{"id":177,"date":"2012-07-18T02:06:03","date_gmt":"2012-07-18T07:06:03","guid":{"rendered":"http:\/\/daylateanddollarshort.com\/bloog\/?p=177"},"modified":"2012-07-28T22:47:57","modified_gmt":"2012-07-29T03:47:57","slug":"heron-like-strategies-for-hyperbolic-tetrahedral-volume","status":"publish","type":"post","link":"http:\/\/daylateanddollarshort.com\/bloog\/heron-like-strategies-for-hyperbolic-tetrahedral-volume\/","title":{"rendered":"Heron-like Strategies for Hyperbolic Tetrahedral Volume"},"content":{"rendered":"\n

With two Heron-like formulas for tetrahedral volume in Euclidean space<\/a>, it makes sense to investigate what happens in non-Euclidean —specifically, hyperbolic— space. After all, a number of hedronometric formulas (for instance, the Laws of Cosines) have counterparts in both geometries.<\/p>\n

Unfortunately, the search for a hyperbolic hedronometric formula for volume is hindered by the fact that the most-accessible general formula (due to Derevnin and Mednykh) for hyperbolic volume is a thorny integral parameterized by the dihedral angles \\(A,B,C,D,E,F\\) (with \\(A, B, C\\) meeting at a vertex and opposing, respectively, \\(D, E, F\\)):<\/p>\n

$$V = -\\frac{1}{4}\\int_{\\theta_0-\\phi}^{\\theta_0+\\phi}\\log \\frac{\\cos\\frac{A+B+C+\\theta}{2} \\cos\\frac{A+E+F+\\theta}{2} \\cos\\frac{D+B+F+\\theta}{2} \\cos\\frac{D+E+C+\\theta}{2}}{\\sin\\frac{A+D+B+E+\\theta}{2} \\sin\\frac{A+D+C+F+\\theta}{2} \\sin\\frac{B+E+C+F+\\theta}{2} \\sin\\frac{\\theta}{2}} d\\theta $$<\/p>\n

where \\(\\theta_0\\) and \\(\\phi\\) themselves are given by elaborate expressions in the dihedral angles.<\/p>\n

The good news is that, because the integral is parameterized by dihedral angles, and because dihedral angles are related neatly to face and pseudoface areas by a Law of Cosines<\/p>\n

$$\\cos\\frac{Y}{2} \\cos\\frac{Z}{2} + \\sin\\frac{Y}{2} \\sin\\frac{Z}{2} \\cos A = \\cos \\frac{H}{2} = \\cos\\frac{W}{2} \\cos\\frac{X}{2} + \\sin\\frac{W}{2}\\sin\\frac{X}{2} \\cos D$$<\/p>\n

with dihedrals \\(A, D\\) between respective face pairs \\(\\{Y,Z\\}\\) and \\(\\{W,X\\}\\), we’re\u00a0pretty close<\/em> to Pseudo-Heronic formula: we just have to solve for the angles from the areas, and then plug the solutions in as the integral parameters. This turns out to be a symbolic nightmare (for the most part), so that expressing the Derevnin-Mednykh formula in terms of \\(W, X, Y, Z, H, J, K\\) seems pretty hopeless; but, we can, at least, do the solving and plugging-in numerically<\/em> for tetrahedra we handle on a case-by-case basis. Of course, that amounts to a strategy<\/em>, not a formula<\/em>, for computing volume, but at this point, that’s the best I know how to do.<\/p>\n

The\u00a0Open Question<\/strong>:<\/p>\n

What’s better than the best I know how to do?<\/p><\/blockquote>\n

Interestingly, there’s an aspect of tetrahedral volume analysis that has a very nice hedronometric counterpart. The determinant of (angular) Gram matrix of a tetrahedron,<\/p>\n

$$G := \\left[ \\begin{array}{cccc}1 & -\\cos A & -\\cos B & -\\cos F \\\\\u00a0– \\cos A & 1 & -\\cos C & -\\cos E \\\\\u00a0-\\cos B & -\\cos C & 1 & -\\cos D \\\\\u00a0-\\cos F & -\\cos E & -\\cos D & 1\u00a0\\end{array}\\right]$$<\/p>\n

is a fundamental quantity in any geometry. (Fun fact: in Euclidean, hyperbolic, and spherical space, the determinant’s value is respectively zero, non-positive, or non-negative.) The quantity actually figures into the Derevnin-Mednykh formula. Hedronometrically,\u00a0it factors!<\/em> \u00a0Specifically, in hyperbolic space:<\/p>\n

$$-4 \\overline{W_2}^2\\overline{X_2}^2\\overline{Y_2}^2\\overline{Z_2}^2\\cdot \\det G = \\begin{array}{c}\\left( \\ddot{H_2} + \\ddot{J_2}\\ddot{K_2} – \\ddot{W_2}\\ddot{X_2} – \\ddot{Y_2} \\ddot{Z_2}\\right) \\\\ \\cdot\\left( \\ddot{J_2} + \\ddot{K_2}\\ddot{H_2} – \\ddot{W_2}\\ddot{Y_2} – \\ddot{Z_2} \\ddot{X_2}\\right) \\\\ \\cdot \\left( \\ddot{K_2} + \\ddot{H_2}\\ddot{J_2} – \\ddot{W_2}\\ddot{Z_2} – \\ddot{X_2} \\ddot{Y_2}\\right)\u00a0\\end{array}$$<\/p>\n

Here, I’m using my “Morse code” shorthand for sine and cosine:<\/p>\n

$$\\overline{W_2} := \\sin\\frac{W}{2} \\qquad \\ddot{W_2} := \\cos\\frac{W}{2}$$<\/p>\n

Even better, one can show that, for a perfect<\/em> tetrahedron (where opposite edges are perpendicular), the three factors on the right are\u00a0nicely proportional<\/em> in a way that defines a symmetric value, \\(M\\),
\n$$\\begin{align}
\n\\ddot{H_2} \\left( \\ddot{H_2} + \\ddot{J_2}\\ddot{K_2} – \\ddot{W_2}\\ddot{X_2} – \\ddot{Y_2} \\ddot{Z_2}\\right) &= \\ddot{J_2} \\left( \\ddot{J_2} + \\ddot{K_2}\\ddot{H_2} – \\ddot{W_2}\\ddot{Y_2} – \\ddot{Z_2} \\ddot{X_2}\\right) \\\\
\n&= \\ddot{K_2}\\left( \\ddot{K_2} + \\ddot{H_2}\\ddot{J_2} – \\ddot{W_2}\\ddot{Z_2} – \\ddot{X_2} \\ddot{Y_2}\\right) \\\\
\n&=: \\ddot{H_2}\\ddot{J_2}\\ddot{K_2} + M
\n\\end{align}
\n$$ so that we have<\/p>\n

$$-4 \\overline{W_2}^2\\overline{X_2}^2\\overline{Y_2}^2\\overline{Z_2}^2\\cdot \\ddot{H_2}\\ddot{J_2}\\ddot{K_2} \\cdot \\det G = \\left( \\ddot{H_2}\\ddot{J_2}\\ddot{K_2} + M \\right)^3$$<\/p>\n

(In theory, a perfect tetrahedron is completely determined by its face areas, so we “could” eliminate references to \\(H, J, K\\) in this formula. However, the result here isn’t at all pretty.)<\/p>\n

This factorization may (or may not) speak to a hedronometric dynamic locked deep within the Derevnin-Mednykh volume integral, but I suspect that the analysis requires more expertise in differential geometry than I possess at this time.<\/p>\n

A complete discussion of all of this stuff —including an example deriving a hedronometric volume integral tailored to very specific types of tetrahedra— is in my note from 2009: “Heron-like Strategies for Non-Euclidean Tetrahedral Volume”<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"

With two Heron-like formulas for tetrahedral volume in Euclidean space, it makes sense to investigate what happens in non-Euclidean —specifically, hyperbolic— space. After all, a number of hedronometric formulas (for instance, the Laws of Cosines) have counterparts in both geometries. Unfortunately, the search for a hyperbolic hedronometric formula for volume is hindered by the fact […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4,11],"tags":[],"class_list":["post-177","post","type-post","status-publish","format-standard","hentry","category-hedronometry","category-open-question"],"_links":{"self":[{"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/posts\/177","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/comments?post=177"}],"version-history":[{"count":10,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/posts\/177\/revisions"}],"predecessor-version":[{"id":200,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/posts\/177\/revisions\/200"}],"wp:attachment":[{"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/media?parent=177"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/categories?post=177"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/tags?post=177"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}