# ConicHullRegion

ConicHullRegion[{p1,,pm+1}]

represents the m-dimensional affine hull region passing through points pi.

ConicHullRegion[p,{v1,,vm}]

represents the m-dimensional affine hull region passing through the point p and parallel to vi.

ConicHullRegion[{p1,,pm+1},{w1,,wn}]

represents the m-dimensional affine hull plus the conic hull generated by the vectors wj.

ConicHullRegion[p,{v1,,vm},{w1,,wn}]

represents the m-dimensional affine hull plus the conic hull generated by the vectors wj.

# Details  • ConicHullRegion is also known as affine space, half-space, and affine hull in special cases.
• ConicHullRegion can be used as a geometric region and graphics primitive.
• • The cases ConicHullRegion[{p1,,pm+1}] and ConicHullRegion[p,{v1,,vm}] represent an affine hull, which is commonly known as an infinite line, infinite plane, or infinite space.
• The conic directions wj represent a pure conic hull that is added to each point in the affine hull, also known as a Minkowski sum of an affine hull and a conic hull.
• Parametric representations are given by:
•  ConicHullRegion[{p1,…,pm+1}] ConicHullRegion[p,{v1,…,vm}] ConicHullRegion[{p1,…,pm+1},{w1,…,wn}] ConicHullRegion[p,{v1,…,vm},{w1,…,wn}] • Low-dimensional versions of ConicHullRegion have special representations:
•  ConicHullRegion[{p1,p2}] InfiniteLine[{p1,p2}] ConicHullRegion[{p1,p2,p3}] InfinitePlane[{p1,p2,p3}] ConicHullRegion[{p1},{w1}] HalfLine[p1,w1] ConicHullRegion[{p1,p2},{w1}] HalfPlane[{p1,p2},w1]
• ConicHullRegion[p,{v1,,vm}] represents an m-dimensional region if the vi are linearly independent.
• ConicHullRegion can be used in Graphics and Graphics3D.
• In graphics, the points p, pi and vectors vi, wj can be Dynamic expressions.
• Graphics rendering is affected by directives such as FaceForm, EdgeForm, Opacity, and color.

# Examples

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## Basic Examples(3)

A ConicHullRegion in 2D:

And in 3D:

Different styles applied to a conic hull region:

Determine if points belong to a given conic hull region:

## Scope(25)

### Graphics(15)

#### Specification(7)

Define an infinite line passing through {1,1} and {3,2}:

Define the same line passing through {1,1} in the direction {2,1}:

Define the upper half-plane using a point, a vector, and a conic direction:

Or a 2D infinite space using a point and two conic directions:

Define an infinite line in 3D passing through the points and :

Define the same line using the point and a direction vector:

Define the plane passing through the points , , and :

Define the same plane using a point and direction vectors:

Define a 2D infinite cone with tip and directions and :

A 3D half-space can be defined with a point, two vectors, and a conic direction:

A 3D infinite space can be defined with a point and some conic directions:

#### Styling(7)

A 1D conic hull region with varying thickness:

Thickness in scaled size:

Thickness in printer's points:

1D conic hull regions can be rendered in dashed or dotted styles:

Color directives specify the edge color for 1D conic hull regions:

Color directives specify the face color for higher-dimensional regions:

FaceForm and EdgeForm can be used to specify the styles of the faces and edges:

In 3D, different properties can be specified for the front and back of faces using FaceForm:

#### Coordinates(1)

Scaled coordinates can be used in 2D:

And in 3D:

### Regions(10)

Embedding dimension is the dimension in which the ConicHullRegion lives:

Geometric dimension is the dimension of the region itself:

Point membership test:

Get conditions for point membership:

A ConicHullRegion has infinite measure:

And indeterminate centroid:

Distance from a point:

Distance to the nearest point:

Signed distance from a point:

Nearest point in the region:

Visualize it:

A conic hull region is unbounded:

But may not be unbounded in all dimensions:

Integrate over a ConicHullRegion:

Optimize over it:

Solve equations over a conic hull:

## Applications(4)

Define a region that occupies each quadrant:

Define a region that occupies each octant:

Construct a conic hull region from a center point and points on a circle:

Construct a conic hull region from a center point and points on a parametric curve:

## Properties & Relations(5)

InfiniteLine is a special case of ConicHullRegion:

HalfLine is a special case of ConicHullRegion:

InfinitePlane is a special case of ConicHullRegion:

HalfPlane is a special case of ConicHullRegion:

ImplicitRegion can represent any ConicHullRegion:

## Neat Examples(1)

A collection of random affine cones: