geometor.wildberger.solvers

Exact polynomial triangle and surveying solvers for Rational Trigonometry.

Replaces classical inverse trigonometric solvers (arcsin, arccos, arctan) with exact linear and quadratic polynomial equations over SymPy rationals and algebraic extensions, covering QQQ, QQS, QSS, SQQ, SSS, Regiomontanus’ problem, inaccessible-base triangulation, Snellius-Pothenot resection, and Hansen’s two-station problem.

Classes

SolvedTriangle

Exact algebraic solution of a planar triangle (Q1, Q2, Q3, s1, s2, s3).

Functions

solve_qqq(q1, q2, q3) → SolvedTriangle

Solve a non-collinear triangle uniquely from three side quadrances (Q1, Q2, Q3).

solve_qqs(q1, q2, s3) → list[SolvedTriangle]

Solve for both acute/obtuse triangle branches given (Q1, Q2, s3).

solve_qss(q1, s1, s2) → list[SolvedTriangle]

Solve for triangle branches given (Q1, s1, s2) via Triple Spread + Spread Law.

solve_sqq(s1, q1, q2) → list[SolvedTriangle]

Solve the rational SSA case (s1, Q1, Q2) via Spread Law + Triple Spread.

solve_sss_proportional(s1, s2, s3[, q1]) → SolvedTriangle

Validate a spread triple (s1, s2, s3) and solve the similar triangle with Q1 = q1.

two_spread_triples_common(a, b, c, d) → sympy.Expr

Compute the unique common spread x = P(a, b, c, d) of {a, b, x} and {c, d, x}.

spread_from_ratio(s_total, ratio_a) → list[tuple[sympy.Expr, sympy.Expr]]

Solve {x, y, s_total} being a spread triple with x / y = ratio_a (Theorem 48, p. 91).

solve_inaccessible_height(step_quadrance, s1, s2) → list[sympy.Expr]

Solve Surveying Problem 24.2: vertical cliff height quadrance H from two station spreads.

solve_regiomontanus(lower_height, upper_height) → dict[str, sympy.Expr]

Solve Regiomontanus' maximum-spread statue problem algebraically without calculus (Section 24.4).

solve_snellius_pothenot_coordinates(beacon_a, ...) → dict[str, sympy.Expr]

Execute Wildberger's Collins-Point / Subtended-Spread resection verification (Section 25.1).

solve_hansen_benchmark() → dict[str, sympy.Expr]

Execute Wildberger's complete rational solution of Hansen's Two-Station Problem (Section 25.2).

Module Contents

class geometor.wildberger.solvers.SolvedTriangle

Exact algebraic solution of a planar triangle (Q1, Q2, Q3, s1, s2, s3).

q1

Quadrance opposite vertex 1.

q2

Quadrance opposite vertex 2.

q3

Quadrance opposite vertex 3.

s1

Spread at vertex 1.

s2

Spread at vertex 2.

s3

Spread at vertex 3.

quadrea

Triangle quadrea A(Q1, Q2, Q3).

circumquadrance

Circumquadrance K = Q1*Q2*Q3 / A.

q1: sympy.Expr
q2: sympy.Expr
q3: sympy.Expr
s1: sympy.Expr
s2: sympy.Expr
s3: sympy.Expr
quadrea: sympy.Expr
circumquadrance: sympy.Expr
geometor.wildberger.solvers.solve_qqq(q1: geometor.wildberger.invariants.CoordScalar, q2: geometor.wildberger.invariants.CoordScalar, q3: geometor.wildberger.invariants.CoordScalar) → SolvedTriangle

Solve a non-collinear triangle uniquely from three side quadrances (Q1, Q2, Q3).

Uses the Cross Law s_3 = 1 - (Q_1 + Q_2 - Q_3)^2 / (4*Q_1*Q_2) = A / (4*Q_1*Q_2).

Parameters:

  • q1 – First side quadrance Q1.

  • q2 – Second side quadrance Q2.

  • q3 – Third side quadrance Q3.

Returns:

Exact SolvedTriangle.

Raises:

ValueError – If A(Q1, Q2, Q3) == 0 (collinear/degenerate triple) or Qi == 0.

geometor.wildberger.solvers.solve_qqs(q1: geometor.wildberger.invariants.CoordScalar, q2: geometor.wildberger.invariants.CoordScalar, s3: geometor.wildberger.invariants.CoordScalar) → list[SolvedTriangle]

Solve for both acute/obtuse triangle branches given (Q1, Q2, s3).

Solves the quadratic Cross Law (Q1 + Q2 - Q3)^2 = 4*Q1*Q2*(1 - s3) for Q3, returning one solution when s3 == 1 (right vertex) and two solutions (acute and obtuse sectors) when 0 < s3 < 1.

Parameters:

  • q1 – First side quadrance Q1.

  • q2 – Second side quadrance Q2.

  • s3 – Included vertex spread s3.

Returns:

List of 1 or 2 exact SolvedTriangle branches.

geometor.wildberger.solvers.solve_qss(q1: geometor.wildberger.invariants.CoordScalar, s1: geometor.wildberger.invariants.CoordScalar, s2: geometor.wildberger.invariants.CoordScalar) → list[SolvedTriangle]

Solve for triangle branches given (Q1, s1, s2) via Triple Spread + Spread Law.

Solves (s1 + s2 + s3)^2 = 2*(s1^2 + s2^2 + s3^2) + 4*s1*s2*s3, which in normal quadratic form is (s3 - (s1 + s2 - 2*s1*s2))^2 = 4*s1*s2*(1 - s1)*(1 - s2).

Parameters:

  • q1 – Known side quadrance Q1 opposite s1.

  • s1 – Non-zero spread s1 opposite Q1.

  • s2 – Second spread s2.

Returns:

List of exact SolvedTriangle branches.

geometor.wildberger.solvers.solve_sqq(s1: geometor.wildberger.invariants.CoordScalar, q1: geometor.wildberger.invariants.CoordScalar, q2: geometor.wildberger.invariants.CoordScalar) → list[SolvedTriangle]

Solve the rational SSA case (s1, Q1, Q2) via Spread Law + Triple Spread.

Parameters:

  • s1 – Non-zero spread s1 opposite Q1.

  • q1 – Side quadrance Q1.

  • q2 – Adjacent side quadrance Q2.

Returns:

List of exact SolvedTriangle branches.

geometor.wildberger.solvers.solve_sss_proportional(s1: geometor.wildberger.invariants.CoordScalar, s2: geometor.wildberger.invariants.CoordScalar, s3: geometor.wildberger.invariants.CoordScalar, q1: geometor.wildberger.invariants.CoordScalar = 1) → SolvedTriangle

Validate a spread triple (s1, s2, s3) and solve the similar triangle with Q1 = q1.

Parameters:

  • s1 – First spread.

  • s2 – Second spread.

  • s3 – Third spread.

  • q1 – Scale quadrance Q1 (default 1).

Returns:

Exact SolvedTriangle at scale Q1 = q1.

Raises:

ValueError – If S(s1, s2, s3) != 0.

geometor.wildberger.solvers.two_spread_triples_common(a: geometor.wildberger.invariants.CoordScalar, b: geometor.wildberger.invariants.CoordScalar, c: geometor.wildberger.invariants.CoordScalar, d: geometor.wildberger.invariants.CoordScalar) → sympy.Expr

Compute the unique common spread x = P(a, b, c, d) of {a, b, x} and {c, d, x}.

Implements Wildberger’s Two Spread Triples Theorem (Theorem 53, p. 98 & p. 256): P(a, b, c, d) = ((a - b)^2 - (c - d)^2) / (2*(a + b - c - d - 2*a*b + 2*c*d)).

Parameters:

  • a – First spread of first triple.

  • b – Second spread of first triple.

  • c – First spread of second triple.

  • d – Second spread of second triple.

Returns:

Exact common spread x.

geometor.wildberger.solvers.spread_from_ratio(s_total: geometor.wildberger.invariants.CoordScalar, ratio_a: geometor.wildberger.invariants.CoordScalar) → list[tuple[sympy.Expr, sympy.Expr]]

Solve {x, y, s_total} being a spread triple with x / y = ratio_a (Theorem 48, p. 91).

Returns both algebraic branches y = s / (a + 1 +/- 2*sqrt(a*(1 - s))) and x = a * y.

Parameters:

  • s_total – Third spread s in {x, y, s}.

  • ratio_a – Known spread ratio a = x / y.

Returns:

List of exact (x, y) spread pairs.

geometor.wildberger.solvers.solve_inaccessible_height(step_quadrance: geometor.wildberger.invariants.CoordScalar, s1: geometor.wildberger.invariants.CoordScalar, s2: geometor.wildberger.invariants.CoordScalar) → list[sympy.Expr]

Solve Surveying Problem 24.2: vertical cliff height quadrance H from two station spreads.

Two collinear ground stations separated by step_quadrance = D observe the top of a vertical cliff with elevation spreads s1 and s2 relative to the horizontal. Horizontal base quadrances are X1 = H*(1 - s1)/s1 and X2 = H*(1 - s2)/s2, and collinearity A(X1, X2, D) = 0 yields the two exact branches (same side vs opposite sides of the cliff base): H = D * s1 * s2 / ((sqrt(s2*(1 - s1)) -/+ sqrt(s1*(1 - s2)))^2).

Parameters:

  • step_quadrance – Quadrance D between the two observation stations.

  • s1 – Elevation spread at station 1.

  • s2 – Elevation spread at station 2.

Returns:

Exact roots for the vertical height quadrance H.

geometor.wildberger.solvers.solve_regiomontanus(lower_height: geometor.wildberger.invariants.CoordScalar, upper_height: geometor.wildberger.invariants.CoordScalar) → dict[str, sympy.Expr]

Solve Regiomontanus’ maximum-spread statue problem algebraically without calculus (Section 24.4).

A vertical segment from y = a to y = b (0 < a < b, quadrances A = a^2, B = b^2) viewed from [x, 0] (X = x^2) subtends maximum spread at exact horizontal quadrance X_opt = a * b, attaining maximum spread s_max = (b - a)^2 / (b + a)^2.

Parameters:

  • lower_height – Lower height a > 0.

  • upper_height – Upper height b > a.

Returns:

Dictionary with optimal_x_quadrance, optimal_x, and max_spread.

geometor.wildberger.solvers.solve_snellius_pothenot_coordinates(beacon_a: geometor.wildberger.invariants.Point2DInput, beacon_b: geometor.wildberger.invariants.Point2DInput, beacon_c: geometor.wildberger.invariants.Point2DInput, station_p: geometor.wildberger.invariants.Point2DInput) → dict[str, sympy.Expr]

Execute Wildberger’s Collins-Point / Subtended-Spread resection verification (Section 25.1).

Given three known beacons A, B, C and an observer P measuring only the rational spreads r1 = s(PA, PB) and r2 = s(PB, PC), computes the exact measured spreads, beacon quadrances, and verifies the resection equations.

Parameters:

  • beacon_a – First beacon A.

  • beacon_b – Second beacon B.

  • beacon_c – Third beacon C.

  • station_p – Observer station P.

Returns:

Dictionary of exact beacon quadrances, observed spreads, and station quadrances.

geometor.wildberger.solvers.solve_hansen_benchmark() → dict[str, sympy.Expr]

Execute Wildberger’s complete rational solution of Hansen’s Two-Station Problem (Section 25.2).

Reconstructs all unknown quadrances between known baseline A=[0,0], B=[1,5] (Q(A,B) = 26) and unknown stations C=[5,4], D=[4,1] strictly from the four measured spreads at C and D using two_spread_triples_common, Alternate Spreads, spread_from_ratio, and the Spread Law—zero transcendental functions!

Returns:

Dictionary of exact intermediate spreads and recovered quadrances.