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:

A hyperplane has infinite measure and undefined centroid:

Distance from a point:

Signed distance from a point:

Nearest point in the region:

Nearest points:

A hyperplane is unbounded:

Find the region range:

In the axis-aligned case, it is bounded in some directions:

Integrate over a hyperplane:

Optimize over a hyperplane:

Solve equations over a hyperplane:

In a parallel arrangement of hyperplanes, all hyperplanes have the same normal n:

Orthogonal arrangements of hyperplanes:

Grids of hyperplanes:

Random arrangements of hyperplanes:

A pencil of hyperplanes is all hyperplanes through a point:

A sheaf of hyperplanes is all hyperplanes through a line:

A bundle of hyperplanes, where all pass through a common point:

A tangent plane to an implicitly defined curve in 2D is given by its normal at a point on the curve. Start by finding points on the curve:

Find tangent lines at each of the points:

Visualize the solution:

A tangent plane to an implicitly defined surface in 3D is also given by its normal and a point on the surface. Start by finding points on the surface:

Find tangent planes at each of the points:

Visualize the solution:

A tangent line for a parametric curve can be defined by its normal for some value of the parameter . Start by picking parameter values:

Find tangent lines for each parameter value:

Visualize the solution:

A tangent plane for a parametric surface can be defined by its normal for some value of the parameters and . Start by picking parameter values:

Find tangent planes at each of the points:

Visualize the solution: