 A
Geometric structure

A graphical geometry.

Visualizing this geometry. (GSP)
 A
Geometric structure

A triangular geometry.

Visualizing this geometry. (GSP)

Some projective geometric ideas based on a focus.

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.

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) deteremines the line with
equation (X,Y,Z)=(1,0,1)t.

A Pline 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 Pline.
For example, [1,0,1] are homogeneous coordinates
for the Pline determined by the plane with equation X + Z = 0.

A Ppoint 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 Ppoint determined by the line with equation (X,Y,Z) = (1,0,1)
t.

A Ppoint lies on a P line or a P line passes through
the the Ppoint if and only if Aa+Bb+Cc= 0 where [A,B,C] are homogeneous
coordinates for the Pline and <a,b,c> are homogeneous coordinates for
the Ppoint.
For example, the Ppoint <1,0,1> lies on the
Pline [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.

A field with two elements: F_{2}
= {0,1}.
+ 
0 
1 


x 
0 
1 
0 
0 
1 


0 
0 
0 
1 
1 
0 


1 
0 
1 
A projective plane using F_{2} has exactly
7 points:
<0,0,1>, <0,1,0>, <0,1,1>,<1,0,0>,<1,0,1>,<1,1,0>,<1,1,1>.
A projective plane using F_{2} has exactly
7 lines:
[0,0,1], [0,1,0], [0,1,1], [1,0,0], [1,0,1], [1,1,0], [1,1,1].
This projective plane satisfies the geometric structure
properties.
Lines\Points

<0,0,1>

<0,1,0>

<0,1,1>

<1,0,0>

<1,0,1>

<1,1,0>

<1,1,1>

[0,0,1]


X


X


X


[0,1,0]

X



X

X



[0,1,1]



X

X



X

[1,0,0]

X

X

X





[1,0,1]


X



X


X

[1,1,0]

X





X

X

[1,1,1]



X


X

X


We can visualize the projective plane using F_{2}
using 7 points of the unit cube in ordinary 3 dimensional coordinate geometry.
(GSP)
THE END.