origin. More...
class	CartesianProduct
	The Cartesian product of convex sets is a convex set: S = X₁ × X₂ × ⋯ × Xₙ = {(x₁, x₂, ..., xₙ) | x₁ ∈ X₁, x₂ ∈ X₂, ..., xₙ ∈ Xₙ}. More...
class	ConvexHull
	Implements the convex hull of a set of convex sets. More...
class	ConvexSet
	Abstract base class for defining a convex set. More...
class	GraphOfConvexSets
	GraphOfConvexSets (GCS) implements the design pattern and optimization problems first introduced in the paper "Shortest Paths in Graphs of Convex Sets". More...
class	HPolyhedron
	Implements a polyhedral convex set using the half-space representation: {x| A x ≤ b}. More...
class	Hyperellipsoid
	Implements an ellipsoidal convex set represented by the quadratic form {x | (x-center)ᵀAᵀA(x-center) ≤ 1}. More...
class	Hyperrectangle
	Axis-aligned hyperrectangle in Rᵈ defined by its lower bounds and upper bounds as {x| lb ≤ x ≤ ub}. More...
class	ImplicitGraphOfConvexSets
	A base class to define the interface to an implicit graph of convex sets. More...
class	Intersection
	A convex set that represents the intersection of multiple sets: S = X₁ ∩ X₂ ∩ ... More...
struct	IrisOptions
	Configuration options for the IRIS algorithm. More...
class	MinkowskiSum
	A convex set that represents the Minkowski sum of multiple sets: S = X₁ ⨁ X₂ ⨁ ... More...
class	Point
	A convex set that contains exactly one element. More...
class	Spectrahedron
	Implements a spectrahedron (the feasible set of a semidefinite program). More...
class	VPolytope
	A polytope described using the vertex representation. More...