2007-10-30

Plane Affine Algebra (II)

See also Part I and Part III.

As noted before, the algebra permits us to calculate proofs without resorting to coordinate manipulations. As a demonstration, I'll present a very simple calculation, and then a theorem that I didn't know (and I would be very hard-pressed to come up with a classical proof).

Given points P, Q, R, the altitude of the triangle PQR through the vertex Q is the segment QH perpendicular to the base PR. Denoting hgt.P.Q.R = |QH| (we write |•−•| = |‹•−•›| per abusum linguae), it is:

(17)  hgt.P.Q.R = (‹RP⋅‹QP›)/|RP|

Proof:

   hgt.P.Q.R
= { definition }
   |QH|
= { definition of the sine line }
   |QP|·sin.(θ.‹RP›.‹QP›)
= { (16) with u, v := ‹RP›, ‹QP› }
   (‹RP⋅‹QP›)/|RP|

The line through point P with direction vector v is the set of points Pv that, in the words of Euclid, "lies equally on the points of itself":

(18.0)  QPv ≡ |v|⋅‹QP› = |QP|⋅v

The following are theorems:

(18.1)  PPv
(18.2)  ⟨∀ k : k ≠ 0 : P⇒(kv) = Pv(18.2)  Pv = ⟨kR | Pkv

The parametric line through points P and Q is defined as the real function:

(18)  ray.P.Q.k = Pk·‹QP

From (18.2), it is immediate that ray.P.Q maps R to the line P⇒‹QP›.

As a special case, the midpoint of the segment PQ is:

(19)  ray.P.Q.½ = P → ½·‹QP

Given points P, Q, R and S, the intersection of the lines containing the segments PQ and RS is:

(20)  meet.P.Q.R.S = ray.P.Q.((‹SR⋅‹RP›)/(‹SR⋅‹QP›))

Proof: At the intersection point meet.P.Q.R.S, for some k, lR:

   ray.P.Q.k = ray.R.S.l
= { (18) }
   Pk·‹QP› = Rl·‹SR›
= { (6) }
   ‹(Pk·‹QP›) − (Rl·‹SR›)› = 0
= { (9) }
   ‹PR› + k·‹QP› − l·‹SR› = 0
= { heading towards eliminating l; inner product }
   ‹SR⋅(‹PR› + k·‹QP› − l·‹SR›) = 0
= { linearity of inner product }
   ‹SR⋅‹PR› + k·‹SR⋅‹QP› − l·‹SR⋅‹SR› = 0
= { (14) }
   ‹SR⋅‹PR› + k·‹SR⋅‹QP› = 0
= { (5); algebra }
   k = (‹SR⋅‹RP›)/(‹SR⋅‹QP›)

In Part III, the theorem.

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