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Drake
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Drake
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A polytope described using the vertex representation.
The set is defined as the convex hull of the vertices. The vertices are not guaranteed to be in any particular order, nor to be minimal (some vertices could be strictly in the interior of the set).
Note: Unlike the half-space representation, this definition means the set is always bounded (hence the name polytope, instead of polyhedron).
A VPolytope is empty if and only if it is composed of zero vertices, i.e., if vertices_.cols() == 0. This includes the zero-dimensional case. If vertices_.rows() == 0 but vertices_.cols() > 0, we treat this as having one or more copies of 0 in the zero-dimensional vector space {0}. If vertices_.rows() and vertices_.cols() are zero, we treat this as no points in {0}, which is empty.
#include <drake/geometry/optimization/vpolytope.h>
Public Member Functions | |
| VPolytope () | |
| Constructs a set with no vertices in the zero-dimensional space, which is empty (by convention). More... | |
| VPolytope (const Eigen::Ref< const Eigen::MatrixXd > &vertices) | |
| Constructs the polytope from a d-by-n matrix, where d is the ambient dimension, and n is the number of vertices. More... | |
| VPolytope (const HPolyhedron &H, double tol=1e-9) | |
| Constructs the polytope from a bounded polyhedron (using Qhull). More... | |
| VPolytope (const QueryObject< double > &query_object, GeometryId geometry_id, std::optional< FrameId > reference_frame=std::nullopt) | |
| Constructs the polytope from a SceneGraph geometry. More... | |
| ~VPolytope () final | |
| VPolytope | GetMinimalRepresentation (double tol=1e-9) const |
| Creates a new VPolytope whose vertices are guaranteed to be minimal, i.e., if we remove any point from its vertices, then the convex hull of the remaining vertices is a strict subset of the polytope. More... | |
| const Eigen::MatrixXd & | vertices () const |
| Returns the vertices in a d-by-n matrix, where d is the ambient dimension, and n is the number of vertices. More... | |
| void | WriteObj (const std::filesystem::path &filename) const |
Uses qhull to compute the Delaunay triangulation and then writes the vertices and faces to filename in the Wavefront Obj format. More... | |
| bool | PointInSet (const Eigen::Ref< const Eigen::VectorXd > &x, double tol=0) const |
Returns true if the point is within tol of the set under the L∞-norm. More... | |
| double | CalcVolume () const |
| Computes the volume of this V-Polytope. More... | |
| bool | IsBounded (Parallelism parallelism=Parallelism::None()) const |
| Every VPolytope is bounded by construction. More... | |
Implements CopyConstructible, CopyAssignable, MoveConstructible, MoveAssignable | |
| VPolytope (const VPolytope &)=default | |
| VPolytope & | operator= (const VPolytope &)=default |
| VPolytope (VPolytope &&)=default | |
| VPolytope & | operator= (VPolytope &&)=default |
 Public Member Functions inherited from ConvexSet | |
| 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 (Parallelism parallelism=Parallelism::None()) 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... | |
Static Public Member Functions | |
| static VPolytope | MakeBox (const Eigen::Ref< const Eigen::VectorXd > &lb, const Eigen::Ref< const Eigen::VectorXd > &ub) |
| Constructs a polyhedron as an axis-aligned box from the lower and upper corners. More... | |
| static VPolytope | MakeUnitBox (int dim) |
Constructs the L∞-norm unit box in dim dimensions, {x | |x|∞ <= 1 }. More... | |
Additional Inherited Members | |
| Protected Member Functions inherited from ConvexSet | |
| 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::optional< bool > | DoIsBoundedShortcutParallel (Parallelism) const |
| Non-virtual interface implementation for DoIsBoundedShortcutParallel(). 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 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... | |
| 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... | |
| ConvexSet (const ConvexSet &)=default | |
| ConvexSet & | operator= (const ConvexSet &)=default |
| ConvexSet (ConvexSet &&)=default | |
| ConvexSet & | operator= (ConvexSet &&)=default |
| Static Protected Member Functions inherited from ConvexSet | |
| static std::unique_ptr< ConvexSet > | AffineHullShortcut (const ConvexSet &self, std::optional< double > tol) |
| When there is a more efficient strategy to compute the affine hull of this set, returns affine hull as an AffineSubspace. More... | |
| VPolytope | ( | ) |
Constructs a set with no vertices in the zero-dimensional space, which is empty (by convention).
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explicit |
Constructs the polytope from a d-by-n matrix, where d is the ambient dimension, and n is the number of vertices.
The vertices do not have to be ordered, nor minimal (they can contain points inside their convex hull).
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explicit |
Constructs the polytope from a bounded polyhedron (using Qhull).
If the HPolyhedron is not full-dimensional, we perform computations in a coordinate system of its affine hull. tol specifies the numerical tolerance used in the computation of the affine hull. See the documentation of AffineSubspace for more details. A loose tolerance is necessary for the built-in solvers, but a tighter tolerance can be used with commercial solvers (e.g. Gurobi and Mosek).
| std::runtime_error | if H is unbounded or if Qhull terminates with an error. |
| VPolytope | ( | const QueryObject< double > & | query_object, |
| GeometryId | geometry_id, | ||
| std::optional< FrameId > | reference_frame = std::nullopt |
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| ) |
Constructs the polytope from a SceneGraph geometry.
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final |
| double CalcVolume |
Computes the volume of this V-Polytope.
Creates a new VPolytope whose vertices are guaranteed to be minimal, i.e., if we remove any point from its vertices, then the convex hull of the remaining vertices is a strict subset of the polytope.
In the 2D case the vertices of the new VPolytope are ordered counter-clockwise from the negative X axis. For all other cases an order is not guaranteed. If the VPolytope is not full-dimensional, we perform computations in a coordinate system of its affine hull. tol specifies the numerical tolerance used in the computation of the affine hull.
| bool IsBounded |
Every VPolytope is bounded by construction.
| parallelism | Ignored – no parallelization is used. |
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static |
Constructs a polyhedron as an axis-aligned box from the lower and upper corners.
Constructs the L∞-norm unit box in dim dimensions, {x | |x|∞ <= 1 }.
This is an axis-aligned box, centered at the origin, with edge length 2.
| bool PointInSet |
Returns true if the point is within tol of the set under the L∞-norm.
Note: This requires the solution of a linear program; the achievable tolerance may be dependent on your specific solver and solver parameters.
| const Eigen::MatrixXd& vertices | ( | ) | const |
Returns the vertices in a d-by-n matrix, where d is the ambient dimension, and n is the number of vertices.
| void WriteObj | ( | const std::filesystem::path & | filename | ) | const |
Uses qhull to compute the Delaunay triangulation and then writes the vertices and faces to filename in the Wavefront Obj format.
Note that the extension .obj is not automatically added to the filename.