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:
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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:
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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
open allclose allBasic Examples (3)
A ConicHullRegion in 2D:
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 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:
Regions (10)
Embedding dimension is the dimension in which the ConicHullRegion lives:
Geometric dimension is the dimension of the region itself:
Get conditions for point membership:
A ConicHullRegion has infinite measure:
Distance to the nearest point:
A conic hull region is unbounded:
But may not be unbounded in all dimensions:
Integrate over a ConicHullRegion:
Applications (4)
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:
Text
Wolfram Research (2014), ConicHullRegion, Wolfram Language function, https://reference.wolfram.com/language/ref/ConicHullRegion.html.
CMS
Wolfram Language. 2014. "ConicHullRegion." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ConicHullRegion.html.
APA
Wolfram Language. (2014). ConicHullRegion. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ConicHullRegion.html