QA-PF-1: QA-Perspective Fields
A Perspective Field is a field of linearly related points.
All points in a Perspective Field can be identified from 3+1 points P1, P2, P3 and P4, no three of which are collinear.
This identification is by a weighted sum of 3 of this four points, where the 4th points is used as reference (calibrating/conditioning/tuning/standardizing) point so that the weights (compliance factors) are tuned in for the situation.
P4 = cf1.P1 + cf2.P2 + cf3.P3,
and cf1, cf2, cf3 are compliance factors, dependent on the coordinates of P1, P2, P3, P4:
- cf1 = Det[P4,P2,P3] / Det[P1,P2,P3]
- cf2 = Det[P1,P4,P3] / Det[P1,P2,P3]
- cf3 = Det[P1,P2,P4] / Det[P1,P2,P3]
See [26b, page 40].
Perspective Fields in fact reveal the linear relations between points.
They not only occur in Triangle Geometry (see PF-2) but also in in Quadri Geometry.
The coordinates of some point Px occurring in a Perspective Field algebraically can be written as:
Px = n1.cf1.P1 + n2.cf2.P2 + n3.cf3.P3,
where P1, P2, P3 will be random non collinear points in the Field and n1, n2, n3 will be real numbers.
(n1 : n2 : n3) are called the Perspective Coordinates of Px in the Perspective Field.
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