F.1 Albert’s Parabola

Tangent, Normal, and Depressed Cubic for Minimum Distance to a Parabola

Given a parabola y = x^2 and a point P = (P_x, P_y) on it, the tangent line T and normal line N at P satisfy y = 2 P_x x - P_x^2 and y = -x/(2 P_x) + P_x^2 + 1/2; requiring N to pass through (M, 0) yields the depressed cubic P_x^3 + (1/2) P_x - M/2 = 0.

Interactive GEOMETOR construction for F.1 (Albert’s Parabola). Hover any geometric label in the text or figure to highlight the corresponding element.

V : Point F : Point directrix : Line P : Point ===== Q = foot(P, directrix) s_fp = segment(F, P) s_pq = segment(P, Q) tangent = bisector(F, P, Q)

Mathematical Synthesis

In his April 2015 study Albert’s :term:`Parabola` Problem (Attempts #2 and #3, SeeligNotebooks), Fred W. Seelig investigates the tangent tangent and normal N to a parabola in two coordinate frames: the vertex frame \(y = x^2\) and the shifted directrix frame \(y = x^2 + 1\) (where the directrix is \(y = 0\), vertex V is \((0, 1)\), and focus F is \((0, 2)\)).

For \(y = x^2\) at P \(= (P_x, P_x^2)\), the tangent T has double-intercept form \(x/(P_x/2) + y/(-P_x^2) = 1\), while the normal N has \(x\)-intercept \(M = 2 P_x (P_x^2 + 1/2)\). Setting \(M = 4\) produces the depressed cubic \(P_x^3 + \frac{1}{2}P_x - 2 = 0\), which simultaneously governs the minimum distance from a fixed point \((M, 0)\) on the directrix to a point Q on the parabola.

Key Algebraic Identities

\[T:\; y = 2 P_x x - P_x^2, \qquad N:\; y = -\frac{x}{2 P_x} + \left(P_x^2 + \frac{1}{2}\right)\]

\[P_x^3 + \frac{1}{2} P_x - \frac{M}{2} = 0, \qquad r = \frac{P_x + 4 P_x^3}{4}\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.1 Albert’s Parabola — Folios (Folios 0001–0002, 2 scanned manuscript pages in resources/seelig/01_alberts-parabola)
Euclid’s Elements	I.11, I.12 (EuclidElements)
GEOMETOR Elements (G-Index)	G.3 extremities of a segment, G.4 segment
Encyclopedia of Triangle Centers	General projective / metric locus (Kimberling1998)
Seelig Spreadsheets	General Reference (Spreadsheets)

Constructive Dependency Graph

Required for: F.17 Ellipse, F.37 Euler’s Inradius Parabola