W.21 Taylor Conics, Algebraic Tangents & Optical Reflection Laws

Derivative-Free Tangent Proportions and Exact Equal-Spread Focal Reflections

Translating any conic f(x, y) = 0 to a point A = [x_0, y_0] on the conic isolates a unique linear term l_1(x - x_0, y - y_0) = 0 defining its algebraic tangent t_tan without limits; for a parabola, t_tan is the perpendicular bisector of focus F_0 and directrix foot D_0 (giving equal spreads with F_0 A and the axis), and for a quadrola, t_tan makes equal spreads with both focal lines F_1 P_q and F_2 P_q.

Interactive GEOMETOR construction for W.21 (Taylor Conics, Algebraic Tangents & Optical Reflection Laws). Hover any geometric label in the text or figure to highlight the corresponding element.

F_0 : Point D_1 : Point D_2 : Point A : Point ===== l_dir = line(D_1, D_2) parabola = parabola_locus(F_0, l_dir) D_0 = foot(A, l_dir) t_tan = perp_bisector(F_0, D_0) l_in = line(F_0, A) l_out = line(D_0, A)

Mathematical Synthesis

Over finite or discrete fields there are no limits \(\lim_{h \to 0}\), yet conics possess exact tangents! Expanding \(f(x_0 + u, y_0 + v) = f(x_0, y_0) + \ell_1(u, v) + q_2(u, v)\) (the Taylor Conic Translation of Brook Taylor, Wildberger2005) produces the algebraic tangent line \(\ell_A = \langle 2 a x_0 + b y_0 + d : b x_0 + 2 c y_0 + e : d x_0 + e y_0 + 2 f \rangle\) whenever \(A = [x_0, y_0]\) is non-singular.

Applied to parabola (focus F0, directrix ldir), the tangent ttan at A is literally the perpendicular bisector of F0 and the directrix foot D0! Since \(D_0 A\) (lout) is perpendicular to ldir (hence parallel to the parabola’s axis), reflection symmetry across ttan immediately proves the Parabola Reflection Theorem \(s(t_{\text{tan}}, F_0 A) = s(t_{\text{tan}}, D_0 A)\) over every field! Analogously, at any point Pq on a quadrola quadrola with foci F1, F2, the algebraic tangent tq satisfies \(s(t_q, F_1 P_q) = s(t_q, F_2 P_q)\).

Key Algebraic Identities

\[\ell_A = \langle 2 a x_0 + b y_0 + d :\; b x_0 + 2 c y_0 + e :\; d x_0 + e y_0 + 2 f \rangle\]

\[s(t_{\text{tan}}, F_0 A) = s(t_{\text{tan}}, D_0 A), \qquad s(t_q, F_1 P_q) = s(t_q, F_2 P_q)\]

Symbolic SymPy Verification

Invariant / Theorem Check	Exact SymPy Value	Verification Status
Taylor conic constant term at A=[4,5]	0	Verified == 0 (A lies on parabola)
Tangent t_tan == perp_bisector(F_0, D_0)	s(t_tan, pb) = 0	Exact line coincidence
Parabola reflection spreads at A	1/5 == 1/5	Equal focal and axial spreads
Quadrola reflection spreads at P_q	25/34 == 25/34	Equal bifocal tangent spreads

Cross-Repository Canon

Executable Demonstration	W.21 Taylor Conics, Algebraic Tangents & Optical Reflection Laws — Code (Standalone model.py & w-21.svg)
Thematic Part	Part III: Universal Triangle, Conic & Chromogeometry
Classical Canon	Apollonius (Conics III.48) & Diocles (On Burning Mirrors)
GEOMETOR Elements	G.3 extremities of a segment, G.4 segment
GEOMETOR Fundamentals	F.1 Albert’s Parabola, F.39 Pole and Polar with Respect to a Triangle
Algebraic Geometry	Purely polynomial tangent space over Q and F_p without calculus

Constructive Dependency Graph

Depends on: W.02 Lineations, Central Rotations & Reflections in Lines, W.03 Quadrance, Midpoints & Point-to-Line Separation, W.20 Metrical Conics: Circles, Ribbons, Parabolas, Quadrolas & Grammolas

Required for: W.26 Algebraic Physics: Projectile Parabolas, Snell’s Law & Relativity