Unit IV: Chapter II, Parts A & B — Parallel Translation & Replacing — Notes

Historical, Mathematical & Cross-Repository Commentary (pp. 46–60)

Historical, Mathematical & GEOMETOR Commentary

1. Group-Theoretic View of Petersen’s Chapter II

Written six years before Felix Klein’s Erlanger Programm (1872), Petersen’s Chapters II and III organize geometric constructions according to the transformation hierarchy of Euclidean plane geometry:

  1. Translation subgroup \(\mathbb{R}^2\) (Chapter II.A: Parallel Translation, preserving direction and length);

  2. Orientation-reversing isometries & similarity flips (Chapter II.B: Replacing & Axial Reflection);

  3. Orientation-preserving spiral similarity group \(\mathrm{Sim}^+(\mathbb{E}^2) \cong \mathbb{C}^\times \ltimes \mathbb{C}\) (Chapter III: Theory of Revolution).

In each case, an unknown configuration \(\mathcal{F}\) whose data are separated by unknown vectors is mapped to \(\mathcal{F}' = T(\mathcal{F})\) so the invariants of \(T\) weld the given elements into a rigid triangle.

2. Circle Shrinking (Problems 275–278) & Apollonius Reduction

Petersen’s subsection “To diminish the radius of a circle until it becomes naught” (pp. 52–53)—famous since Viète’s Apollonius Gallus (1600)—dilates tangent lines parallel to themselves by \(\pm r\) and concentric-shifts circle radii by \(\pm r\). Combined with Problem 238 (PPC) or homothety/inversion, it reduces the three-circle Problem of Apollonius to the simpler point-circle-circle cases (see Appendix II, p. 94).

3. Exact geometor.model Realization: trapezoid_from_four_sides

In geometor.petersen.constructions.trapezoid_from_four_sides() (Problem 259, p. 49), given bases \(a = 19, b = 5\) and non-parallel legs \(c = 13, d = 15\):

  • Shifting leg \(BC\) (13) by vector \(\overrightarrow{CD} = (-5, 0)\) forms the classic Heronian \((13, 14, 15)\) auxiliary triangle \(\triangle AED\) with \(A = (0, 0)\), \(E = (14, 0)\), and \(D = (9, 12)\) (altitude 12!).

  • Completing parallelogram \(EBCD\) yields \(C = (14, 12)\) and \(B = (19, 0)\) entirely in \(\mathbb{Z}^2\), with line_spread([A, B], [D, C]) == 0 and line_spread([E, D], [B, C]) == 0.