F.17 Ellipse

Seven-Step Five-Circle Compass Oval Construction and Focus-Directrix Conic Relations

Given semi-major axis a and semi-minor axis b, a four-center circular-arc oval approximating the ellipse x^2/a^2 + y^2/b^2 = 1 is constructed by a 7-step five-circle compass procedure, while the exact conic satisfies F = a*e, d = a/e, and vertex mean proportional V = a = sqrt(d*F).

Interactive GEOMETOR construction for F.17 (Ellipse). Hover any geometric label in the text or figure to highlight the corresponding element.

O : Point A_1 : Point A_2 : Point B_1 : Point ===== F_1 = focus_left(O, A_2, B_1) F_2 = focus_right(O, A_2, B_1) P = point_on_ellipse(A_2, B_1) r_1 = segment(F_1, P) r_2 = segment(F_2, P)

Mathematical Synthesis

In Topic 17 (SeeligNotebooks), Seelig documents a classical 7-step compass-and-straightedge procedure titled ‘FIVE CIRCLE CONSTRUCTION :term:`ELLIPSE`’ inside a bounding box of semi-axes \(a\) and \(b\), locating major-axis centers \(C_2, C_3\) and minor-axis centers \(C_1, C_4, C_5\) so that the circular end-caps of radius \(r = |C_2 P|\) join smoothly with the broad top/bottom arcs.

In Folio 4, he derives the exact analytic relations among focal distance \(F\), directrix distance \(d = a/e\), eccentricity \(e\), semi-latus rectum \(p = b^2/a\), and the geometric mean relation \(V = a = \sqrt{d F}\).

Key Algebraic Identities

\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \qquad r_1 + r_2 = 2 a, \qquad V = a = \sqrt{d \cdot F}, \qquad L = \frac{2 b^2}{a}\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.17 Ellipse — Folios (Folios 0118–0121, 4 scanned manuscript pages in resources/seelig/17_ellipse)
Euclid’s Elements	I.47, VI.13 (EuclidElements)
GEOMETOR Elements (G-Index)	G.3 extremities of a segment
Encyclopedia of Triangle Centers	General projective / metric locus (Kimberling1998)
Seelig Spreadsheets	General Reference (Spreadsheets)

Constructive Dependency Graph

Prerequisites: F.1 Albert’s Parabola, F.10 Circles and Geometric Mean