[Paper Review] A multidimensional Law of Sines

Igor Rivin|ArXiv.org|Nov 17, 2002

Mathematics and Applications3 citations

TL;DR

This paper presents a multidimensional generalization of the classical Law of Sines for simplices in Euclidean $\mathbb{E}^n$, using linear algebra and the Gram matrix of face normals. The key result establishes a ratio identity between products of face areas and minors of the Gram matrix, extending trigonometric duality to higher dimensions via adjugate matrices and null-space properties.

ABSTRACT

We extend the Law of Sines to simplices in Euclidean spaces of any number of dimensions.

Motivation & Objective

• To generalize the classical Law of Sines from 2D triangles to n-dimensional simplices in $\mathbb{E}^n$.
• To establish a linear-algebraic framework using face areas and outward normals for higher-dimensional trigonometric identities.
• To prove that the ratio of products of face areas equals the ratio of corresponding minors of the Gram matrix.
• To demonstrate that this generalization holds via properties of the adjugate matrix and null-space of the Gram matrix.

Proposed method

• Define the Gram matrix $G$ from the outer products of outward unit normals to the $n$-dimensional simplex faces.
• Use Theorem 0.1 stating that the sum of area-weighted outward normals is zero, implying the null-space of $G$ is spanned by the area vector $\mathbf{a} = (A_1, \dots, A_{n+1})$.
• Apply Cramer’s rule and adjugate matrix theory to relate minors of $G$ to area ratios.
• Leverage the fact that $G$ has rank $n$ and one zero eigenvalue, with the null vector $\mathbf{a}$, to derive the main identity.
• Use the adjugate matrix $\widehat{G}$ to express the ratio of area products as ratios of minors $\widehat{G}_{ij}/\widehat{G}_{kl}$.
• Verify the identity in 2D and 3D cases, showing consistency with known trigonometric identities and spherical geometry.

Experimental results

Research questions

• RQ1 Can the classical Law of Sines be generalized to simplices in $\mathbb{E}^n$ using linear algebraic methods?
• RQ2 What is the relationship between the areas of faces of a simplex and the minors of its Gram matrix?
• RQ3 How does the null-space of the Gram matrix relate to the face areas in higher dimensions?
• RQ4 Can the multidimensional Law of Sines recover known trigonometric identities in 2D and 3D?
• RQ5 What role does the adjugate matrix play in expressing area ratios in higher-dimensional simplices?

Key findings

• The multidimensional Law of Sines is established as $\frac{A_i A_j}{A_k A_l} = \frac{\widehat{G}_{ij}}{\widehat{G}_{kl}}$, where $\widehat{G}_{ij}$ is the $ij$-th minor of the Gram matrix.
• In 2D, the identity reduces to the classical Law of Sines, with $\frac{|BC|^2}{|AC|^2} = \frac{\sin^2\alpha}{\sin^2\beta}$, confirming consistency.
• In 3D, the identity yields a complex expression involving cosines of dihedral angles, which simplifies using spherical Heron’s formula to $\frac{A_4}{A_3} = \frac{\sin(S_4/2)\cos(\alpha_{13}/2)\cos(\alpha_{23}/2)}{\sin(S_3/2)\cos(\alpha_{14}/2)\cos(\alpha_{24}/2)}$, where $S_i$ is the spherical area of the vertex link.
• The null-space of the Gram matrix $G$ is spanned by the vector of face areas $\mathbf{a} = (A_1, \dots, A_{n+1})$, which is essential to the derivation.
• The adjugate matrix $\widehat{G}$ is proportional to the outer product of the area vector and the null vector of $G^t$, with proportionality constant related to the product of non-zero eigenvalues.
• The method provides a new proof of the classical Law of Sines as a special case of the multidimensional result, using vector projection and Stokes’ theorem intuition.