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Theorems on Quadriplanar Coordinates

Published online by Cambridge University Press:  03 November 2016

R. T Robinson
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Extract

Given a tetrahedron of reference ABCD, let λ, μ, ν, π be the cosines of the angles made by a line with the perpendiculars to the faces BCD, ACD, ABD, ABC respectively, these perpendiculars being drawn all inwards or all outwards; where

Aλ1 + Bμ1 + C½1 + Dπ1 = 0

Aλ2 + Bμ2 + C½2 + Dπ2 = 0

A, B, C, D being the areas of the faces BCD, .. If P1 (α1, ²1, γ1, δ1) and P2 (α2, β2, γ2, δ2) are any two points, the lines PP1 and PP2 can be written

(α - α1)/λ1 = (β - β1)/μ1 = (γ - γ1)/ν1 = (δ - δ1)/π1 = θ1,

(α - α2)/λ2 = (β - β2)/μ2 = (γ - γ2)/ν2 = (δ - δ2)/π2 = θ2,

to find the cosine of the angle between PP1 and PP2.

Type
Research Article
Information
The Mathematical Gazette , Volume 27 , Issue 273 , February 1943 , pp. 13 - 19
Copyright
Copyright © Mathematical Association 1943

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