Rational Trigonometry: Algebraic Foundations

• Rational trigonometry is a reformulation that replaces distances and angles with algebraic invariants, quadrance and spread, enabling exact computations in arbitrary fields.
• It systematically develops trigonometric laws using determinant-based invariants and generalized vector products to extend analyses to multidimensional, non-Euclidean, and finite geometries.
• Its significance lies in offering precise, field-independent tools that enhance computational clarity and broaden applications in algebraic and projective geometric contexts.

Rational trigonometry is a reformulation of classical trigonometric theory, replacing the transcendental notions of distance and angle with purely algebraic quantities: quadrance and spread. Developed by N. J. Wildberger, this algebraic approach generalizes to arbitrary symmetric bilinear forms and fields of characteristic not 2, enabling coherent trigonometric analysis in non-Euclidean, finite, and arbitrary affine and projective geometries. In rational trigonometry, traditional metrics are supplanted by rational functions, facilitating exact computations and compatibility with fields beyond the real numbers. Recent research has extended these concepts to multidimensional settings, notably for tetrahedra in three-dimensional affine space, and connected them to projective geometric algebra for enhanced structural and computational clarity.

1. Core Invariants: Quadrance and Spread

Rational trigonometry is centered on two core invariants:

• Quadrance ($Q$): Given two points $A,B$ in an affine space equipped with a symmetric bilinear form $B$, the $B$-quadrance is $$Q_B(A,B) = (B\text{-scalar}) = (A-B)\,B\,(A-B)^\top$$ (Notowidigdo et al., 2019). Quadrance generalizes the notion of squared distance and is always a rational function of coordinates and the bilinear form.
• Spread ($s$): For two lines with direction vectors $v_1, v_2$, the $B$-spread is $$S_B(\ell_1,\ell_2) = 1 - (v_1\cdot_B v_2)^2/Q_B(v_1)Q_B(v_2)$$, a purely algebraic counterpart of $\sin^2$ of the angle between lines (Notowidigdo et al., 2019). For planes, the dihedral spread is defined using the $A,B$0-inner product of their normals.

This shift underpins all further developments in the theory, including trigonometric laws for figures in arbitrary dimensions and metrics (Notowidigdo et al., 2019, Notowidigdo, 2020).

2. Structural Framework: Bilinear Forms and Vector Products

The rational trigonometric framework relies on background structures:

• Symmetric Bilinear Form: The fundamental structure is a symmetric, invertible $A,B$1 matrix $A,B$2, defining the geometry's metric and scalar product via $A,B$3.
• Generalised Cross Product: The $A,B$4-twisted cross product, $A,B$5, generalizes the usual vector product to arbitrary bilinear forms. It ensures orthogonality properties and underpins rational determinants (volumes) and scalar triple products (Notowidigdo et al., 2019, Notowidigdo, 2020).
• Standardization by Affine Maps: Any tetrahedron in affine space can be mapped (via affine transformations) to a standard tetrahedron, transferring the original metric structure to a new bilinear form $A,B$6 and thereby reducing structural investigations to a canonical position (Notowidigdo, 2020, Notowidigdo, 2019).

3. Rational Trigonometry in Three Dimensions

The extension to tetrahedra yields a hierarchy of algebraic invariants and laws:

3.1 Primary Invariants

• Edge Quadrance: $A,B$7, for vertices $A,B$8.
• Face Quadrea: $A,B$9, for a triangle with edge quadrances $B$0 (Notowidigdo, 2019). Over $B$1 and $B$2, this is $B$3.
• Quadrume (Volume Analog): For a tetrahedron with edge quadrances $B$4, the quadrume $B$5 is a quartic determinant (Euler's four-point function): $B$6 (Notowidigdo, 2019).
• Dihedral Spread ($B$7): The algebraic analog of the squared sine of a dihedral angle, defined by $B$8, where the indices specify the faces adjacent to edge $B$9 (Notowidigdo, 2019).

3.2 Trigonometric Laws

• Solid Spread Law: For each vertex $B$0, the ratio of the solid spread $B$1 at $B$2 to the product $B$3 (over the three incident edges) is constant across all vertices (Notowidigdo et al., 2019).
• Generalized Pythagoras (de Gua): For a tri-rectangular tetrahedron at $B$4, $B$5 (Notowidigdo et al., 2019).
• Dihedral Spread Law: At a vertex, $B$6 is constant for the three incident edges (Notowidigdo et al., 2019).
• Quadrance–Spread Reciprocity: For opposite edge pairs, $B$7 is constant—equal for all three such pairs—and connected to face quadreas (Notowidigdo et al., 2019, Notowidigdo, 2019).
• Crelle's Circumquadrance Formula: Relates the quadrume $B$8, the circumradius quadrance $B$9, and products of edge quadrances: $$Q_B(A,B) = (B\text{-scalar}) = (A-B)\,B\,(A-B)^\top$$0 (Notowidigdo, 2019).

These laws are expressible entirely in terms of quadrances and spreads, requiring only rational or determinantal computations.

4. Coordinate Representations and Universal Geometry

• Standard Tetrahedron Formalism: By reducing a tetrahedron to a standard position via affine transformations, computations are performed in a fixed coordinate system, with all geometry encoded via the symmetric matrix $$Q_B(A,B) = (B\text{-scalar}) = (A-B)\,B\,(A-B)^\top$$1 (the inverse Gram matrix of edge vectors). In this frame, classic Euclidean identities are recovered when $$Q_B(A,B) = (B\text{-scalar}) = (A-B)\,B\,(A-B)^\top$$2; otherwise, the geometry is tracked by $$Q_B(A,B) = (B\text{-scalar}) = (A-B)\,B\,(A-B)^\top$$3 (Notowidigdo, 2020).
• Projective Geometric Algebra (PGA): Rational trigonometry can be reframed within the metric-neutral algebraic formalism of PGA (Gunn, 2014). Points and lines are unified as elements in the Clifford algebra, and quadrance and spread acquire exact dual expressions: $$Q_B(A,B) = (B\text{-scalar}) = (A-B)\,B\,(A-B)^\top$$4 and $$Q_B(A,B) = (B\text{-scalar}) = (A-B)\,B\,(A-B)^\top$$5. This approach reveals a structural duality between quadrance and spread not apparent in conventional vector frameworks and supports direct generalization to non-Euclidean geometries.

5. Worked Examples and Special Tetrahedra

Explicit computation for tri-rectangular and standard tetrahedra demonstrates the theory's algebraic nature:

• Tri-rectangular Tetrahedron: Analytical expressions for all spreads, quadrances, quadreas, and quadrume are rational functions of three parameters $$Q_B(A,B) = (B\text{-scalar}) = (A-B)\,B\,(A-B)^\top$$6 (the incident edge quadrances at the right vertex). Key sum and reciprocal laws, such as $$Q_B(A,B) = (B\text{-scalar}) = (A-B)\,B\,(A-B)^\top$$7 for solid spreads and $$Q_B(A,B) = (B\text{-scalar}) = (A-B)\,B\,(A-B)^\top$$8 for dihedral spreads, are satisfied (Notowidigdo et al., 2019).
• Standard Tetrahedron: For the canonical tetrahedron with vertices $$Q_B(A,B) = (B\text{-scalar}) = (A-B)\,B\,(A-B)^\top$$9 and $s$0, all quadrances and spreads are 1, and the rational laws reduce to classical Euclidean relationships (Notowidigdo, 2020).

These examples underscore that all computations involve polynomials and determinants, without recourse to transcendental or irrational structures.

6. Field Independence and Generalization

All core definitions and laws of rational trigonometry are field-independent and extend to arbitrary non-null symmetric bilinear forms over fields of characteristic $s$1 (or $s$2 for certain constructions) (Notowidigdo, 2019, Notowidigdo et al., 2019). This universality enables genuine rational trigonometry for classical, hyperbolic, elliptic, and even finite geometries. The projective geometric algebra formalism further facilitates investigations in metric-neutral and universal settings, providing a unified language compatible with modern geometric algebra approaches (Gunn, 2014).

7. Significance and Outlook

Rational trigonometry replaces the transcendental functions of classical trigonometry with purely algebraic quantities, enabling exact manipulation, compatibility with arbitrary fields, and deep structural insights—such as the algebraic duality of quadrance and spread in projective geometric algebra. Its field-general formulations extend trigonometric principles to diverse mathematical contexts beyond the real numbers. The recent research spectrum includes affine and projective contexts, determinants identities for tetrahedra, connections to universal geometry, and explicit computational frameworks in three or more dimensions (Notowidigdo, 2020, Notowidigdo, 2019, Notowidigdo et al., 2019, Gunn, 2014). The theory's compatibility with computational environments and cryptographic applications (this suggests increased applicability for computer-based symbolic reasoning in arbitrary metric geometries) is a plausible implication. The ability to formulate geometric results in a purely rational form makes rational trigonometry a powerful framework for trigonometric investigations in modern mathematics.