Grassmann algebra is an algebra for Geometry

## Constructing conics in space

This graphic shows the conic formed from two points, a line, and two planes according to an
equation in which P is the variable point describing the conic. Note that since a conic is of second
degree, the variable point P occurs twice in the product. The product itself involves both the
exterior product operation for building entities, and the regressive product operation for
intersecting them.
© John Browne 2018.

## Complements of points and lines in the plane

In the plane, the complement of a point is a line, and the complement of a line is a point. If a line is
defined by two points, then its complement is the intersection of the complements of the points. The
graphic below shows these relationships, with x, y and z the position vectors of three points.
Inverses of points and vectors are indicated with *-superscripts.
In higher dimensional spaces, the same relationships hold but their interpretations are different. For
example in a bound 3-space the complement of a point is a plane and the complement of a plane is a
point. The complements of two points P and Q yield two planes which intersect in a line R whose
complement passes through the points P and Q.

## The triangle components

Given a simple unit multivector, any vector x may always be decomposed into the sum of two
components: one orthogonal to the multivector, and one parallel to it.

## Desargues’ Theorem

Desargues' Theorem states that
If two triangles have corresponding vertices joined by concurrent lines, then the intersections of
corresponding sides are collinear. That is, if PU, QV and RW all pass through one point O, then the
intersections X = PQ.UV, Y = PR.UW and Z = QR.VW al lie on one line.
As Grassmannian equations, these conditions translate to

## Pappus’ Theorem

Pappus' Theorem states that
Given two sets of collinear points P, Q, R and U, V, W, then the intersection points X, Y, Z of the
lines PV and QU, PW and RU, QW and RV, are collinear.
As Grassmannian equations, these conditions translate to

## Regions defined by two lines in the plane

To define the four regions of a plane defined by two non-parallel lines we only need to know the
location of one point, Q say. All products of a point and a line in the plane are congruent, so
quotients of them can be defined as the quotients of their scalar multiples. These quotients may be
positive or negative. The four regions of the plane correspond to the four combinations of the signs
+ and - with the region containing Q corresponding to (+, +).
To see if a point P lies in the same quadrant as a point Q evaluate the conjunction of their
inequalities.

## Creating a 5-star pyramid

We are not constricted to the plane when it comes to defining regions. In an n-space, a hyperplane
divides its space into two regions. If two points lie on the same side of a hyperplane H, then the
ratio R of the exterior products of the two points with H is positive. If the points lie on opposite
sides of H then R is negative. If the point in the numerator lies on the hyperplane then R is zero.
Regions of space may be defined by logical functions of the predicates R > 0, R < 0, R = 0, R >= 0,
and R <= 0.
The graphic below shows a pyramid with a 5-star base defined by 5 points, and 5 hyperplanes (in
this case planes) through two points and the vertex.

## The geometric equation as a prescription to construct

A geometric equation, either as a series of steps, or in its final form can be used as a prescription to
construct a curve or surface. In the case below, the equation can be viewed as a prescription to
construct a conic section in the plane. An exterior product of points means "draw a line through the
points". A regressive product of lines means "find their point of intersection".
Construct an ellipse from the equation below by working from left to right.