interiorPoint

Purpose

Compute a point in the relative interior of the Polyhedron.

Syntax

sol = P.interiorPoint
sol = P.interiorPoint(facetIndex)

Description

Compute a point in the relative interior of the polyhedron. If ../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/interiorpoint1.png is specified, then a point in the relative interior of ../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/interiorpoint2.png is returned.

Input Arguments

P

Polyhedron in any format

Class: Polyhedron

facetIndex

Index of an inequality of ../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/interiorpoint3.png (row of P.H).

Class: integer

Output Arguments

sol

Class: struct

sol.x

The interior point

Class: double vector

sol.isStrict

The output is true if ../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/interiorpoint4.png is in the strict relative interior, false otherwise.

Class: logical

sol.r

Radius of the largest ball centered at ../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/interiorpoint5.png that is still within ../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/interiorpoint6.png

../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/interiorpoint9.png

Note : ../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/interiorpoint7.png is empty if ../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/interiorpoint8.png is empty or only has a V-rep.

Class: double

Example(s)

Example 1

Compute interior point in unbounded polyhedra
P = Polyhedron('V', randn(20,3), 'R', -[1 0 0]);

sol = P.interiorPoint
sol = 

           x: [3x1 double]
    isStrict: 1
           r: []

Example 2

Compute a point in the relative interior of the fourth facet
P = Polyhedron('H',[sin([0:0.5:2*pi])' cos([0:0.5:2*pi])' ones(13,1)]);
 Polyhedron must be in its minimal representation to compute facets. Perform redundancy elimination.
 P.minHRep(); 
 Compute the center of the fourth facet 
sol = P.interiorPoint(4)
sol = 

           x: [2x1 double]
    isStrict: 1
           r: 0.122417438109627

plot(P); hold on;
		pplot(sol.x, 'ro', 'markerfacecolor', 'r', 'markersize', 10); 

../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/interiorpoint_img_1.png

Example 3

Compute a point in the relative interior of a lower-dimensional polyhedron.
P = Polyhedron('H',[randn(20,3) ones(20,1)], 'He', [0 0 1 0]);

sol = P.interiorPoint
sol = 

           x: [3x1 double]
    isStrict: 0
           r: 0.39490124253373

plot(P); hold on;
		pplot(sol.x, 'ro', 'markerfacecolor', 'r', 'markersize', 10); 

../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/interiorpoint_img_2.png

See Also

chebycenter, facetinteriorpoints


© 2010-2013 Colin Neil Jones: EPF Lausanne, colin.jones@epfl.ch