HEDRONOMETRIC FORMULAS FOR A HYPERBOLIC TETRAHEDRON

HEDRONOMETRIC FORMULAS

FOR A HYPERBOLIC TETRAHEDRON

B.D.S. “BLUE” [email protected]

Abstract. This “living document” will serve as an ever-expanding resource of resultsin hyperbolic (tetra)hedronometry. To maximize information density, I minimize discus-sion and proof (at least in early drafts), but the reader can readily verify the geometrywith basic plane trigonometry and the algebra with standard identities; the only trulysophisticated notions are the Schalfli and Derevnin-Mednykh formulas for volume.

(Refer to Appendix A for notation used throughout.)

1. Fundamentals: The Laws of Cosines

At the core of hedronometry lie three Laws of Cosines that epitomize the field as“dimensionally-enhanced” trigonometry by relating, not the edges and planar angles ofa triangle, but the faces (W , X, Y , Z) and dihedral angles (A, B, C, D, E, F ) of a tetra-hedron. The Second Law, in particular, also formally defines the figure’s geometrically-amorphous1 but algebraically-rewarding2 pseudo-faces (H, J , K).

Theorem 1.1 (The First Law of Cosines).

W2 = X2Y2Z2 −X2Y2Z2

»−4[···A,B,C···] + X2Y2Z2A+X2Y2Z2B +X2Y2Z2C(1)

Theorem 1.2 (The Second Law of Cosines, and the Definition of 0 ≤ H,J,K ≤ 2π).

Y2Z2 + Y2Z2A = H2 = W2X2 +W2X2D

Z2X2 + Z2X2B = J2 = W2Y2 +W2Y2E(2)

X2Y2 +X2Y2C = K2 = W2Z2 +W2Z2F

Theorem 1.3 (The Third Law of Cosines).

0 = 1− W22 − X2

2 − Y22 − Z2

2 − 4W2X2Y2Z2 − H22 − J2

2 − K22 − 2H2J2K2

+ 2H2

ÄW2X2 + Y2Z2

ä+ 2J2

ÄW2Y2 + Z2X2

ä+ 2K2

ÄW2Z2 + X2Y2

ä(3)

1In Euclidean space, a pseudo-face is a quadrilateral determined by projection of the tetrahedron into aplane parallel to a pair of opposite edges; no geometric interpretation of a general hyperbolic pseudo-face isknown. (See Section 6 for some special cases.) Nevertheless, we declare that the element’s “area” satisfiesthe Second Law of Cosines, and that the area is bounded as if the element were a hyperbolic quadrilateral.

2I doubt that the significance of pseudo-faces in catalyzing tetrahedral analysis can be overstated.

1

2 B.D.S. “BLUE” MCCONNELL

These give rise to symbolic abbreviations3 that we’ll find helpful:

〈W2〉 := −W2 − 2X2Y2Z2 + H2X2 + J2Y2 + K2Z2 ≥ 0(4a)

〈H2〉 := H2 + J2K2 − W2X2 − Y2Z2 ≥ 0(4b)

We also note this consequence of (2)

JH2,W2, X2K =1

4W2

2X2

2D

2 ≥ 0(5)

and this Law-of-Cosines-like equality that follows upon reduction modulo (3):

(6) 〈J2〉2 + 〈K2〉2 − 2H2〈J2〉〈K2〉 = 16 JH2,W2, X2KJH2, Y2, Z2K

2. From Lengths to Areas, and Back Again

Standard formulas give the area of a tetrahedron’s face from lengths of bounding edges

(7) W2 =d2

2+ e2

2 + f22 − 1

2d2e2f2

W2 =

»Jd, e, fK

2d2e2f2

and we can determine pseudo-face areas from lengths of edges in conspicuous pairs

H2 =−a2

2d22

+ b22e2

2 + c22f2

2

2 b2e2 c2f2

H2 =

√î2a2d2, 2b2e2, 2c2f2

ó2 b2e2 c2f2

(8a)

H42

=−a2

2d22

+Äb2e2 + c2f2

ä24 b2e2 c2f2

H42

=a2

2d22 −Äb2e2 − c2f2

ä24 b2e2 c2f2

(8b)

From these, we can verify

〈W2〉a2b2c2 = 〈X2〉a2e2f2 = 〈Y2〉d2b2f2 = 〈Z2〉d2e2c2(9)

= 〈H2〉a2d2 = 〈J2〉b2e2 = 〈K2〉c2f2 =−Êg

64a2b2c2d2e2f2

Consequently, we have hedronometric formulas4 for edge lengths:

a22 =

〈J2〉〈K2〉〈W2〉〈X2〉

a22 =

4〈H2〉JH2Y2Z2K〈W2〉〈X2〉

a22 =

4〈H2〉JH2Y2Z2K〈J2〉〈K2〉

(10a)

d22

=〈J2〉〈K2〉〈Y2〉〈Z2〉

d22

=4〈H2〉JH2W2X2K〈Y2〉〈Z2〉

d22

=4〈H2〉JH2W2X2K〈J2〉〈K2〉

(10b)

3Non-negativity follows from (9).4The sine formulas derive from the cosine formulas, reduced modulo the Third Law of Cosines (3).

HEDRONOMETRIC FORMULAS FOR A HYPERBOLIC TETRAHEDRON 3

and, in turn, a symmetric equality involving opposing pairs of edges5

a2d2

JH2Y2Z2KJH2W2X2K=

b2e2

JJ2Z2X2KJJ2W2Y2K=

c2f2

JK2X2Y2KJK2W2Z2K(11)

=

Ç16〈H2〉〈J2〉〈K2〉〈W2〉〈X2〉〈Y2〉〈Z2〉

å2

3. Gram Relations

Consider the face-indexed (“angle”) Gram matrix, G, and conjugate (“edge”) Grammatrix, g, detailed in Appendix A.3. Their determinants —and the conjugation ratio Γ—have the following hedronometric forms:ÁG := detG =

−4〈H2〉〈J2〉〈K2〉W2

2X2

2Y2

2Z2

2 ≤ 0 Êg := det g =−43〈H2〉3〈J2〉3〈K2〉3

〈W2〉2〈X2〉2〈Y2〉2〈Z2〉2≤ 0(12a)

Γ :=ÁGÊg =

Ç〈W2〉〈X2〉〈Y2〉〈Z2〉

4〈H2〉〈J2〉〈K2〉W2X2Y2Z2

å2

(12b)

Cofactors of entries along the diagonals of the matrices have these forms:

MW = 4[···A,B,C···] =〈W2〉2

X22Y2

2Z2

2(13a)

mW = −4Jd, e, fK = −16W22d2

2e2

2f22

= −ÇW2

4〈H2〉〈J2〉〈K2〉〈X2〉〈Y2〉〈Z2〉

å2

(13b)

while, via (70b), the remaining cofactors of G and g have these forms:

MWX = a√MWMX =

Äa2

2 + a22ä√

MWMX =〈J2〉〈K2〉+ 4〈H2〉JH2Y2Z2K

W2X2 Y22Z2

2(14a)

mWX = D√mWmX =

ÄH2 − W2X2

ä 42〈H2〉2〈J2〉2〈K2〉2

〈W2〉〈X2〉〈Y2〉2〈Z2〉2(14b)

One can readily verify identities6,7 presented by Mednykh and Paskevich in [3], such as:

5A cleaner version of this result appears in (15b).6(15a) This follows from our expressions for Γ, MW , mW . M&P’s equation (2) —prior to Lemma

1— asserts equality between the ratios and positive Γ; M&P’s derivation includes factors such as (in ournotation)

√mW , but, by (69b), mW ≤ 0. Someone among us has a sign error. (15b) M&P’s Theorem 2,

which follows from (5) and (11). (15c) M&P’s Theorem 4. (15d) M&P’s Corollary 1, which follows from(15b) and (15c). The “±i”s match for a given i, but are otherwise independent.

7M&P’s result relating altitude lengths to Γ appears to be in error; see the footnote for equation (19).


MW

mW=MX

mX=MY

mY=MZ

mZ= −Γ(15a)

AD

ad=BE

be=CF

cf=√

Γ(15b)

AD − BEad− be

=BE − CFbe− cf

=CF − ADcf − ad

= −√

Γ(15c)

(A±1 D) − (B ±2 E)

(a±1 d) − (b±2 e)=

(B ±3 E) − (C ±4 F )

(b±3 e) − (c±4 f)=

(C ±5 F ) − (A±6 D)

(a±5 d) − (b±6 e)= −√

Γ(15d)

We can often (as in Section 4) use the conjugation ratio to catalyze the conversionbetween equations in edge-related lower-case symbols (a, m?, g, etc.) and ones in angle- orarea-related upper-case symbols (A, M?, G, etc.), effectively doubling the utility of eachresult. We call these conjugate relations.

For example, write P := MWMXMYMZ and p := mWmXmYmZ . Then

(16) P =

Ç〈W2〉〈X2〉〈Y2〉〈Z2〉W2

3X2

3Y2

3Z2

3

å2

p =

Ç44〈H2〉4〈J2〉4〈K2〉4 W2X2Y2Z2

〈W2〉3〈X2〉3〈Y2〉3〈Z2〉3

å2

and, again, one can readily verify these conjugate identities8 from [3, Proposition 5]:

(17) ÊgP = ËG 3↔ ÁGp = Ág 3 P 3p = ËG 8

↔ p3P = Ág 8

4. Altitudes and Pseudo-altitudes

Let w, x, y, z be altitudes to respective faces W , X, Y , Z. We have conjugate relations9

w2MW = x2MX = y2MY = z2MZ = −ÁG(18a)

w2mW = x2mX = y2mY = z2mZ = Êg(18b)

8Given the interplay of factors S := W2X2Y2Z2, T := 〈W2〉〈X2〉〈Y2〉〈Z2〉, U := 4〈H2〉〈J2〉〈K2〉, one canshow that any vanishing product of integer powers of G = −U/S2, g = −U3/T 2, P = T 2/S6, p = S2U8/T 6,

can be written (ÊGp/Êg 3)α(Pp3/Êg 8

)β = 1 for integers α, β. Abusing terminology, the relations ÊGp = Êg 3

and Pp3 = Êg 8have all possible relations of this kind in their integer span. Likewise, their conjugates. For

one relation and its conjugate to span all relations in this way, however, requires non-integer exponents.9Equation (18b) agrees with Mednykh and Pashkevich [3, Proposition 4(ii)], which appears in this note

as equation (70b). The second set of equations arises from dividing the first set through by −Γ. See (15a).


whence10

(19) wxyz =ÁG2

√MWMXMYMZ

=Êg2

√mWmXmYmZ

=42〈H2〉2〈J2〉2〈K2〉2

W2X2Y2Z2 〈W2〉〈X2〉〈Y2〉〈Z2〉By (14a) and (10), we can write

(20)w2x2M2

WX

a2=w2y2M2

WX

b2=w2z2M2

WX

c2=y2z2M2

WX

d2=z2x2M2

WX

e2=x2y2M2

WX

f2

Also,

(21) w2W22〈W2〉2 =

Êg W22〈W2〉2

mW=Êg〈W2〉2〈X2〉2〈Y2〉2〈Z2〉2

42〈H2〉2〈J2〉2〈K2〉2

(22) w2W22〈W2〉2 = w2W2

2X2

2Y2

2Z2

2(−4[···A,B,C···])

Let h, j, k be pseudo-altitudes11 between respective edge pairs (a, d), (b, e), (c, f); andlet τh, τj , τk be the corresponding twists about those pseudo-altitudes. Equation (74a)from Appendix B gives these conjugate relations

adhτh = bejτj = cfkτk =»−Êg(23a)

ADhτh =BEjτj =CFkτk =»−ÁG(23b)

Recall now from (69b) that m? is a Heronic sine product; invoking the product’s “side-angle-side” form (63), we see that equalities (18b) and (23a) have an identical structure,giving rise to a result that neatly accommodates both standard and pseudo elements.12,13

Theorem 4.1 (Law of “Side-Angle-Side-Altitude” Sines). The product

(24) sinh ( edge ) · sinh ( edge ) · sin ( angle ) · sinh ( altitude )

is a metric invariant for hyperbolic tetrahedra, with value√−Êg for any two distinct “edge”s

and the “angle” and “altitude” they determine.14

10This disagrees with Mednykh and Paskevich [3, Proposition 5(iv)], which claims that the product isÊG2/MW /MX/MY /MZ = 1/Γ.11 A pair of skew lines in hyperbolic space admits an orthogonal transversal. (See [2].) A pseudo-altitude

is such a transversal, joining lines through opposing edges of a tetrahedron. The twist is the (supplementarilyambiguous) angle between an edge and the plane of the opposing edge and their mutual pseudo-altitude.

12Mednykh and Pashkevich [3, Proposition 6] invoke the area-based form of the Heronic sine product(69b) to arrive at this alternative result

sin ( half-face ) · cosh ( half-edge ) cosh ( half-edge ) cosh ( half-edge ) · sinh ( altitude ) =√−Êg/4

for any (standard) “face”, the “edge”s surrounding it, and the “altitude” to it. No pseudo elements appear.13The analogous product for Euclidean tetrahedra is “edge · edge · sin(angle) · altitude = 6 · volume”,

which is valid for both standard and pseudo elements.14For adjacent edges, the “angle” lies between them, and the “altitude” drops to the face containing them;

for opposite edges, the “altitude” is the pseudo-altitude joining them, and the “angle” is the correspondingtwist. Note that taking the sine of a twist renders the supplementary ambiguity of the twist moot.


Moving to cosines, we expand Appendix B’s equation (74b) into conjugate relations∣∣∣ cf − be ∣∣∣ = adh |τh|∣∣∣ ad− cf ∣∣∣ = bej |τj |

∣∣∣ be− ad ∣∣∣ = cfk |τk|(25a) ∣∣∣ CF − BE ∣∣∣ = ADh |τh|∣∣∣ AD − CF ∣∣∣ = BEj |τj |

∣∣∣ BE − AD ∣∣∣ = CFk |τk|(25b)

Isolating either of h or τh in (23a) and (25a) gives the same result; that is, we find that

u = h2 and u = τh2 are the two roots in each of these conjugate quadratic equations:

u2a2d2 − u

(a2d

2+Äcf − be

ä2 − Êg) +Äcf − be

ä2= 0(26a)

u2A2D

2 − u(A

2D

2+ÄCF − BE

ä2 − ÁG)+ÄCF − BE

ä2= 0(26b)

As h2 ≥ 1 ≥ τh2 for hyperbolic cosine h and circular cosine τh, we can write specifically

h2 =a2d

2+Äcf − be

ä2 − Êg + δ

2a2d2 =

A2D

2+ÄCF − BE

ä2 − ÁG+ ∆

2A2D

2(27a)

τh2 =

a2d2

+Äcf − be

ä2 − Êg − δ2a2d

2 =A

2D

2+ÄCF − BE

ä2 − ÁG−∆

2A2D

2(27b)

for non-negative discriminants, δ and ∆, which a convenient imaginary unit allows us toexpress as Heronoic products:

(27c) δ2 = −[

2ad, 2Äcf − be

ä, 2i»−Êg ] ∆2 = −

ï2AD, 2

ÄCF − BE

ä, 2i»−ÁG ò

Note that the above implies

(28) h2 − τh2 =

√−î2ad, 2

Äbe− cf

ä, 2i√−Êgó

a2d2 =

−ï2AD, 2

ÄBE − CF

ä, 2i»−ÁGò

A2D

2

5. Orthogonal Twists and Perfect Tetrahedra

A right-angle twist about a pseudo-altitude is naturally called orthogonal. A tetrahedronis called perfect15 when all three pairs of opposing edges are orthogonally twisted abouttheir pseudo-altitudes. For example, a right-corner tetrahedron (A = B = C = π/2) isorthogonal, as is a regular tetrahedron whose faces are equilateral triangles.

By (25a) and (25b), a perfect tetrahedron’s opposite elements combine symmetrically:

ad = be = cf AD = BE = CF(29)

Using the Second Law of Cosines (2) to re-write the dihedral cosines above, we derive thishedronometric characterization of perfection:

(30) H2〈H2〉 = J2〈J2〉 = K2〈K2〉

15Also orthocentric or orthogonal. Perfect is my own term, coined decades ago, so I’ll keep it.


An orthogonal twist simplifies the quadratics in (26), providing concise expression of thelength of the corresponding pseudo-altitude; for example,

(31) τh =π

2=⇒ h =

√−Êgad

=

»−ÁGAD

The pseudo-altitudes of a perfect tetrahedron, then, have this symmetric property

h2j

2k

2=

−Êg3

a2b2c2d

2e2f

2 =−ÁG3

A2B

2C

2D

2E

2F

2(32)

=〈H2〉3〈J2〉3〈K2〉3

JH2W2X2KJH2Y2Z2KJJ2W2Y2KJJ2Z2X2KJK2W2Z2KJK2X2Y2K

The dependencies of perfection reduce the degrees of freedom in a tetrahedron’s ele-ments from six to four. That is to say, up to symmetry, a perfect tetrahedron is uniquelydetermined by the areas of its faces. In theory, we can remove all references to pseudo-facesin our formulas by solving for H, J , K in terms of W , X, Y , Z via the system of equationscomprising (30) and (3); in practice, the results seem symbolically intractable, as with this

degree-12 polynomial equation for H2:

0 = 16H212 − 64H2

11+ 24H2

10 Ä4s2X − 3

ä− 8H2

9 Ä8s3X − 7sY sZ − 42sX

ä(33a)

+ · · ·+ sÄs+ s2

Y

ä Äs+ s2

Z

ä Äs+ s2

Y + s2Z

äwhere

s := 1− W22 − X2

2 − Y22 − Z2

2 − 4W2Z2Y2Z2(33b)

sX := W2X2 + Y2Z2 sY := W2Y2 + Z2X2 sZ := W2Z2 + X2Y2

This equation and its counterparts may be helpful numerically, but what their sprawlingsymbolic structure is trying to say about the tetrahedron remains a mystery.16

6. Pseudo-Faces: Shadows and the Search for Substance

For a Euclidean tetrahedron, the pseudo-face associated with pair of opposite edges is ashadow: the quadrilateral projection of the remaining edges into any plane parallel to thatpair. Projection and parallelism are tricky concepts in hyperbolic space, so the appropriateanalog of a pseudo-face shadow is non-obvious. We consider here a seemingly-promisingconstruction.

Take a tetrahedron having vertices P , Q, R, S, with a := |PQ| and d := |RS|. Supposethe pseudo-altitude h meets PQ and RS at M and N , respectively, and define p := |MP |,q := |MQ|, r := |NR|, s := |NS|. Let O be the midpoint of the pseudo-altitude.

16Much the same is true in Euclidean space, which has its own expansive polynomials for pseudo-faceareas, but at least perfect Euclidean tetrahedra admit a quartic formula for volume in terms of face areas.


Emulating the notion of a shadowy projection into a plane parallel to a and d, let P ′,Q′, R′, S′ be the feet of perpendiculars dropped from respective points P , Q, R, S intothe plane perpendicular to h at O. Write p′ := |OP ′|, q′ := |OQ′|, r′ := |OR′|, s′ := |OS′|.

We consider the candidacy of �P ′R′Q′S′ for the geometric realization of pseudo-faceH. For simplicity, we will assume that edges a and d are orthogonally twisted aboutpseudo-altitude h, and that p = r (so that also p′ = r′).

Observe that �PMOP ′ is a Lambert quadrilateral with legs of length h2 and p′, andwith a co-leg of length p. A little trigonometry gives

p′2

=p2h2

2

1 + p2h22 p′

2=

p2

1 + p2h22

Moreover, 4P ′OR′ is an isosceles right triangle with leg length p′ and area given by

sin(4P ′OR′) =p′p′

1 + p′p′=

p2

1 + p2h

We consider three cases.

• M and N are midpoints of PQ and RS. This tetrahedron is symmetric aboutpseudo-altitude h, and we have that p = q = r = s = a2 and b = c = e =f = p2h = a2h, and that �P ′R′Q′S′ comprises four identical copies of 4P ′OR′.Consequently, sin(1

4�P′R′Q′S′) = sin(4P ′OR′) = a2

2/2/b22

= H4 by (8b). Our“mid-plane shadow” is a geometric realization of pseudo-face H!

• M = Q and N = S. This tetrahedron is an orthoscheme, with q = s = 0 andp = a = d = r; assign b := |PR|, c := |PS|, e := |QS| = h, f := |QR|, so that

b = a2h and c = f = ah. The points Q′ and S′ coincide at O, so that 4P ′OR′ is

�P ′R′Q′S′, with area given by sin(4P ′OR′) = a2/2/b22. On the other hand, as

the reader can verify, H via (8) has an incompatible, far-less-concise expression, sothat this “mid-plane shadow” is not a realization of pseudo-face H.

• h = 0. This tetrahedron is degenerate, with coplanar vertices. We can abandonthe orthogonal twist condition, and the supposition p = r, but let us assume thatedges a and d themselves —not merely the lines determined by them— intersectas diagonals of convex quadrilateral �PRQS. (Non-intersecting circumstances arehandled similarly.) Then the quadrilateral is the union of the tetrahedron’s facesin two ways: |�PRQS| = Y + Z = W + X. But also, A = D = π, so that

H2 = cos (Y2 + Z2) = cos (W2 +X2). The quadrilateral —arguably, the shadow ofthe degenerate tetrahedron into a plane “perpendicular” to pseudo-altitude h— re-alizes pseudo-face H. Moreover, the bow-tie quadrilateral �PSRQ, with diagonals(say) b and e, has area17 |�PSRQ| = |Z −X| = |W − Y |, whereas B = E = 0 and

J2 = cos (Z2 −X2) = cos (W2 − Y2); the bow-tie realizes pseudo-face J . Bow-tie

17We take a bow-tie’s area as the absolute value of the difference of its triangular regions, as is consistentwith tracing the four edges of the quadrilateral in order, traversing the triangles in different orientations.


�PRSQ, likewise, realizes K. Note that, while the bow-ties may be consideredshadows of the tetrahedron, and in a plane (degenerately) “parallel” to pairs ofopposing edges, they are not in a plane perpendicular to the pseudo-altitudes cor-responding to J and K.

The symmetric case demonstrates that a straightforward construction can realize apseudo-face as a “shadow”, and the degenerate case further correlates pseudo-faces withshadows; the orthoscheme case, however, indicates that there is work yet to do in devising ageneral construction of pseudo-face shadows, if indeed pseudo-faces are shadows in general.

7. Special Tetrahedra

7.1. Regular Tetrahedra, A = B = C = D = E = F . If the dihedral angles of ahyperbolic tetrahedron match (so that the figure is necessarily perfect, by (29)), then itsside lengths match (a = b = c = d = e = f), its face areas match (W = X = Y = Z), andits pseudo-face areas match (H = J = K). Moreover, any one metric —angle measure A,side length a, face area X, pseudo-face area H— completely determines the tetrahedron.A sampling of relations among these metrics follows.

The most straightforward route to relating A and X passes through the realms of hyper-bolic and spherical trigonometry. We recall that, by the definition of hyperbolic area, theplane angle θ at any vertex of our equilateral faces satisfies X = π−3θ, so that θ = π3−X3.The dihedral angle A relates to face angle θ by the spherical Law of Cosines for Sides:

(34) A =θ − θθθθ

=θ

1 + θ=

cos(π3 −X3)

1 + cos (π3 −X3)=

U3

1 + U3

where U := π −X. By the Second Law of Cosines (2),

(35) H2 = X22

+X22A =

1

2

(1 + 4U3 − 4U3

2)

=⇒ H4 =1

2

Ä2U3 − 1

äWe can also derive

X22

= H42 Ä

3 + 2H4

äX2

2=Ä1 +H4

ä2 Ä1− 2H4

ä(36a)

〈X2〉 = 4H43 Ä

1 +H4

ä/»

1− 2H4 〈H2〉 = 4H43 Ä

1 +H4

ä(36b)

JH2, X2, X2K = 2H44 Ä

1 +H4

ä(36c)

(36d) a22 =

1

1− 2H4a2

2 =2H4

1− 2H4a2

2 = 2H4 = 2U3 − 1 =3A− 1

1− A

7.2. Isohedral Right-Corner Tetrahedra, A = B = C = π/2, D = E = F . Here,three areas match (X = Y = Z), all pseudo-face areas match (H = J = K), and triples

of side lengths match (a = b = c and d = e = f , with d = a2). The First Law of Cosinesrelates hypotenuse-face W and leg-face X

(37) W2 = X23 −X2

3=

1

2

Ä2 +X

ä»1−X W2

2=

1

4X

2 Ä3 +X

ä


but the spherical Law of Cosines applied to plane angle θ := π3 −W3 in hypotenuse Wand (acute) plane angle φ := π4 −X2 in leg X gives something more direct:

(38) 0 = A =θ − φφφφ

=⇒ φ2 = θ =⇒ X = 2(π3 −W3)− 1

From the Second Law of Cosines (2),

(39) X22

= H2 = W2X2 +W2X2D

so that

(40) D =T2»

1 + T22

=

»2U3

2U6

D =1»

1 + T22

=

√2

2U6

where T := π2 −X and U := π −W . Also,

〈H2〉 = X2X23 〈W2〉 = X2

3 〈X2〉 = X2X23(X2 −X2)(41a)

JH2X2X2K =1

4X2

4 JH2W2X2K =1

2X2

2X2

4(41b)

a22 =

X2

X2 −X2

a22 =

X2

X2 −X2

a22 = ”X2(41c)

d22

=1

1−Xd2

2=

X

1−Xd2

2= X = 2U3 − 1 =

3D2 − 1

D2(41d)

8. Volume

Volume seems the least-scrutable measurement in hyperbolic space, even for tetrahedra.The principle formula is a differential equation from Schlafli [4] in 1950:

(42) dV = −a2 dA− b2 dB − c2 dC − d2 dD − e2 dE − f2 dF

Derevnin and Mednykh’s 2005 solution [1] takes the form of a monolithic integral:

(43) V = −1

4

∫ α+β

α−βlog

cos A+B+C+θ2 cos A+E+F+θ

2 cos D+B+F+θ2 cos D+E+C+θ

2

sin A+D+B+E+θ2 sin B+E+C+F+θ

2 sin C+F+A+D+θ2 sin θ

2

dθ

where

α := atanpαqα

β := atanpβqβ


and

pα := sin(A+B + C +D + E + F ) + sin(A+D) + sin(B + E) + sin(C + F )

+ sin(D + E + F ) + sin(D +B + C) + sin(A+ E + C) + sin(A+B + F )

qα := − (cos(A+B + C +D + E + F ) + cos(A+D) + cos(B + E) + cos(C + F )

+ cos(D + E + F ) + cos(D +B + C) + cos(A+ E + C) + cos(A+B + F ))

pβ :=√p2α + q2

α − p2β = 2

√−detG

qβ := 2 (sinA sinD + sinB sinE + sinC sinF )

We note that

p2α + q2

α = p2β + q2

β(44)

= 8Ä1 + ABC + AEF + DBF + DEC

+ADBE + BECF + CF AD +ADBE +BECF + CFADä

Also, with considerable trigonometric effort, one finds that the difference of the numer-ator and denominator in the logarithm’s argument reduces thusly:

(45) cos(·) cos(·) cos(·) cos(·)− sin(·) sin(·) sin(·) sin(·) = qβ − (pα sin θ + qα cos θ)

The values θ = α± β are roots of this expression and thus are also roots of the integrand.

Unfortunately —unlike in the Euclidean case18— these formula, in general, resist directhedronometric re-parameterization in terms of face and pseudo-face areas. (Below, weexamine a couple of less-resistant special cases, and we discuss the complexity of the generalcase.) For the most part, the best we can do currently is invoke the Second Law of Cosines(2) to convert areas into dihedral angle measures, and substitute those measures into theDerevnin-Mednykh formula (43).

8.1. Regular Tetrahedra, A = B = C = D = E = F . Since a = b = c = d = e = f andA = B = C = D = E = F , we have dV = −6 a2 dA. We can express this differential interms of angle A, face area X, or pseudo-face area H:

(46a) −dV6

= atanh

√3A− 1

1− AdA = atanh

»2U3 − 1

dA

dXdX = atanh

»2H4

dA

dHdH

dA

dX= −

”U6

3»

1 + 3U3

dA

dH= − L8

2Ä3 + 2L4

ä(46b)

18The volume, V , of a Euclidean tetrahedron with faces W , X, Y , Z and pseudo-faces H, J , K satisfiesa formula bearing a striking resemblance to the Third Law of Cosines in hyperbolic space (3).

81V 4 = 2W 2X2Y 2 + 2W 2Y 2Z2 + 2W 2Z2X2 + 2X2Y 2Z2 +H2J2K2

−H2(W 2X2 + Y 2Z2)− J2(W 2Y 2 + Z2X2)−K2(W 2Z2 +X2Y 2)


with U := π −X and L := 2π −H. So

V = −6

∫ A

acos 13

atanh

3 cos θ − 1

1− cos θdθ(47a)

= 2

∫ X

0

tan(π6 − ξ6)»1 + 3 cos(π3 − ξ3)

atanh»

2 cos(π3 − ξ3)− 1 dξ(47b)

= 3

∫ H

0

sin(π4 − η8)

3 + 2 sin η4atanh

√2 sin η4 dη(47c)

8.2. Isohedral Right-Corner Tetrahedra, A = B = C = π/2, D = E = F . Limitingourselves (and our paths of integration) to the universe of right-corner tetrahedra withA = B = C = π/2, we can assert dA = dB = dC = 0; then, with D = E = F , we havedD = dE = dF , so that dV = −3 d2 dD. We can write:

−dV3

= atanh

»3D2 − 1

DdD = atanh

√X

dD

dXdX = atanh

»2U3 − 1

dD

dWdW(48a)

(48b)dD

dX=

−T2

4(1 + T2

2) dD

dW=−”U6

6»U3

with T := π2 −X and U := π −W . So

V = −3

∫ D

acos√

13

atanh

√3 cos2 θ − 1

sin θdθ(49a)

=3

2

∫ X

0

sin(π4 − ξ2)

1 + cos2(π4 − ξ2)atanh

√sin ξ dξ(49b)

=1

2

∫ W

0

tan(π6 − ω6)»cos(π3 − ω3)

atanh»

2 cos(π3 − ω3)− 1 dω(49c)

8.3. General Tetrahedra. A key complication in deriving a face-based formula for tetra-hedral volume is that our seven hedronometric parameters —faces areas W , X, Y , Z andpseudo-face areas H, J , K— aren’t independent (and shouldn’t be, as there are only six de-grees of freedom in determining a tetrahedron). Nevertheless, the “differentialized” SecondLaw of Cosines (2) converts the Schlafli angular differentials to hedronometric ones:

4»

JH2Y2Z2K dA = H2 dH − Z2 − Y2H2

Y2dY − Y2 − Z2H2

Z2dZ(50a)

4»

JH2W2X2K dD = H2 dH−X2 − W2H2

W2dW − W2 − X2H2

X2dX


while the differentialized Third Law of Cosines (3) expresses the dependency among thehedronometric differentials:

〈W2〉W2 dW + 〈X2〉X2 dX + 〈Y2〉Y2 dY + 〈Z2〉Z2 dZ(50b)

= 〈H2〉H2 dH + 〈J2〉J2 dJ + 〈K2〉K2 dK

While we hope for a monolithic integral in the spirit of Derevnin-Mednykh that respectsthe inherent symmetries of the face and pseudo-face area parameters, for now we investigatean asymmetric solution.

8.3.1. Using an auxiliary parameter. Consider the universe of tetrahedra with constant W ,X, Y , Z, H. Variable pseudo-face areas J and K have a dependency via the Third Lawof Cosines (3). We can interpret the Law as a quadratic in J2 and K2 that describes anellipse (rotated by π/4 from the coordinate axes), allowing us to parameterize the values interms of a single (and remarkably well-suited) “polar” angle, θ. Translating and rotatingthe ellipse, we arrive at these relations, where we indicate dependence of J and K on θwith a superscript:

Jθ2 H22

= W2Y2 + Z2X2 − H2

ÄW2Z2 + X2Y2

ä(51a)

+ 4 cos(θ −H4)»

JH2W2X2K JH2Y2Z2K

Kθ2 H2

2= W2Z2 + X2Y2 − H2

ÄW2Y2 + Z2X2

ä(51b)

− 4 cos(θ +H4)»

JH2W2X2K JH2Y2Z2K

Corresponding formulas for 〈Jθ2 〉 and 〈Kθ2〉 reduce nicely

〈Jθ2 〉 H2 = 4 sin(θ +H4)»

JH2W2X2K JH2Y2Z2K(52a)

〈Kθ2〉 H2 = 4 sin(θ −H4)

»JH2W2X2K JH2Y2Z2K(52b)

and provide the insight that, given the non-negative nature of each other factor in theproducts, we must have sin(θ ±H4) ≥ 0, so that we can take H4 ≤ θ ≤ π −H4.

Abusing notation slightly, we define

(53) 〈Hθ2 〉 := H2 + Jθ2 K

θ2 − W2X2 − Y2Z2

and observe that 〈Hθ2 〉 does not appear to admit insightful simplification, so that its be-

havior with respect to θ remains unclear. Numerical experiments in Mathematica suggestthat prudent (re-)labeling of face areas can guarantee 〈Hθ

2 〉 to be strictly positive acrossthe domain determined by 〈Jθ2 〉 and 〈Kθ

2〉 in the preceding paragraph. The validity of thisphenomenon would help streamline Theorem 8.2; unfortunately, proof is elusive, so westate it here as a conjecture:

Conjecture 8.1. Given a hyperbolic tetrahedron, one can assign W , X, Y , Z to its faceareas —and H to the corresponding pseudo-face area— in such a way that 〈Hθ

2 〉 > 0 forall H4 < θ < π −H4.


The relations (52) provide compact representations of the trigonometric functions of θ:19

4θ»

JH2W2X2K JH2Y2Z2K = H4(〈Jθ2 〉+ 〈Kθ2〉)(54a)

4θ»

JH2W2X2K JH2Y2Z2K = H4(〈Jθ2 〉 − 〈Kθ2〉)(54b)

θ = ”H4〈Jθ2 〉+ 〈Kθ

2〉〈Jθ2 〉 − 〈Kθ

2〉(54c)

With most face areas constant, the differentials of Jθ and Kθ are straightforward:20

Jθ2H22dJθ = 8 sin(θ −H4)

»JH2W2X2K JH2Y2Z2K dθ(55a)

Kθ2H2

2dKθ = −8 sin(θ +H4)

»JH2W2X2K JH2Y2Z2K dθ(55b)

Moreover, the various angular differentials (as in (50a)) simplify. Of course, dA and dDvanish outright; otherwise, we have, for instance,

(55c) dB =Jθ2 dJ

θ

4»

JJθ2Z2X2K=

2»

JH2W2X2K JH2Y2Z2K

H22

sin(θ −H4)»JJθ2Z2X2K

dθ

We may therefore give this formula for the Schafli volume differential (42):(56a)

dV =

»JH2W2X2K JH2Y2Z2K

H22

− sin(θ −H4)

Ñbθ»

JJθ2Z2X2K+

eθ»JJθ2W2Y2K

é+ sin(θ +H4)

Ñcθ»

JKθ2X2Y2K

+fθ»

JKθ2W2Z2K

é dθ

for non-negative bθ, cθ, eθ, fθ satisfying“bθ22= 4

JJθ2Z2X2K〈Hθ

2 〉sin(θ +H4)

sin(θ −H4)“eθ22

= 4JJθ2W2Y2K〈Hθ

2 〉sin(θ +H4)

sin(θ −H4)(56b) “cθ22

= 4JKθ

2X2Y2K〈Hθ

2 〉sin(θ −H4)

sin(θ +H4)fθ2

2= 4

JKθ2W2Z2K〈Hθ

2 〉sin(θ −H4)

sin(θ +H4)

Now, note that our constant areas make the variable aspect of ÁG, the tetrahedron’sGram determinant (see (12a)), proportional to 〈Hθ

2 〉〈Jθ2 〉〈Kθ2〉, which is in turn propor-

tional to 〈Hθ2 〉 sin(θ+H4) sin(θ−H4). Consequently, ÁG vanishes —indicating a degenerate

tetrahedron— at the endpoints H4 and π−H4 of the bounds on θ imposed after (52); in theabsence of proof of Conjecture 8.1, we recognize the possibility of degenerate tetrahedradetermined by some θ = θ? between these endpoints.

Interpreting the above appropriately, we find that we can compute the volume of atetrahedron via a hedronometric formula that integrates through our universe of tetrahedra

19Squaring and adding the sine and cosine relations re-confirms equation (6).20With (52), these are evidently consistent with the differentialized Third Law of Cosines (50b).


with constant W , X, Y , Z, H, from the degenerate form (corresponding to θ = H4, orpossibly θ = θ?) to our target (corresponding to the particular θ satisfying (54), withJθ = J and Kθ = K).

Theorem 8.2. A hyperbolic tetrahedron with face areas W , X, Y , Z and correspondingpseudo-face areas H, J , K has volume, V , given by(57)

V =

»JH2W2X2K JH2Y2Z2K

H22

∫ Θ

Φ

− sin(θ −H4)

Ñbθ»

JJθ2Z2X2K+

eθ»JJθ2W2Y2K

é+ sin(θ +H4)

Ñcθ»

JKθ2X2Y2K

+fθ»

JKθ2W2Z2K

é dθ

with

• non-negative bθ, cθ, eθ, fθ satisfying (56b);

• H4 ≤ Θ ≤ π −H4, such that θ = Θ satisfies (54) with Jθ = J and Kθ = K;

• Φ is the largest root of 〈Hθ2 〉〈Jθ2 〉〈Kθ

2〉 = 0 such that H4 ≤ Φ ≤ Θ.21

9. What is the Pythagorean Theorem for Right-Corner 4-Simplices?

In Euclidean any-dimensional space, our understanding of the Pythagorean Theorem forright-corner simplices is complete: always, the square of the content of the hypotenuse isequal to the sum of the squares of the contents of the legs. In hyperbolic geometry, ourknowledge seems embarrassingly limited:

Dimension 2 : cosh hyp = cosh leg1 cosh leg2

Dimension 3 : coshyp

2= cos

leg1

2cos

leg2

2cos

leg3

2− sin

leg1

2sin

leg2

2sin

leg3

2

Dimension 4+ : ??? = ???

Consider the case of a right-corner simplex in hyperbolic 4-space: as Section 8 indicates,any relation between the volume of the simplex’s hypotenuse-tetrahedron and the volumesof its leg-tetrahedra involves comparing complicated integrals. Even the most-special ofspecial cases (below) defies this author’s attempts at comprehension.

21By Conjecture 8.1, appropriate face and pseudo-face labeling would allow us to take Φ = H4.


9.1. Isosceles Right-Corner 4-Simplices. Consider right-corner 4-simplex such thateach of its four congruent legs are isohedral right-corner tetrahedra determined by dihedralangles (A,B,C,D,E, F ) = (π2 ,

π2 ,

π2 , P, P, P ); its hypotenuse is a regular tetrahedron with

congruent dihedral angles (Q,Q,Q,Q,Q,Q). One can verify that

(58)1

3≤ cos2 P = χ = cosQ ≤ 1

2for a parameter χ that we use to write tantalizingly-similar formulas for the volume H ofthe hypotenuse (via (47a)) and volume L of each leg (via (49a)):

H := −3

∫ acos√χ

acos√

13

atanh

3 cos2 θ − 1

1− cos2 θdθ =

3

2

∫ χ

13

1»t (1− t)

atanh

3t− 1

1− tdt(59a)

L := −6

∫ acosχ

acos 13

atanh

3 cos θ − 1

1− cos θdθ = 6

∫ χ

13

1√1− t2

atanh

3t− 1

1− tdt(59b)

Although Pythagoras beckons, a connection between these volumes remains elusive.Indeed, the rather exotic —but not unfamiliar22— values for χ = 1/2 (defining a simplexwith a quadrupally-asymptotic hypotenuse and triply-asymptotic legs)

(60) H? :=∞∑k=1

1

k2sin

πk

3= 1.01494 . . . L? :=

1

2

∞∑k=1

1

k2sin

πk

2= 0.45978 . . .

suggest that the corresponding Pythagorean Theorem is quite non-trivial.

22L? is half of Catalan’s constant, and H? is known, e.g., as the maximum of the Clausen function, Cl2.


A. Standard Notation

This note (in the tradition of most notes on hedronometry by this author) reserves much—in fact, just-over half— of the Latin alphabet for specific notational use, as describedbelow. Generally, lower-case letters indicate “trigonometric” elements (sides and planarangles, and their measures), while upper-case letters indicate “hedronometric” elements(faces and dihedral angles, and their measures); the lower- and upper-case counterparts ofa letter indicate elements in a natural correspondence.

We avoid the use of integer subscripts as indices; rather, we reserve them for compactrepresentation of fractions in formulas. (See Section A.2.)

Note. Without fear of confusion, a symbol represents both a geometric element and itsmeasure. (E.g., edge a has length a; angle B has measure B; face Z has area Z.)

A.1. Tetrahedral Elements (a, b, c, d, e, f ;W,X, Y, Z;H,J,K). A tetrahedron’s edges a,b, c concur at a vertex and oppose respective edges d, e, f . The dihedral angles along theedges are, respectively, A, B, C and D, E, F .

The tetrahedron’s faces are

W := 4def X := 4dbc Y := 4aec Z := 4abf

When discussing “right-corner” tetrahedra, we take edges a, b, c to be mutually-orthogonal,and angles A, B, C to be right angles. This makes W the figure’s “hypotenuse-face” andX, Y , Z its right-triangular “leg-faces”.

We sometimes (as in Section 4) assign symbols w, x, y, z to the altitudes to faces W ,X, Y , Z, but other times revert using these symbols for generic quantities and coordinatesin the traditional way.

With regard to pseudo-elements23 (and their measures), we make the following associa-tions between various pairs of traditional elements and trios comprising a pseudo-face (H,J , K), a pseudo-altitude (h, j, k), and a twist angle (τh, τj , τk):

a, d ↔ A,D ↔ (Y,Z), (W,X) ↔ H,h, τhb, e ↔ B,E ↔ (Z,X), (W,Y ) ↔ J, j, τjc, f ↔ C,F ↔ (X,Y ), (W,Z) ↔ K, k, τk

A.2. Trigonometry. To remove visual clutter from trigonometric expressions, we adoptthis “Morse Code” (and hat) notation:

x := coshx and x := sinhx and x := tanhx , for lengths x

θ := cos θ and θ := sin θ and θ := tan θ , for angles θ

X := cosX and X := sinX and “X := tanX , for areas X

Context should make clear whether hyperbolic or circular functions are intended.

23See footnotes 1 and 11 for definitions of pseudo-face and pseudo-altitude.


“Half-measures” (half-angles, half-lengths, half-areas) and even “quarter-measures” ap-pear so often in formulas that writing (and reading) them as fractions can be a strain. Fortypographic simplicity, we write “xn” for “x/n”.

Finally, we define the “Heronic sine and cosine products”24 via

Jx, y, zK := (x2 + y2 + z2) (−x2 + y2 + z2) (x2 − y2 + z2) (x2 + y2 − z2)(61a)

=1

4

Ä2xyz − x2 − y2 − z2 + 1

ä[···x, y, z···] := (x2 + y2 + z2) (−x2 + y2 + z2) (x2 − y2 + z2) (x2 + y2 − z2)(61b)

=1

4

Ä2xyz + x2 + y2 + z2 − 1

äThese take their name and symbology from the so-called and -symbolized (by me) alge-

braic Heronic product

(62) [x, y, z] := (x2 + y2 + z2) (−x2 + y2 + z2) (x2 − y2 + z2) (x2 + y2 − z2)

that gives Heron’s formula for the square of the area of a Euclidean triangle with edges x,y, z. This note occasionally uses (62) to abbreviate the characteristic four-part product.25

We’ll note the “side-angle-side” form of the Heronic (hyperbolic) sine product of edgesx, y, z of a hyperbolic triangle, with angles θ, φ, ψ opposite respective sides x, y, z:

(63) Jx, y, zK =1

4y2z2θ

2=

1

4z2x2φ

2=

1

4x2y2ψ

2

This equation inspires the Law of “Side-Angle-Side-Altitude” Sines (Theorem 4.1).

A.3. Matrices, Determinants, Conjugation Ratio, and Minors (G,M). Define thetraditional (“angle”) Gram matrix, G, and conjugate (“edge”) Gram matrix, g, with rowsand columns indexed by faces W , X, Y , Z (in order).26

(64) G :=

1 −D −E −F−D 1 −C −B−E −C 1 −A−F −B −A 1

g :=

−1 −a −b −c−a −1 −f −e−b −f −1 −d−c −e −d −1

24The equivalence of factored and expanded forms holds for both circular and hyperbolic functions.25See, for instance, (8a) and (27c). Note that, because I incorporate the halving of arguments in building

the factors (to provide uniformity with the sine and cosine products), I must double arguments —or elsemultiply the result by 16— when I simply want to use the “[·]” notation to compress these four-fold products.

26The angles in a row (column) of G surround the index-face, while remaining angles oppose that face;conjugately, the edges in a row (column) of g oppose the index-face, while the remaining edges surround it.


The determinants of these matrices are as follows:

ÁG := detG =

â1− A2 − B2 − C2 − D2 − E2 − F 2

−2ABC − 2AEF − 2DBF − 2DEC

+A2D2 + B2E2 + C2F 2

−2ADBE − 2BECF − 2CF AD

ì≤ 0(65a)

Êg := det g =

Ü1− a2 − b2 − c2 − d2 − e2 − f2

+2dbc+ 2aec+ 2abf + 2def

+a2d2 + b2e2 + c2f2 − 2adbe− 2adcf − 2becf

ê≤ 0(65b)

and we define (what I call) the conjugation ratio27 thusly:

(66) Γ :=ÁGÊg

For face-indices P and Q, define the (PQ)-th cofactors, MPQ and mPQ:

(67) MPQ := (−1)P+Q minorPQ G mPQ := (−1)P+Q minorPQ g

Thus,

MWX = A2D + A

ÄBE + CF

ä+ BF + CE(68a)

mWX = −ad2+ dÄbe+ cf

ä−Äbf + ce

ä(68b)

When P = Q, we use a single index. For instance,

MW = −2ABC − A2 − B2 − C2 + 1 = −4[···A,B,C···](69a)

mW = −2def + d2 + e2 + f2 − 1 = −4Jd, e, fK = −16W22d2

2e2

2f22

(69b)

We note that Mednykh and Pashkevich [3, Proposition 4] provide cofactor formulas forthe lengths of a tetrahedron’s edges and altitudes:

(70a) a2 =M2WX

MWMXd2 =

M2Y Z

MYMZw2 =

−ÁGMW

=Êg

mW

Likewise, we have cofactor formulas for dihedral angles:

(70b) D2 =m2WX

mWmXA2 =

m2Y Z

mYmZ

27Note the use of upper-case gamma as a reminder that the numerator of the conjugation ratio contains

the upper-cased —that is, angle-based— determinant, ÊG. I’m tempted to introduce “γ” to denote thereciprocal ratio, just for the sake of completeness.


B. Coordinates

In coordinatized hyperbolic space,28 let P (xp, yp, zp) be such that

• zp measures the signed length of a perpendicular dropped from P to a point Pxy inthe xy-plane;• yp measures the signed length of a perpendicular dropped from Pxy to a point Px

on the x-axis; and• xp measures the signed distance from Px to the origin.

Every coordinate system has a distance formula, and here’s the one for this context:

Theorem B.1 (The Distance Formula in Coordinatized Hyperbolic Space). The distance,|AB|, from point A(xa, ya, za) to point B(xb, yb, zb) is given by

(71) cosh |AB| = zazb ( yayb ( xaxb − xaxb )− yayb )− zazb

We wish to describe a coordinatization of a tetrahedron. Let the tetrahedron haveelements with standard29 edge, (pseudo)face, and pseudo-altitude labels, and let verticesP , Q, R, S be opposite respective faces W , X, Y , Z. We align pseudo-altitude h with thex-axis and an adjacent edge PQ with the y-axis; say that the pseudo-altitude separatesPQ into segments of length aP and aQ, and RS into segments of length dR and dS . Wewrite

P (0,−aP , 0) Q(0, aQ, 0) R(h,−yR,−zR) S(h, yS , zS)

where

tanh yR = tanh dR cos τh tanh yS = tanh dS cos τh

sinh zR = sinh dR sin τh sinh zS = sinh dS sin τh

for τh the angular twist of edge RS out of the xy-plane.This arrangement, and the Distance Formula, provide these formulas for edge lengths.30

a = cosh |PQ| = aP aQ + aPaQ d = cosh |RS| = dRdS + dRdS

b = cosh |PR| = aP dRh− aPdRτh e = cosh |SQ| = aQdS h− aQdS τh(72)

c = cosh |PS| = aP dS h+ aPdS τh f = cosh |QR| = aQdRh+ aQdRτh

With these, the conjugate Gram determinant, Êg, (see (65b)) collapses from over a dozenterms to just one:

(73) Êg = −a2d2h

2τh

2

28Here, we enlist x, y, z as symbols for standard coordinates instead of tetrahedral altitudes.29See Appendix A.30For instance,

b = zR(aP yR

(h− 0

)− aP yR

)− 0 = aP (yRzR)h− aP (yRzR) = aP dRh− aP dRτh

where the simplifications yRzR = dR and yRzR = dRτh follow from the trigonometry of the right trianglewith hypotenuse dR, legs yR and zR, and angle τh.


whence, all un-squared sines being non-negative while Êg being non-positive,

(74a) a d h τh =»−Êg

Also, one readily verifies this identity:

(74b) a d h |τh| =∣∣∣ cf − be ∣∣∣

where the supplementary ambiguity of τh necessitates the use of absolute values.With a bit of work, one can extract the sub-segment lengths aP , etc, writing

sinh 2aP =2a (mWX + amX)√

−î2ad, 2

Äbe− cf

ä, 2i√−Êgó(75a)

sinh 2aQ =2a (mWX + amW )√

−î2ad, 2

Äbe− cf

ä, 2i√−Êgó(75b)

so that

(76)sinh 2aPsinh 2aQ

=mWX + amX

mWX + amW

References

[1] Derevnin, D. A., and Mednykh, M. G. “A formula for the volume of a hyperbolic tetrahedron”.Communications of the Moscow Mathematical Society, Vol. 60, No. 2, pp. 346-348, 2005. http:

//mathlab.snu.ac.kr/~top/articles/MednykhDerevninHyperbolicTetrahedron.pdf

[2] Horvath, Akos. “Skew Lines in Hyperbolic Space”. Periodica Polytechnica Ser. Mech. Eng., Vol. 47, No.1, pp. 2530, 2003. http://www.pp.bme.hu/me/2003_1/pdf/me2003_1_03.pdf

[3] Mednykh, A. D., and Pashkevich, M. G. “Elementary Formulas for a Hyperbolic Tetrahedron”. SiberianMathematical Journal, Vol. 47, No. 4, pp. 687-695, 2006. http://mathlab.snu.ac.kr/~top/articles/Elementary_Formulae_for_tetrahedron.pdf

[4] Schlafli, L. “Theorie der vielfachen Kontinuitat”. Gesammelte mathematishe Abhandlungen, Basel:Birkhauser, 1950.

HEDRONOMETRIC FORMULAS FOR A HYPERBOLIC TETRAHEDRON

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