Unit III: Chapter I, Part B — Loci of Lines & Tangential Envelopes — Notes

Historical, Mathematical & Cross-Repository Commentary (pp. 40–45)

Historical, Mathematical & GEOMETOR Commentary

1. Plücker–Poncelet Line Duality Rendered Elementary

In nineteenth-century projective geometry (Poncelet, Gergonne, Plücker), a curve may be viewed either punctually as a locus of points or tangentially as an envelope of straight lines. Petersen’s masterstroke in Chapter I, Part B (pp. 40–45) is to translate tangential envelopes into strictly elementary compass-and-straightedge terms:

  • A point \(O\) in Loci m, o, p acts as a degenerate circle of radius zero (the envelope of all lines through \(O\)).

  • Parallel lines of fixed slope in Locus n (difference) act as a pencil through an infinitely distant point.

  • Concentric circles in Locus l and midpoint circles in Locus n (sum) act as true metric circular envelopes.

  • Consequently, every problem determined by two tangential line-loci collapses uniformly into drawing common tangents to two circles of radii \(r_1, r_2 \ge 0\)!

2. Problem 238 (PPC Apollonius Problem) via Locus p

Compare Petersen’s Problem 238 (p. 44: circle through two points \(A, B\) tangent to circle \(\Gamma\)) with lengthy coordinate derivations: drawing any auxiliary circle through \(A, B\) cutting \(\Gamma\) in \(C, D\) invokes Locus p (radical axis theorem) to pinpoint the radical centre \(O = AB \cap CD\) in three steps, reducing circle tangency to drawing tangents from \(O\) to \(\Gamma\) (Euclid III.17 / III.36)!

3. Exact geometor.model Realization: equal_chords_line_locus

In geometor.petersen.constructions.equal_chords_line_locus() (Problem 220, p. 42), we set outer circle \((O, 17)\) with desired chord length \(L = 16\) (\(c = 8\)) and external point \(P = (25, 0)\):

  • Concentric envelope radius \(r = \sqrt{17^2 - 8^2} = 15\) (Pythagorean \(8\text{–}15\text{–}17\) triple);

  • Euclid III.17 Thales semicircle on \(OP\) yields rational contact points \(T_1 = (9, -12)\) and \(T_2 = (9, 12)\) (\(9\text{–}12\text{–}15\text{–}20\text{–}25\) right-triangle cascade);

  • Tangent \([P, T_1]\) cuts \((O, 17)\) in \(X_1 = (\tfrac{13}{5}, -\tfrac{84}{5})\) and \(Y_1 = (\tfrac{77}{5}, -\tfrac{36}{5})\), with exact chord quadrance \((\tfrac{64}{5})^2 + (\tfrac{48}{5})^2 = \tfrac{6400}{25} = 256 = 16^2\).