represents the affine space passing through the points pi.
represents the affine space passing through p in the directions vi.
- AffineSpace is also known as a point, line, plane, -flat, -plane, etc.
- AffineSpace can be used as a geometric region and a graphics primitive.
- AffineSpace represents the region or . The dimension is k if the pi are affinely independent or the vi are linearly independent.
- AffineSpace can be used in Graphics and Graphics3D.
- AffineSpace will be clipped by PlotRange when rendering.
- Graphics rendering is affected by directives such as Opacity and color as well as:
PointSize 0-dimensional () Thickness,Dashing 1-dimensional () FaceForm 2-dimensional ()
- For a two-dimensional AffineSpace, FaceForm[front,back] can be used to specify different styles for the front and back, where the front is defined to be in the direction of the normal Cross[v1,v2] or Cross[p2-p1,p3-p1], depending on which input form is used.
Examplesopen allclose all
Basic Examples (3)
An AffineSpace in 2D:
Different styles applied to an affine space region:
Specify coordinates by fractions of the plot range:
Specify scaled offsets from the ordinary coordinates:
Points and vectors can be Dynamic:
Embedding dimension is the dimension of the coordinates:
Geometric dimension is the dimension of the region itself:
Get the conditions for membership:
An affine space has infinite measure and undefined centroid:
Integrate over an affine space:
Optimize over an affine space:
Coordinate Systems (4)
Visualizing Transformations (3)
Visualize the axis of rotation for RotationTransform:
Visualize rotated coordinate axes in 3D:
Define a ReflectionTransform using a point on the plane and its normal vector:
Illustrating Plots (3)
Partition space in a BubbleChart:
Finding Intersections (10)
Find the intersection of two lines:
Find the intersections of a line and a circle:
Find all pairwise intersections between five random lines:
Use BooleanCountingFunction to express that exactly two conditions are true:
Find the intersection of a line and a plane:
Find the intersections of a line and a sphere:
Find the intersections of a line and the boundary of a tetrahedron:
Find the altitude of a triangle:
Find the plane in which a triangle is embedded:
AffineSpace can use the same parametrization as Triangle:
Find the plane in which a polygon is embedded:
To find the plane, take the first three points (or any three points not on a line):
Find the intersection points of a sphere, a plane, and a surface defined by :
Arrangements of Lines, Planes and Spaces (4)
Parallel lines have parallel direction vectors:
Parallel vectors have angle or :
Parallel planes in 3D have parallel normal vectors:
Perpendicular lines have orthogonal tangent vectors and orthogonal normal vectors:
Tangent vectors are orthogonal:
Perpendicular planes have orthogonal normal vectors:
The normal vectors are orthogonal:
AffineSpace[p,vv1] is parallel to AffineSpace[q,vv2] if either all vectors belong to the linear space generated by or all vectors belong to the linear space generated by :
To test whether two affine spaces are parallel, check that the rank of the union of and is equal to the maximum of ranks of and :
Test whether a plane and a 3D affine subspace of the 4D space are parallel:
Properties & Relations (6)
AffineSpace is a special case of ConicHullRegion:
InfiniteLine is a special case of AffineSpace:
InfinitePlane is a special case of AffineSpace:
Hyperplane is a special case of AffineSpace:
ParametricRegion can represent any AffineSpace in :
ImplicitRegion can represent any AffineSpace in :
Wolfram Research (2015), AffineSpace, Wolfram Language function, https://reference.wolfram.com/language/ref/AffineSpace.html.
Wolfram Language. 2015. "AffineSpace." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/AffineSpace.html.
Wolfram Language. (2015). AffineSpace. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/AffineSpace.html