Some informal projective geometric ideas (with related algebra)
based on a focus in two and three dimensions

Think of your eye or a point-light source as a focus.
• A line through the focus appears as a point.
• A plane in space through the focus appears as a line.
• *Two points in space together with the focus determine a plane in space including the focus. This plane will appear as a line.
• *Any two planes in space through the focus determine a line in space through the focus. This line will appear as a point.
• Definitions:

• A line through the focus is called a projective point (a P-point) and
A plane through the focus is called a projective line (a P-line).

A projective plane is the geometric object made up of the collection of P-points and P-lines.

• As a consequence of the informal * statements above, in a projective plane we have the following:

Any two P-points determine a unique P-line and
any two P-lines determine a unique P-point.

• Some algebraic descriptions of lines and planes in 3 dimensions.

• A plane through the origin has an equation of the form Ax + By + Cz  = 0, where [A,B,C] is not [0,0,0].
The triple [uA,uB,uC] will determine the same plane as long as u is not 0.

• For example, [1,0,1] determines the plane with equation X + Z = 0. This plane is also determined by [3,0,3].

• A line through the origin has the equation of the form (X,Y,Z) = (a,b,c) t where (a,b,c) is not (0,0,0).
The triple (ua,ub,uc)will determine the same line as long as u is not 0.

• For example, (1,0,-1) determines the line with equation (X,Y,Z)=(1,0,-1)t. This point is also determined by (3,0,-3).
• A P-point has the equation (X,Y,Z) = (a,b,c) t where (a,b,c) is not (0,0,0).The triple (ua,ub,uc) will determine the same line as long as u is not 0.

We'll call <a,b,c> homogeneous coordinates of the P- point.

For example, <1,0,-1> are homogeneous coordinates for the P-point determined by the line with equation (X,Y,Z) = (1,0,-1) t. <3,0,-3> are homogeneous coordinates for the same P-point.

• A P-line has an equation of the form Ax + By + Cz  = 0, where [A,B,C] is not [0,0,0].

• The triple [uA,uB,uC] will determine the same plane as long as u is not 0.

We'll call [A,B,C] homogeneous coordinates of the P-line.

For example, [1,0,1] are homogeneous coordinates for the P-line determined by the plane with equation X + Z = 0.
[3,0,3] are homogeneous coordinates for the same P-line.

• A P-point lies on a P line or a P -line passes through the the P-point if and only if  Aa+Bb+Cc= 0 where [A,B,C] are homogeneous coordinates for the P-line and <a,b,c> are homogeneous coordinates for the P-point.

• For example, the P-point <1,0,-1> lies on the P-line [1,0,1].

• NOTE: All of the discussion works as long as the symbols A,B,C, a,b, and c represent elements of a field, that is, a set with two operations that work like the real or rational numbers in terms of addition and multiplication.