Half Space Number System

Intro

Half spaces and their interaction with points can be viewed as a generalization of the $>$ $<$ $\leq$ $\geq$ $==$ $\neq$ relational operators. For example is a point above a plane can be expresses as $p > H$ .

Here I bring together some of the maths of half spaces. While 2D examples are given the benefit of half spaces is of course they generalize to any dimension so can be used anywhere.

Half Spaces

A Point divides a line in two.
A Line divides a plane in two.
A Plane divides a 3D space in two.

A half space has a cutter one dimension less than the space it is cutting. Further the half space has a direction perpendicular to the cutter.

The half space are a type of partition because it partitions the space in two. Another partition is a circle where all the points are either inside or not inside the circle.

Partitions have issues on the boundary - is a point inside or on the line, is the line included in the partition, can we even determine this, does it matter in an algorithm or is this ambiguity why the algorithm works.

Half Space Number System

A half space number system can be defined as binary operations between a point and a half space, one on each side respectively.

The operators are the familiar relational operators.

$>$ $<$ $\leq$ $\geq$ $==$ $\neq$

Let a half space be composed of a partition in the space of 1 less in dimension to the space and a normal vector in the space.
Let the partition be called a cutter.
The normal is perpendicular to the cutter.

In the special case where the normal of the half space is zero the half space is a partition. This is the == operator.

I call the partition a cutter as it cuts the space.

The operators are dependent on the half space. They indicate the nature of the half space. This makes writing the equation easy.

When combined with other notation the algebra becomes clear. For example to test if point q is below the line $p_{0}$ to $p_{1}$ using the Constructing Half Spaces from Ordered Points notation here is the maths.
$q < H (p_{0}, p_{1})$

I also express this loosely by saying that the half space $H (p_{0}, p_{1})$ does not see the point q.

Test if a Point is Above the Half Space

$p_{0}$ is a point on the half space.
$N$ is a normal vector pointing in the positive region of the half space.

$(x - p_{0}) \cdot N > 0$

Half Space Multiplication and Addition

Let multiplication of two half spaces be defined as having a point q belong to both partitions.
Let addition be that the point q is in either partition.

Shapes can be composed by multiplying and adding half spaces. Here are some example in 2D. First testing a point inside and or on a triangle boundary. Then a non convex shape.

See Convexity and Shape Representation with Algebra.

Constructing Half Spaces from Ordered Points

A very nice feature of half spaces is that they can be constructed from a list of ordered points. We have to define what the order means.

For example in 2D I have defined the half spaces as a left of the directed line from p₀ to p₁.

In 3D let the normal from the half space point outwards from the anti clockwise point ordering.

Another variable could include whether or not the boundary is considered to be part of the half space.

Point and Half Space Subtraction

By defining the half space and point relation $q > H$ and corresponding this with the equivalent dot product subtraction can be defined.

$(q - p_{0}) \cdot N > 0$
$q > H$
$q - H > 0$
$∴ q - H := (q - p_{0}) \cdot N$

Half Space Number System

Intro

Half Spaces

Half Space Number System

Test if a Point is Above the Half Space

Half Space Multiplication and Addition

Constructing Half Spaces from Ordered Points

Point and Half Space Subtraction

Intersection and Union