PR-3 The Harmonic Conjugate
There exists a fundamental relationship between perspective geometry and harmonic conjugation. Harmonic conjugates arise naturally in configurations involving collinear points and proportional segments, and they play a key role in constructing and interpreting perspective scales.
Definition
Let W, X, Y, Z be four collinear points. Then W and X are said to be harmonic conjugates with respect to Y and Z if:
This condition defines a harmonic division of the line segment, where the cross-ratio of the four points equals -1.
Geometric Contexts
Harmonic conjugates frequently appear in perspective constructions.
For example, in PR-Tf1, PR-Tf2, and PR-Tf3, three consecutive points P₁, P₂, P₃ on a perspective scale, together with the vanishing point V, form a harmonic configuration: P₁ and P₃ are harmonic conjugates with respect to P₂ and V.
However, harmonic conjugation is not limited to linear perspective. It manifests in various geometric situations. In the examples in the table below, H1 and H2 are harmonic conjugates with respect to H3 and H4.
| Description | (H1, H2) | (H3, H4) |
|---|---|---|
| 3 Consecutive points P₁, P₂, P₃ on a perspective scale including vanishing point V | P₁, P₃ | P₂, V |
| Centers of 2 circles M₁, M₂ and Insimili-/Exsimili-Centers Sᵢ, Sₑ | M₁, M₂ | Sᵢ, Sₑ |
| 2 Diameter points on circle S₁, S₂ and point P at diameter + its inverse Pᵢ | S₁, S₂ | P, Pᵢ |
| 2 Diameter points on circle S₁, S₂ and 2 points P₁, P₂ at orthogonal intersecting circle | S₁, S₂ | P₁, P₂ |
| Point P in or outside triangle ABC with respect to Cevian Triangle A′B′C′ and Anticevian Triangle A″B″C″ | P, A″ | A′, A |
| Vertices A, B and intersection points of Internal and External Angle Bisector from C with AB | A, B | Bi-c, Be-c |
| Intersection points AB, BC, AC, BD of a quadrilateral with 3rd diagonal | Sab, Sbc | Sac, Sbd |
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