## CGAL::Polytope_distance_d<Traits>

### Definition

An object of the class Polytope_distance_d<Traits> represents the (squared) distance between two convex polytopes, given as the convex hulls of two finite point sets in d-dimensional Euclidean space d. For point sets P and Q we denote by pd(P,Q) the distance between the convex hulls of P and Q. Note that pd(P,Q) can be degenerate, i.e. pd(P,Q)= if P or Q is empty.

Two inclusion-minimal subsets SP of P and SQ of Q with pd(SP,SQ)=pd(P,Q) are called pair of support sets, the points in SP and SQ are the support points. A pair of support sets has size at most d+2 (by size we mean |SP|+|SQ|). The distance between the two polytopes is realized by a pair of points p and q lying in the convex hull of SP and SQ, repectively, i.e. sqrt(||p-q||)=pd(P,Q). In general, neither the support sets nor the realizing points are necessarily unique.

The underlying algorithm can cope with all kinds of input, e.g. P and Q may be in non-convex position or points may occur more than once. The algorithm computes a pair of support sets SP and SQ with realizing points p and q which remain fixed until the next set, insert, or clear operation.

#include <CGAL/Polytope_distance_d.h>

### Requirements

The template parameter Traits is a model for OptimisationDTraits.

We provide the models Optimisation_d_traits_2, Optimisation_d_traits_3, and Optimisation_d_traits_d using the two-, three-, and d-dimensional CGAL kernel, respectively.

### Types

Polytope_distance_d<Traits>::Point
| typedef to Traits::Point_d. Point type used to represent the input points.
Polytope_distance_d<Traits>::FT
| typedef to Traits::FT. Number type used to return the squared distance between the two polytopes.
Polytope_distance_d<Traits>::ET
| typedef to Traits::ET. Number type used to do the exact computations in the underlying solver for quadratic programs (cf. Implementation).
Polytope_distance_d<Traits>::Point_iterator
| non-mutable model of the STL concept RandomAccessIterator with value type Point. Used to access the points of the two polytopes.
Polytope_distance_d<Traits>::Support_point_iterator
| non-mutable model of the STL concept RandomAccessIterator with value type Point. Used to access the support points.
Polytope_distance_d<Traits>::Coordinate_iterator
| non-mutable model of the STL concept RandomAccessIterator with value type ET. Used to access the coordinates of the realizing points.

### Creation

Polytope_distance_d<Traits> poly_dist ( | Traits traits = Traits(), int verbose = 0, std::ostream& stream = std::cout);

| initializes poly_dist to pd(Ø,Ø).
template < class InputIterator1, class InputIterator2 >

Polytope_distance_d<Traits> poly_dist ( | InputIterator1 p_first, InputIterator1 p_last, InputIterator2 q_first, InputIterator2 q_last, Traits traits = Traits(), int verbose = 0, std::ostream& stream = std::cout);

| initializes poly_dist to pd(P,Q) with P and Q being the sets of points in the range [p_first,p_last) and [q_first,q_last), respectively. Requirement: The value type of InputIterator1 and InputIterator2 is Point. Precondition: All points have the same dimension.

| advanced |

If verbose is set to 1, 2, or 3 then some, more, or full verbose output of the underlying solver for quadratic programs is written to stream, resp.

| advanced |

### Access Functions

int | poly_dist.ambient_dimension ()
|  | returns the dimension of the points in P and Q. If poly_dist is pd(Ø,Ø), the ambient dimension is -1.

int | poly_dist.number_of_points ()
|  | returns the number of all points of poly_dist, i.e. |P|+|Q|.
int | poly_dist.number_of_points_p ()
|  | returns the number of points in P.
int | poly_dist.number_of_points_q ()
|  | returns the number of points in Q.

int | poly_dist.number_of_support_points ()
|  | returns the number of support points of poly_dist, i.e. |SP|+|SQ|.
int | poly_dist.number_of_support_points_p ()
|  | returns the number of support points in SP.
int | poly_dist.number_of_support_points_q ()
|  | returns the number of support points in SQ.

Point_iterator | poly_dist.points_p_begin ()
|  | returns an iterator referring to the first point in P.
Point_iterator | poly_dist.points_p_end ()
|  | returns the corresponding past-the-end iterator.

Point_iterator | poly_dist.points_q_begin ()
|  | returns an iterator referring to the first point in Q.
Point_iterator | poly_dist.points_q_end ()
|  | returns the corresponding past-the-end iterator.

Support_point_iterator
| poly_dist.support_points_p_begin ()
|  | returns an iterator referring to the first support point in SP.
Support_point_iterator
| poly_dist.support_points_p_end ()
|  | returns the corresponding past-the-end iterator.

Support_point_iterator
| poly_dist.support_points_q_begin ()
|  | returns an iterator referring to the first support point in SQ.
Support_point_iterator
| poly_dist.support_points_q_end ()
|  | returns the corresponding past-the-end iterator.
Point | poly_dist.realizing_point_p ()
|  | returns the realizing point of P. Requirement: An implicit conversion from ET to RT is available. Precondition: pd(P,Q) is finite.
Point | poly_dist.realizing_point_q ()
|  | returns the realizing point of Q. Requirement: An implicit conversion from ET to RT is available. Precondition: pd(P,Q) is finite.
FT | poly_dist.squared_distance ()
|  | returns the squared distance of poly_dist, i.e. (pd(P,Q))2. Requirement: An implicit conversion from ET to RT is available. Precondition: pd(P,Q) is finite.

Coordinate_iterator
| poly_dist.realizing_point_p_coordinates_begin ()
|  | returns an iterator referring to the first coordinate of the realizing point of P. Note: The coordinates have a rational representation, i.e. the first d elements of the iterator range are the numerators and the (d+1)-st element is the common denominator.
Coordinate_iterator
| poly_dist.realizing_point_p_coordinates_end ()
|  | returns the corresponding past-the-end iterator.

Coordinate_iterator
| poly_dist.realizing_point_q_coordinates_begin ()
|  | returns an iterator referring to the first coordinate of the realizing point of Q. Note: The coordinates have a rational representation, i.e. the first d elements of the iterator range are the numerators and the (d+1)-st element is the common denominator.
Coordinate_iterator
| poly_dist.realizing_point_q_coordinates_end ()
|  | returns the corresponding past-the-end iterator.
ET | poly_dist.squared_distance_numerator ()
|  | returns the numerator of the squared distance of poly_dist.
ET | poly_dist.squared_distance_denominator ()
|  | returns the denominator of the squared distance of poly_dist.

### Predicates

bool | poly_dist.is_finite ()
|  | returns true, if pd(P,Q) is finite, i.e. none of the two polytopes is empty.
bool | poly_dist.is_zero ()
|  | returns true, if pd(P,Q) is zero, i.e. the two polytopes intersect (this implies degeneracy).
bool | poly_dist.is_degenerate ()
|  | returns true, iff pd(P,Q) is degenerate, i.e. pd(P,Q) is not finite.

### Modifiers

void | poly_dist.clear () | resets poly_dist to pd(Ø,Ø).
template < class InputIterator1, class InputIterator2 >

void

poly_dist.set ( | InputIterator1 p_first, InputIterator1 p_last, InputIterator2 q_first, InputIterator2 q_last)

|  | sets poly_dist to pd(P,Q) with P and Q being the sets of points in the ranges [p_first,p_last) and [q_first,q_last), respectively. Requirement: The value type of InputIterator1 and InputIterator2 is Point. Precondition: All points have the same dimension.
template < class InputIterator >

void

poly_dist.set_p ( | InputIterator p_first, InputIterator p_last)

|  | sets poly_dist to pd(P,Q) with P being the set of points in the range [p_first,p_last) (Q remains unchanged). Requirement: The value type of InputIterator is Point. Precondition: All points in P have dimension poly_dist.ambient_dimension() if Q is not empty.
template < class InputIterator >

void

poly_dist.set_q ( | InputIterator q_first, InputIterator q_last)

|  | sets poly_dist to pd(P,Q) with Q being the set of points in the range [q_first,q_last) (P remains unchanged). Requirement: The value type of InputIterator is Point. Precondition: All points in Q have dimension poly_dist.ambient_dimension() if P is not empty.
void | poly_dist.insert_p ( Point p)
|  | inserts p into P. Precondition: The dimension of p equals poly_dist.ambient_dimension() if poly_dist is not pd(Ø,Ø).
void | poly_dist.insert_q ( Point q)
|  | inserts q into Q. Precondition: The dimension of q equals poly_dist.ambient_dimension() if poly_dist is not pd(Ø,Ø).
template < class InputIterator1, class InputIterator2 >

void

poly_dist.insert ( | InputIterator1 p_first, InputIterator1 p_last, InputIterator2 q_first, InputIterator2 q_last)

|  | inserts the points in the range [p_first,p_last) and [q_first,q_last) into P and Q, respectively, and recomputes the (squared) distance. Requirement: The value type of InputIterator1 and InputIterator2 is Point. Precondition: All points have the same dimension. If poly_dist is not pd(Ø,Ø), this dimension must be equal to poly_dist.ambient_dimension().
template < class InputIterator >

void

poly_dist.insert_p ( | InputIterator p_first, InputIterator p_last)

|  | inserts the points in the range [p_first,p_last) into P and recomputes the (squared) distance (Q remains unchanged). Requirement: The value type of InputIterator is Point. Precondition: All points have the same dimension. If poly_dist is not empty, this dimension must be equal to poly_dist.ambient_dimension().
template < class InputIterator >

void

poly_dist.insert_q ( | InputIterator q_first, InputIterator q_last)

|  | inserts the points in the range [q_first,q_last) into Q and recomputes the (squared) distance (P remains unchanged). Requirement: The value type of InputIterator is Point. Precondition: All points have the same dimension. If poly_dist is not empty, this dimension must be equal to poly_dist.ambient_dimension().

### Validity Check

An object poly_dist is valid, iff ...

- poly_dist contains all points of its defining set P,
- poly_dist is the smallest sphere containing its support set S, and
- S is minimal, i.e. no support point is redundant.

bool

poly_dist.is_valid ( | bool verbose = false, int level = 0)

|  | returns true, iff poly_dist is valid. If verbose is true, some messages concerning the performed checks are written to standard error stream. The second parameter level is not used, we provide it only for consistency with interfaces of other classes.

### Miscellaneous

const Traits& | poly_dist.traits ()
|  | returns a const reference to the traits class object.

### I/O

std::ostream& | std::ostream& os << poly_dist
|  | writes poly_dist to output stream os. Requirement: The output operator is defined for Point_d.
std::istream& | std::istream& is >> poly_dist&
|  | reads poly_dist from input stream is. Requirement: The input operator is defined for Point_d.

### See Also

CGAL::Optimisation_d_traits_2<K,ET,NT>
CGAL::Optimisation_d_traits_3<K,ET,NT>
CGAL::Optimisation_d_traits_d<K,ET,NT>
OptimisationDTraits

### Implementation

The problem of finding the distance between two convex polytopes given as the convex hulls of two finite point sets can be formulated as an optimization problem with linear constraints and a convex quadratic objective function. The solution is obtained using our exact solver for quadratic programs [GS00].

The creation time is almost always linear in the number of points. Access functions and predicates take constant time, inserting a point might take up to linear time. The clear operation and the check for validity each take linear time.