W.02 Lineations, Central Rotations & Reflections in Lines

Square-Root-Free Point and Line Reflections Across Arbitrary Non-Null Axes

For any non-null line l = <a : b : c> over a field F (char F != 2), the reflection sigma_l maps every point A = [x, y] and every line m = <a_2 : b_2 : c_2> to exact rational images whose denominators depend only on the quadratic norm a^2 + b^2, without radicals or angles.

Interactive GEOMETOR construction for W.02 (Lineations, Central Rotations & Reflections in Lines). Hover any geometric label in the text or figure to highlight the corresponding element.

P_1 : Point P_2 : Point A : Point B : Point ===== l = line(P_1, P_2) F = foot(A, l) A_r = reflect_point(A, l) m = line(A, B) m_r = reflect_line(m, l)

Mathematical Synthesis

Classical orthogonal reflection across an oblique axis typically invokes unit normal vectors containing \(\sqrt{a^2 + b^2}\) or trigonometric double-angle matrices. In Universal Geometry (Wildberger2005, SnapperTroyer1971), because the foot F of the altitude from A to l is rational in \((a, b, c)\), the reflected point Ar \(= 2 F - A\) is always rational over the ground field \(F\)!

Even more remarkably, reflecting an entire line m across l produces the reflected line mr \(= \Sigma_l(m)\) via homogeneous quadratic polynomials in \((a, b)\). Iterating line reflections generates cyclic reflection sequences that directly govern spread polynomials and regular star symmetries over both \(\mathbb{Q}\) and finite fields.

Key Algebraic Identities

\[\sigma_\ell([x, y]) = \left[\frac{(b^2 - a^2)x - 2 a b y - 2 a c}{a^2 + b^2},\; \frac{-2 a b x + (a^2 - b^2)y - 2 b c}{a^2 + b^2}\right]\]

\[\Sigma_\ell(\langle a_2 : b_2 : c_2 \rangle) = \langle (a^2 - b^2)a_2 + 2 a b b_2 :\; 2 a b a_2 - (a^2 - b^2)b_2 :\; 2 c(a a_2 + b b_2) - (a^2 + b^2)c_2 \rangle\]

Symbolic SymPy Verification

Invariant / Theorem Check	Exact SymPy Value	Verification Status
Mirror axis l = line(P_1, P_2)	<-1 : 2 : 0>	Norm a^2 + b^2 = 5
Reflected point sigma_l([1, 4])	[19/5, -8/5]	Exact in Q^2
Reflected line Sigma_l(m)	<-16 : 12 : 80>	Matches line(A_r, B_r)
Quadrance preservation Q(A,B) == Q(A_r,B_r)	16 == 16	Verified
Equal spreads s(l, m) == s(l, m_r)	1/5 == 1/5	Verified

Cross-Repository Canon

Executable Demonstration	W.02 Lineations, Central Rotations & Reflections in Lines — Code (Standalone model.py & w-02.svg)
Thematic Part	Part I: Preliminaries & Foundational Invariants
Euclid’s Elements	I.9, I.10, I.12 (EuclidElements)
GEOMETOR Elements	G.3 extremities of a segment, G.4 segment
GEOMETOR Fundamentals	F.5 Centers of Similitude
Metric Affine Canon	SnapperTroyer1971 (Orthogonal reflections over general fields)

Constructive Dependency Graph

Depends on: W.01 Planar Points, Line Proportions & Incidence Determinants

Required for: W.05 Spread Polynomials S_n(s) & Pentagonal Golden Spreads, W.12 Vertex Bisectors, Square Vertices & The Four Incenters, W.21 Taylor Conics, Algebraic Tangents & Optical Reflection Laws