Unit III: Chapter I, Part B — Loci of Lines & Tangential Envelopes — Summary¶
Structural & Heuristic Overview — Kapitel I.B — Geometriske Steder for Linier (l–p) (pp. 40–45 · Problems 220–245)
Architectural Overview (pp. 40–45)¶
In Chapter I, Part B, Petersen dualizes the Method of Two Loci from unknown points to unknown straight lines. Omitting one condition on a sought line produces a one-parameter family of lines (locus of lines) that envelopes a constructible circle or passes through a fixed point (finite or at infinity). Intersecting two such line-loci reduces every two-condition secant or chord problem to drawing:
The straight line joining two points (two point-pencils);
The tangents from a point to a circle (one point-pencil + one circular envelope, Problem 16 / 220); or
The four common tangents to two circles (two circular envelopes, Problem 137 / 221 / 275).
Complete Dictionary of Line Loci (l – p)¶
Locus |
Detached Condition on Straight Line \(L\) |
Resulting Envelope / Pencil Vertex |
|---|---|---|
l |
Cuts off chords of given length \(2c\) from given circle \((O, R)\) |
Concentric envelope circle \((O, \sqrt{R^2 - c^2})\) (Problem 220) |
m |
Distances from two fixed points \(A, B\) have given ratio \(m : n\) |
Two fixed points dividing \(AB\) internally and externally in ratio \(m : n\) |
n |
Algebraic distances from \(A, B\) have given sum \(s\) (or given difference \(d\)) |
Circle centered at midpoint of \(AB\) with radius \(s/2\) (or two parallel directions at \(\infty\)) |
o |
Divides inscribed angles on arc \(AB\) in a fixed ratio/difference (e.g. bisectors) |
Fixed point on the opposite circular arc \(AB\) (Problems 232, 236) |
p |
Common chords of all circles through \(A, B\) meeting a fixed circle \(\Gamma\) |
Fixed radical-centre point \(O \in AB\) (Problem 238, PPC Apollonius case) |