Detailed Description

Abstract base class for defining a convex set.

#include <drake/geometry/optimization/convex_set.h>

Public Member Functions
virtual	~ConvexSet ()

std::unique_ptr< ConvexSet >	Clone () const
	Creates a unique deep copy of this set. More...

int	ambient_dimension () const
	Returns the dimension of the vector space in which the elements of this set are evaluated. More...

bool	IntersectsWith (const ConvexSet &other) const
	Returns true iff the intersection between `this` and `other` is non-empty. More...

bool	IsBounded () const
	Returns true iff the set is bounded, e.g., there exists an element-wise finite lower and upper bound for the set. More...

bool	IsEmpty () const
	Returns true iff the set is empty. More...

std::optional< Eigen::VectorXd >	MaybeGetPoint () const
	If this set trivially contains exactly one point, returns the value of that point. More...

std::optional< Eigen::VectorXd >	MaybeGetFeasiblePoint () const
	Returns a feasible point within this convex set if it is nonempty, and nullopt otherwise. More...

bool	PointInSet (const Eigen::Ref< const Eigen::VectorXd > &x, double tol=0) const
	Returns true iff the point x is contained in the set. More...

std::pair< VectorX< symbolic::Variable >, std::vector< solvers::Binding< solvers::Constraint > > >	AddPointInSetConstraints (solvers::MathematicalProgram *prog, const Eigen::Ref< const solvers::VectorXDecisionVariable > &vars) const
	Adds a constraint to an existing MathematicalProgram enforcing that the point defined by vars is inside the set. More...

std::vector< solvers::Binding< solvers::Constraint > >	AddPointInNonnegativeScalingConstraints (solvers::MathematicalProgram *prog, const Eigen::Ref< const solvers::VectorXDecisionVariable > &x, const symbolic::Variable &t) const
	Let S be this convex set. More...

std::vector< solvers::Binding< solvers::Constraint > >	AddPointInNonnegativeScalingConstraints (solvers::MathematicalProgram *prog, const Eigen::Ref< const Eigen::MatrixXd > &A, const Eigen::Ref< const Eigen::VectorXd > &b, const Eigen::Ref< const Eigen::VectorXd > &c, double d, const Eigen::Ref< const solvers::VectorXDecisionVariable > &x, const Eigen::Ref< const solvers::VectorXDecisionVariable > &t) const
	Let S be this convex set. More...

std::pair< std::unique_ptr< Shape >, math::RigidTransformd >	ToShapeWithPose () const
	Constructs a Shape and a pose of the set in the world frame for use in the SceneGraph geometry ecosystem. More...

double	CalcVolume () const
	Computes the exact volume for the convex set. More...

SampledVolume	CalcVolumeViaSampling (RandomGenerator *generator, const double desired_rel_accuracy=1e-2, const int max_num_samples=1e4) const
	Calculates an estimate of the volume of the convex set using sampling and performing Monte Carlo integration. More...

std::optional< std::pair< std::vector< double >, Eigen::MatrixXd > >	Projection (const Eigen::Ref< const Eigen::MatrixXd > &points) const
	Computes in the L₂ norm the distance and the nearest point in this convex set to every column of `points`. More...

bool	has_exact_volume () const
	Returns true if the exact volume can be computed for this convex set instance. More...

Protected Member Functions
	ConvexSet (int ambient_dimension, bool has_exact_volume)
	For use by derived classes to construct a ConvexSet. More...

template<typename Archive >
void	Serialize (Archive *a)
	Implements non-virtual base class serialization. More...

virtual std::unique_ptr< ConvexSet >	DoClone () const =0
	Non-virtual interface implementation for Clone(). More...

virtual std::optional< bool >	DoIsBoundedShortcut () const
	Non-virtual interface implementation for DoIsBoundedShortcut(). More...

virtual std::vector< std::optional< double > >	DoProjectionShortcut (const Eigen::Ref< const Eigen::MatrixXd > &points, EigenPtr< Eigen::MatrixXd > projected_points) const
	Non-virtual interface implementation for DoProjectionShortcut(). More...

virtual bool	DoIsEmpty () const
	Non-virtual interface implementation for IsEmpty(). More...

virtual std::optional< Eigen::VectorXd >	DoMaybeGetPoint () const
	Non-virtual interface implementation for MaybeGetPoint(). More...

virtual std::optional< Eigen::VectorXd >	DoMaybeGetFeasiblePoint () const
	Non-virtual interface implementation for MaybeGetFeasiblePoint(). More...

virtual bool	DoPointInSet (const Eigen::Ref< const Eigen::VectorXd > &x, double tol) const
	Non-virtual interface implementation for PointInSet(). More...

virtual std::optional< bool >	DoPointInSetShortcut (const Eigen::Ref< const Eigen::VectorXd > &x, double tol) const
	A non-virtual interface implementation for PointInSet() that should be used when the PointInSet() can be computed more efficiently than solving a convex program. More...

virtual std::pair< VectorX< symbolic::Variable >, std::vector< solvers::Binding< solvers::Constraint > > >	DoAddPointInSetConstraints (solvers::MathematicalProgram *prog, const Eigen::Ref< const solvers::VectorXDecisionVariable > &vars) const =0
	Non-virtual interface implementation for AddPointInSetConstraints(). More...

virtual std::vector< solvers::Binding< solvers::Constraint > >	DoAddPointInNonnegativeScalingConstraints (solvers::MathematicalProgram *prog, const Eigen::Ref< const solvers::VectorXDecisionVariable > &x, const symbolic::Variable &t) const =0
	Non-virtual interface implementation for AddPointInNonnegativeScalingConstraints(). More...

virtual std::vector< solvers::Binding< solvers::Constraint > >	DoAddPointInNonnegativeScalingConstraints (solvers::MathematicalProgram *prog, const Eigen::Ref< const Eigen::MatrixXd > &A, const Eigen::Ref< const Eigen::VectorXd > &b, const Eigen::Ref< const Eigen::VectorXd > &c, double d, const Eigen::Ref< const solvers::VectorXDecisionVariable > &x, const Eigen::Ref< const solvers::VectorXDecisionVariable > &t) const =0
	Non-virtual interface implementation for AddPointInNonnegativeScalingConstraints(). More...

virtual std::pair< std::unique_ptr< Shape >, math::RigidTransformd >	DoToShapeWithPose () const =0
	Non-virtual interface implementation for ToShapeWithPose(). More...

virtual double	DoCalcVolume () const
	Non-virtual interface implementation for CalcVolume(). More...

std::optional< symbolic::Variable >	HandleZeroAmbientDimensionConstraints (solvers::MathematicalProgram prog, const ConvexSet &set, std::vector< solvers::Binding< solvers::Constraint >> constraints) const
	Instances of subclasses such as CartesianProduct and MinkowskiSum can have constituent sets with zero ambient dimension, which much be handled in a special manner when calling methods such as DoAddPointInSetConstraints. More...

Implements CopyConstructible, CopyAssignable, MoveConstructible, MoveAssignable
	ConvexSet (const ConvexSet &)=default

ConvexSet &	operator= (const ConvexSet &)=default

	ConvexSet (ConvexSet &&)=default

ConvexSet &	operator= (ConvexSet &&)=default

Constructor & Destructor Documentation

◆ ~ConvexSet()

virtual ~ConvexSet ( )

virtual

◆ ConvexSet() [1/3]

ConvexSet ( const ConvexSet & )

protecteddefault

◆ ConvexSet() [2/3]

ConvexSet ( ConvexSet && )

protecteddefault

◆ ConvexSet() [3/3]

ConvexSet	(	int	ambient_dimension,
		bool	has_exact_volume
	)

explicitprotected

For use by derived classes to construct a ConvexSet.

Parameters

has_exact_volume Derived classes should pass true if they've implemented DoCalcVolume() to return a value (at least sometimes).

Member Function Documentation

◆ AddPointInNonnegativeScalingConstraints() [1/2]

std::vector<solvers::Binding<solvers::Constraint> > AddPointInNonnegativeScalingConstraints	(	solvers::MathematicalProgram *	prog,
		const Eigen::Ref< const solvers::VectorXDecisionVariable > &	x,
		const symbolic::Variable &	t
	)		const

Let S be this convex set.

When S is bounded, this method adds the convex constraints to imply

x ∈ t S,
t ≥ 0,

where x is a point in ℜⁿ (with n the ambient_dimension) and t is a scalar.

When S is unbounded, then the behavior is almost identical, except when t=0. In this case, the constraints imply t ≥ 0, x ∈ t S ⊕ rec(S), where rec(S) is the recession cone of S (the asymptotic directions in which S is not bounded) and ⊕ is the Minkowski sum. For t > 0, this is equivalent to x ∈ t S, but for t = 0, we have only x ∈ rec(S).

Exceptions

std::exception if ambient_dimension() == 0

◆ AddPointInNonnegativeScalingConstraints() [2/2]

std::vector<solvers::Binding<solvers::Constraint> > AddPointInNonnegativeScalingConstraints	(	solvers::MathematicalProgram *	prog,
		const Eigen::Ref< const Eigen::MatrixXd > &	A,
		const Eigen::Ref< const Eigen::VectorXd > &	b,
		const Eigen::Ref< const Eigen::VectorXd > &	c,
		double	d,
		const Eigen::Ref< const solvers::VectorXDecisionVariable > &	x,
		const Eigen::Ref< const solvers::VectorXDecisionVariable > &	t
	)		const

Let S be this convex set.

When S is bounded, this method adds the convex constraints to imply

A * x + b ∈ (c' * t + d) S,
c' * t + d ≥ 0,

where A is an n-by-m matrix (with n the ambient_dimension), b is a vector of size n, c is a vector of size p, x is a point in ℜᵐ, and t is a point in ℜᵖ.

When S is unbounded, then the behavior is almost identical, except when c' * t+d=0. In this case, the constraints imply

A * x + b ∈ (c' * t + d) S ⊕ rec(S),
c' * t + d ≥ 0,

where rec(S) is the recession cone of S (the asymptotic directions in which S is not bounded) and ⊕ is the Minkowski sum. For c' * t + d > 0, this is equivalent to A * x + b ∈ (c' * t + d) S, but for c' * t + d = 0, we have only A * x + b ∈ rec(S).

Exceptions

std::exception if ambient_dimension() == 0

◆ AddPointInSetConstraints()

std::pair<VectorX<symbolic::Variable>, std::vector<solvers::Binding<solvers::Constraint> > > AddPointInSetConstraints	(	solvers::MathematicalProgram *	prog,
		const Eigen::Ref< const solvers::VectorXDecisionVariable > &	vars
	)		const

Adds a constraint to an existing MathematicalProgram enforcing that the point defined by vars is inside the set.

Returns: (new_vars, new_constraints) Some of the derived class will add new decision variables to enforce this constraint, we return all the newly added decision variables as new_vars. The meaning of these new decision variables differs in each subclass. If no new variables are added, then we return an empty Eigen vector. Also we return all the newly added constraints to prog through this function.

Exceptions

std::exception if ambient_dimension() == 0

◆ ambient_dimension()

int ambient_dimension ( ) const

Returns the dimension of the vector space in which the elements of this set are evaluated.

Contrast this with the affine dimension: the dimension of the smallest affine subset of the ambient space that contains our set. For example, if we define a set using A*x = b, where A has linearly independent rows, then the ambient dimension is the dimension of x, but the affine dimension of the set is ambient_dimension() - rank(A).

◆ CalcVolume()

double CalcVolume ( ) const

Computes the exact volume for the convex set.

Note: Not every convex set can report an exact volume. In that case, use CalcVolumeViaSampling() instead.

Exceptions

std::exception	if `has_exact_volume()` returns `false`.
if	ambient_dimension() == 0.

◆ CalcVolumeViaSampling()

SampledVolume CalcVolumeViaSampling	(	RandomGenerator *	generator,
		const double	desired_rel_accuracy = `1e-2`,
		const int	max_num_samples = `1e4`
	)		const

Calculates an estimate of the volume of the convex set using sampling and performing Monte Carlo integration.

Note: this method is intended to be used for low to moderate dimensions (d<15). For larger dimensions, a telescopic product approach has yet to be implemented. See, e.g., https://proceedings.mlr.press/v151/chevallier22a/chevallier22a.pdf

Parameters

generator	a random number generator.
desired_rel_accuracy	the desired relative accuracy of the volume estimate in the sense that the estimated volume is likely to be within the interval defined by (1±2desired_rel_accuracy)true_volume with probability of at least* 0.95 according to the Law of Large Numbers. https://people.math.umass.edu/~lr7q/ps_files/teaching/math456/Chapter6.pdf The computation will terminate when the relative error is less than rel_accuracy or when the maximum number of samples is reached.
max_num_samples	the maximum number of samples to use.

Precondition: desired_rel_accuracy is in the range [0,1].

Returns: a pair the estimated volume of the set and an upper bound for the relative accuracy

Exceptions

if	ambient_dimension() == 0.
if	the minimum axis-aligned bounding box of the set cannot be computed.

◆ Clone()

std::unique_ptr<ConvexSet> Clone ( ) const

Creates a unique deep copy of this set.

◆ DoAddPointInNonnegativeScalingConstraints() [1/2]

virtual std::vector<solvers::Binding<solvers::Constraint> > DoAddPointInNonnegativeScalingConstraints	(	solvers::MathematicalProgram *	prog,
		const Eigen::Ref< const solvers::VectorXDecisionVariable > &	x,
		const symbolic::Variable &	t
	)		const

protectedpure virtual

Non-virtual interface implementation for AddPointInNonnegativeScalingConstraints().

Precondition: x.size() == ambient_dimension(); ambient_dimension() > 0

◆ DoAddPointInNonnegativeScalingConstraints() [2/2]

virtual std::vector<solvers::Binding<solvers::Constraint> > DoAddPointInNonnegativeScalingConstraints	(	solvers::MathematicalProgram *	prog,
		const Eigen::Ref< const Eigen::MatrixXd > &	A,
		const Eigen::Ref< const Eigen::VectorXd > &	b,
		const Eigen::Ref< const Eigen::VectorXd > &	c,
		double	d,
		const Eigen::Ref< const solvers::VectorXDecisionVariable > &	x,
		const Eigen::Ref< const solvers::VectorXDecisionVariable > &	t
	)		const

protectedpure virtual

Non-virtual interface implementation for AddPointInNonnegativeScalingConstraints().

Subclasses must override to add the constraints needed to keep the point A * x + b in the non-negative scaling of the set. Note that subclasses do not need to add the constraint c * t + d ≥ 0 as it is already added.

Precondition: ambient_dimension() > 0; A.rows() == ambient_dimension(); A.rows() == b.rows(); A.cols() == x.size(); c.rows() == t.size()

◆ DoAddPointInSetConstraints()

virtual std::pair<VectorX<symbolic::Variable>, std::vector<solvers::Binding<solvers::Constraint> > > DoAddPointInSetConstraints	(	solvers::MathematicalProgram *	prog,
		const Eigen::Ref< const solvers::VectorXDecisionVariable > &	vars
	)		const

protectedpure virtual

Non-virtual interface implementation for AddPointInSetConstraints().

Precondition: vars.size() == ambient_dimension(); ambient_dimension() > 0

◆ DoCalcVolume()

virtual double DoCalcVolume ( ) const

protectedvirtual

Non-virtual interface implementation for CalcVolume().

This will only be called if has_exact_volume() returns true and ambient_dimension() > 0

◆ DoClone()

virtual std::unique_ptr<ConvexSet> DoClone ( ) const

protectedpure virtual

Non-virtual interface implementation for Clone().

◆ DoIsBoundedShortcut()

virtual std::optional<bool> DoIsBoundedShortcut ( ) const

protectedvirtual

Non-virtual interface implementation for DoIsBoundedShortcut().

Trivially returns std::nullopt. This allows a derived class to implement its own boundedness checks, to potentially avoid the more expensive base class checks.

Precondition: ambient_dimension() >= 0

◆ DoIsEmpty()

virtual bool DoIsEmpty ( ) const

protectedvirtual

Non-virtual interface implementation for IsEmpty().

The default implementation solves a feasibility optimization problem, but derived classes can override with a custom (more efficient) implementation. Zero-dimensional sets are considered to be nonempty by default. Sets which can be zero-dimensional and empty must handle this behavior in their derived implementation of DoIsEmpty.

◆ DoMaybeGetFeasiblePoint()

virtual std::optional<Eigen::VectorXd> DoMaybeGetFeasiblePoint ( ) const

protectedvirtual

Non-virtual interface implementation for MaybeGetFeasiblePoint().

The default implementation solves a feasibility optimization problem, but derived classes can override with a custom (more efficient) implementation.

◆ DoMaybeGetPoint()

virtual std::optional<Eigen::VectorXd> DoMaybeGetPoint ( ) const

protectedvirtual

Non-virtual interface implementation for MaybeGetPoint().

The default implementation returns nullopt. Sets that can model a single point should override with a custom implementation.

Precondition: ambient_dimension() >= 0.

◆ DoPointInSet()

virtual bool DoPointInSet	(	const Eigen::Ref< const Eigen::VectorXd > &	x,
		double	tol
	)		const

protectedvirtual

Non-virtual interface implementation for PointInSet().

Precondition: x.size() == ambient_dimension(); ambient_dimension() >= 0

◆ DoPointInSetShortcut()

virtual std::optional<bool> DoPointInSetShortcut	(	const Eigen::Ref< const Eigen::VectorXd > &	x,
		double	tol
	)		const

protectedvirtual

A non-virtual interface implementation for PointInSet() that should be used when the PointInSet() can be computed more efficiently than solving a convex program.

Returns: Returns true if and only if x is known to be in the set. Returns false if and only if x is known to not be in the set. Returns std::nullopt if a shortcut implementation is not provided (i.e. the method has not elected to decide whether the point x is in the set).

For example, membership in a VPolytope cannot be verified without solving a linear program and so no shortcut implementation should be provided. On the other hand, membership in an HPolyhedron can be checked by checking the inequality Ax ≤ b and so a shortcut is possible.

◆ DoProjectionShortcut()

virtual std::vector<std::optional<double> > DoProjectionShortcut	(	const Eigen::Ref< const Eigen::MatrixXd > &	points,
		EigenPtr< Eigen::MatrixXd >	projected_points
	)		const

protectedvirtual

Non-virtual interface implementation for DoProjectionShortcut().

This allows a derived class to implement a method which computes the projection of some, but not necessarily all, of the points more efficiently than the generic implementation.

The default implementation checks whether each column of points is in the set using DoPointInSetShortcut. Points in the set are given a distance of 0 and are projected to themselves.

Parameters

[in]	points	are the points which we wish to project to the convex set.
	[in/out]	projected_points are the projection of `points` onto the convex set.

Returns: A vector distances which is the same size as points.cols().These are the distances from points to the convex set. If distances[i] has a value, then projected_points->col(i) is the projection of points.col(i) onto the set. If distances[i] is nullopt, then the projection of points.col(i) has not yet been computed, and so the value at projected_points->col(i) is meaningless.

Precondition: ambient_dimension() >= 0; distances.size() == points.cols()

◆ DoToShapeWithPose()

virtual std::pair<std::unique_ptr<Shape>, math::RigidTransformd> DoToShapeWithPose ( ) const

protectedpure virtual

Non-virtual interface implementation for ToShapeWithPose().

Precondition: ambient_dimension() == 3

◆ HandleZeroAmbientDimensionConstraints()

std::optional<symbolic::Variable> HandleZeroAmbientDimensionConstraints	(	solvers::MathematicalProgram *	prog,
		const ConvexSet &	set,
		std::vector< solvers::Binding< solvers::Constraint >> *	constraints
	)		const

protected

Instances of subclasses such as CartesianProduct and MinkowskiSum can have constituent sets with zero ambient dimension, which much be handled in a special manner when calling methods such as DoAddPointInSetConstraints.

If the set is empty, a trivially infeasible constraint must be added. We also warn the user when this happens, since they probably didn't intend it to occur. If the set is nonempty, then it's the unique zero-dimensional vector space {0}, and no additional variables or constraints are needed. If a new variable is created, return it, to optionally be stored (as in AddPointInSetConstraints), or not be stored (as in DoAddPointInNonnegativeScalingConstraints).

◆ has_exact_volume()

bool has_exact_volume ( ) const

Returns true if the exact volume can be computed for this convex set instance.

Note: This value reasons about to the generic case of the convex set class rather than the specific instance of the convex set. For example, the exact volume of a box is trivival to compute, but if the box is created as a HPolyhedron, then the exact volume cannot be computed.

◆ IntersectsWith()

bool IntersectsWith ( const ConvexSet & other ) const

Returns true iff the intersection between this and other is non-empty.

Exceptions

std::exception if the ambient dimension of other is not the same as that of this.

◆ IsBounded()

bool IsBounded ( ) const

Returns true iff the set is bounded, e.g., there exists an element-wise finite lower and upper bound for the set.

Note: for some derived classes, this check is trivial, but for others it can require solving an (typically small) optimization problem. Check the derived class documentation for any notes.

◆ IsEmpty()

bool IsEmpty ( ) const

Returns true iff the set is empty.

Note: for some derived classes, this check is trivial, but for others, it can require solving a (typically small) optimization problem. Check the derived class documentation for any notes. Zero-dimensional sets must be handled specially. There are two possible sets in a zero-dimensional space – the empty set, and the whole set (which is simply the "zero vector space", {0}.) For more details, see: https://en.wikipedia.org/wiki/Examples_of_vector_spaces#Trivial_or_zero_vector_space Zero-dimensional sets are considered to be nonempty by default. Sets which can be zero-dimensional and empty must handle this behavior in their derived implementation of DoIsEmpty. An example of such a subclass is VPolytope.

◆ MaybeGetFeasiblePoint()

std::optional<Eigen::VectorXd> MaybeGetFeasiblePoint ( ) const

Returns a feasible point within this convex set if it is nonempty, and nullopt otherwise.

◆ MaybeGetPoint()

std::optional<Eigen::VectorXd> MaybeGetPoint ( ) const

If this set trivially contains exactly one point, returns the value of that point.

Otherwise, returns nullopt. By "trivially", we mean that representation of the set structurally maps to a single point; if checking for point-ness would require solving an optimization program, returns nullopt. In other words, this is a relatively cheap function to call.

◆ operator=() [1/2]

ConvexSet& operator= ( ConvexSet && )

protecteddefault

◆ operator=() [2/2]

ConvexSet& operator= ( const ConvexSet & )

protecteddefault

◆ PointInSet()

bool PointInSet	(	const Eigen::Ref< const Eigen::VectorXd > &	x,
		double	tol = `0`
	)		const

Returns true iff the point x is contained in the set.

If the ambient dimension is zero, then if the set is nonempty, the point is trivially in the set, and if the set is empty, the point is trivially not in the set.

◆ Projection()

std::optional<std::pair<std::vector<double>, Eigen::MatrixXd> > Projection ( const Eigen::Ref< const Eigen::MatrixXd > & points ) const

Computes in the L₂ norm the distance and the nearest point in this convex set to every column of points.

If this set is empty, we return nullopt.

Precondition: points.rows() == ambient_dimension().

Exceptions

if	the internal convex optimization solver fails.

◆ Serialize()

void Serialize ( Archive * a )

protected

Implements non-virtual base class serialization.

◆ ToShapeWithPose()

std::pair<std::unique_ptr<Shape>, math::RigidTransformd> ToShapeWithPose ( ) const

Constructs a Shape and a pose of the set in the world frame for use in the SceneGraph geometry ecosystem.

Exceptions

std::exception if ambient_dimension() != 3 or if the functionality for a particular set has not been implemented yet.

The documentation for this class was generated from the following file:

drake/geometry/optimization/convex_set.h

Detailed Description

Public Member Functions

Protected Member Functions

Constructor & Destructor Documentation

◆ ~ConvexSet()

◆ ConvexSet() [1/3]

◆ ConvexSet() [2/3]

◆ ConvexSet() [3/3]

Member Function Documentation

◆ AddPointInNonnegativeScalingConstraints() [1/2]

◆ AddPointInNonnegativeScalingConstraints() [2/2]

◆ AddPointInSetConstraints()

◆ ambient_dimension()

◆ CalcVolume()

◆ CalcVolumeViaSampling()

◆ Clone()

◆ DoAddPointInNonnegativeScalingConstraints() [1/2]

◆ DoAddPointInNonnegativeScalingConstraints() [2/2]

◆ DoAddPointInSetConstraints()

◆ DoCalcVolume()

◆ DoClone()

◆ DoIsBoundedShortcut()

◆ DoIsEmpty()

◆ DoMaybeGetFeasiblePoint()

◆ DoMaybeGetPoint()

◆ DoPointInSet()

◆ DoPointInSetShortcut()

◆ DoProjectionShortcut()

◆ DoToShapeWithPose()

◆ HandleZeroAmbientDimensionConstraints()

◆ has_exact_volume()

◆ IntersectsWith()

◆ IsBounded()

◆ IsEmpty()

◆ MaybeGetFeasiblePoint()

◆ MaybeGetPoint()

◆ operator=() [1/2]

◆ operator=() [2/2]

◆ PointInSet()

◆ Projection()

◆ Serialize()

◆ ToShapeWithPose()