AffineSpace

AffineSpace[{p₁,…,p_k+1}]

represents the affine space passing through the points p_i.

AffineSpace[p,{v₁,…,v_k}]

represents the affine space passing through p in the directions v_i.

Details

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 p_i are affinely independent or the v_i 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.

Examples

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Basic Examples (3)

An AffineSpace in 2D:

And in 3D:

Different styles applied to an affine space region:

Determine if points belong to a given affine space region:

Scope (17)

Graphics (7)

Specification (2)

Define an affine space in 3D using points:

Define the same affine space using a single point and two tangent vectors:

An affine space in 3D defined by a single point and one tangent vector:

Affine spaces varying in direction:

Styling (2)

Color directives specify the color of the affine space:

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

Coordinates (3)

Specify coordinates by fractions of the plot range:

Specify scaled offsets from the ordinary coordinates:

Points and vectors can be Dynamic:

Regions (10)

Embedding dimension is the dimension of the coordinates:

Geometric dimension is the dimension of the region itself:

Point membership test:

Get the conditions for membership:

An affine space has infinite measure and undefined centroid:

Distance from a point:

Signed distance from a point:

Nearest point in the region:

Nearest points:

An affine space is unbounded:

Find the region range:

Integrate over an affine space:

Optimize over an affine space:

Solve equations over an affine space:

Applications (24)

Coordinate Systems (4)

Visualize coordinate axes:

In 3D:

Visualize coordinate planes:

Add drop lines to a plot:

Add drop planes to a plot:

Visualizing Transformations (3)

Visualize the axis of rotation for RotationTransform:

Visualize rotated coordinate axes in 3D:

Visualize a reflection plane:

Define a reflection plane:

Define a ReflectionTransform using a point on the plane and its normal vector:

Visualize the reflection of a unit cube about the plane:

Illustrating Plots (3)

Illustrate asymptotes:

Indicate Mean on a Histogram:

Partition space in a BubbleChart:

Combine the graphics:

Finding Intersections (10)

Find the intersection of two lines:

Plot it:

Find the intersections of a line and a circle:

Plot it:

Find all pairwise intersections between five random lines:

Use BooleanCountingFunction to express that exactly two conditions are true:

Plot it:

Find the intersection of a line and a plane:

Plot it:

Find the intersections of a line and a sphere:

Plot it:

Find the intersections of a line and the boundary of a tetrahedron:

Plot it:

Find the altitude of a triangle:

Visualize altitude in red:

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 :

Visualize intersection points:

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:

The normals are parallel:

Perpendicular lines have orthogonal tangent vectors and orthogonal normal vectors:

Tangent vectors are orthogonal:

And so are their normals:

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 :