W.24 Exact Triangle Solvers & Surveying Triangulation

Polynomial QQQ/QQS/QSS/SQQ Solvers, Inaccessible Heights, and Regiomontanus’ Extremum

Every classical triangle specification (QQQ, QQS, QSS, SQQ, and SSS up to scaling) reduces to linear or quadratic equations in quadrances and spreads with exact discriminant branching; applied to surveying, inaccessible heights and Regiomontanus’ maximum-spread viewing problem resolve by pure rational algebra without trigonometric tables.
t_surv l_ground seg_tower (Q = 36) A = [0, 0] B = [5, 0] D_foot = [9, 0] C_top = [9, 6] P_obs = [3, 0] s_A=4/13, s_B=9/13 => H = 36 Regiomontanus (4,9): X_opt=36, s_max=25/169 AA BB DfootDfoot CtopCtop PobsPobs + − ⟲ Lbl Gde

Interactive GEOMETOR construction for W.24 (Exact Triangle Solvers & Surveying Triangulation). Hover any geometric label in the text or figure to highlight the corresponding element.

A : Point B : Point D_foot : Point C_top : Point ===== l_ground = line(A, D_foot) seg_tower = segment(D_foot, C_top) t_surv = polygon(A, B, C_top) P_obs = regiomontanus_optimal_point(D_foot, a, b)

Mathematical Synthesis

Classical triangle resolution (SSS, SAS, ASA, SSA) alternates awkwardly between inverse cosines, inverse sines, and tangent half-angle formulas, accumulating floating-point rounding at every step. In Rational Trigonometry (Wildberger2005), QQQ and QSS are purely rational in one step, while SQQ (Cross Law quadratic in \(Q_3\)), QQS (Spread Law + Cross Law), and SSS (proportional quadrances \(Q_1 : Q_2 : Q_3 = s_1 : s_2 : s_3\) once \(s_3\) solves the Triple Spread Formula) require at most a single quadratic root!

In field surveying, measuring horizontal baseline \(Q(A, B)\) along lground and the elevation spreads from A and B to tower top Ctop above base Dfoot (segtower, triangle tsurv) determines the vertical quadrance \(Q(D_{\text{foot}}, C_{\text{top}})\) by one rational formula. Likewise, Regiomontanus’ 1471 problem—maximizing the viewing spread of a vertical statue from \(y = a\) to \(y = b\)—completes the square without derivatives to yield optimal quadrance \(Q(D_{\text{foot}}, P_{\text{obs}}) = a b\) at Pobs with maximum spread \((b - a)^2 / (a + b)^2\)!

Key Algebraic Identities

\[\text{SQQ:}\quad (Q_3 - (Q_1 + Q_2))^2 - 4 Q_1 Q_2 (1 - s_3) = \mathcal{A}(Q_1, Q_2, Q_3) - 4 Q_1 Q_2 s_3 = 0\]
\[\text{Regiomontanus }(0 < a < b):\quad x_{\text{opt}}^2 = a b, \qquad s_{\max} = \frac{(b - a)^2}{(a + b)^2}\]

Symbolic SymPy Verification

Invariant / Theorem Check

Exact SymPy Value

Verification Status

QQQ solver on A B C_top (52, 117, 25)

s = (4/13, 9/13, 25/169)

Pure rational Cross Law

SQQ quadratic branches

Q_3 roots = [52, 2116/13]

Recovers exact Q_2 = 117

Inaccessible height H from D=25, s_A=4/13, s_B=9/13

Roots = [900/169, 36]

Recovers exact tower quadrance H = 36

Regiomontanus extremum (a=4, b=9)

X_opt = 36, s_max = 25/169

Zero derivatives

Cross-Repository Canon

Executable Demonstration

W.24 Exact Triangle Solvers & Surveying Triangulation — Code (Standalone model.py & w-24.svg)

Thematic Part

Part IV: Practical Solvers, 3D Platonic Geometry & Polar Curves

Classical Surveying

Regiomontanus (1471 extremal viewing problem)

GEOMETOR Elements

G.3 extremities of a segment, G.4 segment, G.7 plane surface

GEOMETOR Fundamentals

F.10 Circles and Geometric Mean, F.41 Geometric Problems and Extremal Loci, F.54 HP-35s Triangle Program and Coordinate Scripts

Computational Engine

Implemented in geometor.wildberger.solvers

Constructive Dependency Graph

digraph dependency { rankdir=TB; bgcolor="transparent"; node [fontname="Helvetica,Arial,sans-serif", fontsize=10, shape=box, style="rounded,filled", margin="0.15,0.08"]; edge [color="#888888", arrowsize=0.7]; "W.08" [label="W.08\nLaw III: The Spread Law & Exten...", fillcolor="#d8dee9", color="#4c566a", URL="../w-08/index.html"]; "W.09" [label="W.09\nLaw IV: The Cross Law & Quadrea...", fillcolor="#d8dee9", color="#4c566a", URL="../w-09/index.html"]; "W.10" [label="W.10\nLaw V: Triple Spread, Triple Cr...", fillcolor="#d8dee9", color="#4c566a", URL="../w-10/index.html"]; "W.24" [label="W.24\nExact Triangle Solvers & Survey...", fillcolor="#ffff99", color="#d4a017", penwidth=2.0, URL="../w-24/index.html"]; "W.25" [label="W.25\nSnellius–Pothenot Resection & H...", fillcolor="#e5e9f0", color="#777777", URL="../w-25/index.html"]; "W.08" -> "W.24"; "W.09" -> "W.24"; "W.10" -> "W.24"; "W.24" -> "W.25"; }

Depends on: W.08 Law III: The Spread Law & Extended Circumquadrance Law, W.09 Law IV: The Cross Law & Quadrea-Spread Identity, W.10 Law V: Triple Spread, Triple Cross & Quadruple Spread Formulas

Required for: W.25 Snellius–Pothenot Resection & Hansen’s Surveying Problem