Generalised vector products and metrical trigonometry of a tetrahedron
aa r X i v : . [ m a t h . M G ] J un Generalised vector products and metrical trigonometry of atetrahedron
G A NotowidigdoSchool of Mathematics and StatisticsUNSW, Sydney, Australia N J WildbergerSchool of Mathematics and StatisticsUNSW, Sydney, Australia
Abstract
We study the general rational trigonometry of a tetrahedron, based on quadrances, spreadsand solid spreads, using vector products associated to an arbitrary symmetric bilinear form over ageneral field, not of characteristic two. This gives us algebraic analogs of many classical formulas,as well as new insights and results. In particular we derive original relations for a tri-rectangulartetrahedron.
Keywords: scalar product; vector product; symmetric bilinear form; tetrahedron; rational trigonom-etry; affine geometry; projective geometry
In this paper, which is a follow-up to
Generalised vector products applied to affine and projective ra-tional trigonometries in three dimensions (henceforth referred to as [23]), we will apply the frameworkof generalised scalar and B -vector products developed there to set up a framework for the ratio-nal trigonometry of a general tetrahedron in three-dimensional affine space with a general metricalstructure defined over an arbitrary field, not of characteristic two.The metrical structure of the tetrahedron is a much studied topic, since at least the time ofTartaglia, who was the first to give a formula for the volume in terms of the (squared) lengths of thesides, a formula also found by Euler and generalized by Cayley and Menger. This classical treatment ofthe trigonometry of the tetrahedron reached a high point with the large scale summary of Richardson([24]) for the Euclidean case, involving not just lengths and angles of faces but also dihedral anglesbetween faces and solid angles at vertices and transversal lengths between opposite edges. Since locallyat a vertex a tetrahedron determines a tripod, there was also a projective aspect which connectsnaturally with spherical trigonometry ([29]). In modern times both computational geometry in threedimensions ([14]) and the finite element method in space ([5]) have utilized tetrahedral meshes andtheir measurements. And of course there is naturally interest in generalizations to metrical formulasfor more general polytopes, as in the rigidity results of Sabitov ([25]), and to explorations of analogsof trigonometric concepts and laws to simplices in higher dimensions, for example ([12].) But it isclear that our understanding of the n -dimensional story is still very much in its early days.1long with this classical mostly Euclidean orientation, since the 19th century development of non-Euclidean geometry, the question of how trigonometry extends to hyperbolic and spherical settingshas been of keen interest, and an important issue is the three-dimensional situation for hyperbolictetrahedra, where even the volume formula originally due to Lobachevsky is much more complicatedthan in the Euclidean case, and much work has been done in understanding and extending this resultfor example the work of ([18], [9] and [20]).With the advent of rational trigonometry ([31], [32]) the possibility emerged to reframe trigonom-etry in a purely algebraic fashion; where lengths and angles become secondary to purely algebraicconcepts defined in terms of a symmetric bilinear form. A projective version of this theory appliesalso to allow a recasting of hyperbolic geometry ([34], [35], [36] and [2]) which also extends the subjectto general fields, including finite fields.In [23] we laid out a three-dimensional version of this theory which extends classical formulasof Lagrange, Binet, Cauchy and others to this more general setting valid for arbitrary fields (not ofcharacteristic two) and to general quadratic forms. In this follow up paper we intend to apply thistechnology to completely reformulating the classical trigonometry of a tetrahedron, using rationalanalogs also of dihedral angles and solid spreads. We hope that this will provide a bridge also toa projective version which describes the trigonometry of spherical and hyperbolic tetrahedra, and infact some key projective formulas for that project are already contained in this present work.Suppose that V is the three-dimensional vector space, over a field F not of characteristic 2,consisting of row vectors v = ( x, y, z ) and that B is a symmetric non-degenerate 3 × B = 0. Then we may define a symmetric bilinear form, or B -scalar product , by the rule thatfor any two vectors v and w v · B w ≡ vBw T . This number is always an element of the field F .The B -quadrance of a vector v is Q B ( v ) ≡ v · B v and a vector v is B -null precisely when Q B ( v ) = 0 . If v and w are non-null vectors, then the B -spread between them is the number s B ( v, w ) ≡ − ( v · B w ) Q B ( v ) Q B ( w ) . In Euclidean geometry, the quadrance and spread are typically thought of respectively as the squareddistance and the squared sine of an angle; in our framework, we are using quadrances and spreads toallow for extensions of Euclidean geometry to arbitrary symmetric bilinear forms over a general fieldmore easily.In [23], we extended the definition of vector products in Euclidean three-dimensional vector spacealso to this more general three-dimensional situation. Given the usual (Euclidean) vector product v × w , for two vectors v and w in V , we define the B -vector product of v and w to be v × B w ≡ ( v × w ) adj B B = (det B ) B − is the adjugate of the invertible matrix B .After a short review of the properties of B -scalar and B -vector products which were proven in [23],we define the fundamental trigonometric invariants in three-dimensional affine space, denoted by A ,where V , as given above, is its associated vector space. These include the B -quadrance and B -spread,as well as the B -quadrea of a triangle in A which extends the definition of quadrea in [31].While these three quantities featured prominently in [23], this paper introduces four new trigono-metric invariants: the B -quadrume , the B -dihedral spread , the B -solid spread and the B -dualsolid spread . The B -quadrume, which is a quadratic version of volume in our framework, has a closeconnection to the Cayley-Menger determinant, as seen in [4], [11, pp. 285-289] and [27, pp. 124-126].The latter three quantities, which are analogs of angles between two planes or solid angles betweenthree lines in our framework, have origins in projective geometry (see [32] and [33]) and thus high-light the power of using generalised vector products to explain affine rational trigonometry in threedimensions.We can then compute these quantities for a general tetrahedron in A , for which we can thendiscover interesting algebraic relations. For a tetrahedron A A A A with points A , A , A , A , wewill denote, for indices 0 ≤ i < j < k ≤ • by Q ij the B -quadrance between two points A i and A j ; • by A ijk the B -quadrea of the triangle with points A i , A j and A k ; • by V its B -quadrume; • by E ij the B -dihedral spread between the planes through any three points intersecting at a linethrough A i and A j ; • by S i the B -solid spread between three concurrent lines at A i ; and • by D i the B -dual solid spread between three concurrent lines at A i .The quantity R ≡ V A A A A is a key component of our study, whose geometric meaning is yet to be fully understood; we will callthis the Richardson constant . Here are some examples of relations we obtain: E E Q Q = E E Q Q = E E Q Q = RD A = D A = D A = D A = R S S S Q Q Q = S S S Q Q Q = S S S Q Q Q = S S S Q Q Q = V Q Q Q Q Q Q . B -quadrances between opposite lines of a tetrahedron, which are skew, as well as presentingpotential directions for further research. We begin with the framework of the three-dimensional vector space V over a field F not of character-istic 2, consisting of row vectors v = ( x, y, z ) , with the usual arithmetical structures of vector additionand subtraction, together with scalar multiplication. B -scalar product A 3 × B ≡ a b b b a b b b a (1)determines a symmetric bilinear form on V defined by v · B w ≡ vBw T . We will call this the B - scalar product . The associated B - quadratic form on V is defined by Q B ( v ) ≡ v · B v and we call the number Q B ( v ) the B -quadrance of v . A vector v is B -null precisely when Q B ( v ) = 0 . The B -quadrance satisfies the obvious properties that for vectors v and w in V and a number λ in F Q B ( λv ) = λ Q B ( v )as well as Q B ( v + w ) = Q B ( v ) + Q B ( w ) + 2 ( v · B w )and Q B ( v − w ) = Q B ( v ) + Q B ( w ) − v · B w ) . B -scalar product can be expressed in terms of the B -quadratic form by either of the two polarisation formulas v · B w = Q B ( v + w ) − Q B ( v ) − Q B ( w )2 = Q B ( v ) + Q B ( w ) − Q B ( v − w )2 . The B -scalar product is non-degenerate precisely when the condition that v · B w = 0 for any vector v in V implies that w = 0; this will occur precisely when B is invertible. We will assume that the B -scalar product is non-degenerate throughout this paper. Finally, two vectors v and w in V are B - perpendicular precisely when v · B w = 0, in which case we write v ⊥ B w . B -vector product Define the adjugate of B from (1) to be the matrixadj B ≡ a a − b b b − a b b b − a b b b − a b a a − b b b − a b b b − a b b b − a b a a − b . When B is invertible this is adj B = (det B ) B − . For vectors v ≡ ( v , v , v ) and w ≡ ( w , w , w ) in V , the usual Euclidean vector product [13, p.65] is v × w ≡ ( v w − v w , v w − v w , v w − v w ) . So the B -vector product [23] of v and w is defined to be the vector v × B w ≡ ( v × w ) adj B. With the B -scalar and B -vector products defined, following [23] we define the following expressionsinvolving vectors v , v , v and v : • B -scalar triple product : [ v , v , v ] B ≡ v · B ( v × B v ) • B -vector triple product : h v , v , v i B ≡ v × B ( v × B v ) • B -quadruple scalar product :[ v , v ; v , v ] B ≡ ( v × B v ) · B ( v × B v ) • B -quadruple vector product : h v , v ; v , v i B ≡ ( v × B v ) × B ( v × B v )5 .3 Summary of results of B -scalar and vector products We summarize some results from [23] pertaining to B -scalar and B -vector products, B -triple productsand B -quadruple products. The first result allows us to express the B -scalar triple product in termsof determinants. Theorem 1 (Scalar triple product theorem)
Let M ≡ v v v for vectors v , v and v in V .Then [ v , v , v ] B = [ v , v , v ] B = [ v , v , v ] B = − [ v , v , v ] B = − [ v , v , v ] B = − [ v , v , v ] B = (det B ) [ v , v , v ] = det ( M B ) . The following result expresses the B -vector triple product as a linear combination of two vectors. Theorem 2 (Lagrange’s formula)
For vectors v , v and v in V , h v , v , v i B = (det B ) [( v · B v ) v − ( v · B v ) v ] . The B -scalar quadruple product can be computed as a determinantal identity involving B -scalarproducts, as follows. Theorem 3 (Binet-Cauchy identity)
For vectors v , v , v and v in V , [ v , v ; v , v ] B = (det B ) [( v · B v ) ( v · B v ) − ( v · B v ) ( v · B v )] . The following result immediately follows from the Binet-Cauchy identity and links the B -vectorproduct between two vectors to their B -quadrances and their B -scalar product. Theorem 4 (Lagrange’s identity)
For vectors v and v in V , Q B ( v × B v ) = (det B ) h Q B ( v ) Q B ( v ) − ( v · B v ) i . The B -vector quadruple product can be computed by using only the B -scalar triple product, asfollows. Theorem 5 (Vector quadruple product theorem)
For vectors v , v , v and v in V , h v , v ; v , v i B = (det B ) ([ v , v , v ] B v − [ v , v , v ] B v )= (det B ) ([ v , v , v ] B v − [ v , v , v ] B v ) . Immediately following from the B -vector quadruple product theorem, the following corollary willprove very useful in the study of a vector tetrahedron.6 orollary 6 Let M ≡ v v v for vectors v , v and v in V . Then h v , v ; v , v i B = h (det B ) (det M ) i v . A consequence of this corollary gives us a result that will also prove useful in this paper. Wepresent a quick proof of this result as follows.
Theorem 7 (Scalar triple product of products)
For vectors v , v and v in V , [ v × B v , v × B v , v × B v ] B = (det B ) ([ v , v , v ] B ) . Proof.
For M ≡ v v v we use Corollary 6 and the Scalar triple product theorem to obtain[ v × B v , v × B v , v × B v ] B = [ v × B v , v × B v , v × B v ] B = h (det B ) (det M ) i [ v , v , v ] B = (det B ) ([ v , v , v ] B ) as required. For the rest of this paper, we will work over the three-dimensional affine space over a field F not ofcharacteristic two, denoted by A , where V is its associated vector space. While the main objectsin A are points , which we denote as triples enclosed in rectangular brackets, vectors in V can beexpressed as a separation between two points. In other words, for two points X and Y in A , a vectorfrom X to Y is expressed as −−→ XY and computed to be the affine difference Y − X .A line is a pair ( A, v ) containing a point A in A and a vector v in V , so that a point X lies onit precisely when there exists a number λ in F such that X − A = λv. Two lines ( A , v ) and ( A , v ) are equal precisely when v , v and −−−→ A A are all scalar multiples ofeach other. The vector v is a direction vector for the line ( A, v ). Given two points X and X both lying on a line, we can denote such a line by X X , so that the lines X X and Y Y are equalprecisely when Y and Y both lie on the line X X and vice versa.A plane in A is a triple ( A, v, w ) containing a point A in A and two linearly independent vectors v and w in V , so that a point X lies on it if there exists numbers λ and µ in F such that X − A = λv + µw.
7n other words, the vector −−→ AX is a linear combination of v and w , so that two planes ( A , v , w )and ( A , v , w ) are equal when any of v , v , w , w and −−−→ A A are linear combinations of any two ofthese. The vectors v and w are then spanning vectors for the plane ( A, v, w ). We can associate toa plane in A a B -normal vector n so that any two points X and Y in A lying on the plane satisfy( Y − X ) · B n = −−→ XY · B n = 0 . A plane in A with three points X , Y and Z will be denoted by XY Z , with two planes X Y Z and X Y Z being equal precisely when X , Y and Z lie on X Y Z and vice versa.A triangle in A is an unordered collection of three points in A , say { A , A , A } , and is de-noted by A A A . Such a triangle determines two vector triangles [23] n −−−→ A A , −−−→ A A , −−−→ A A o and n −−−→ A A , −−−→ A A , −−−→ A A o where the vectors in the vector triangle sum to . By defining v ij ≡ −−−→ A i A j forany integer i and j between 1 and 3, we denote these vector triangles respectively by v v v and v v v . A tetrahedron in A is an unordered collection of four points in A , say { A , A , A , A } ,and is denoted by A A A A . An unordered collection of any two distinct points of a tetrahedronwill be called an edge of the tetrahedron, and an unordered collection of any three distinct points ofa tetrahedron will be called a triangle of the tetrahedron. Associated to each edge and triangle ofa tetrahedron is the line and plane (respectively) that passes through the collection of points of thetetrahedron; we call these the lines and planes of the tetrahedron. We now define the rational trigonometric quantities that we will use to analyse a general tetrahedronover a general field and symmetric bilinear form.The B - quadrance between two points A and A in A is the number Q B ( A , A ) ≡ Q B (cid:16) −−−→ A A (cid:17) = −−−→ A A · B −−−→ A A . Note that Q B (cid:16) −−−→ A A (cid:17) = Q B (cid:16) −−−→ A A (cid:17) , so that the B -quadrance between two points in A is indepen-dent of order.Define Archimedes’ function [31, p. 64] as A ( a, b, c ) ≡ ( a + b + c ) − (cid:0) a + b + c (cid:1) so that we also have A ( a, b, c ) = 4 ab − ( a + b − c ) = 4 ac − ( a + c − b ) = 4 bc − ( b + c − a ) . A A A with B -quadrances Q ≡ Q B ( A , A ) Q ≡ Q B ( A , A ) and Q ≡ Q B ( A , A )the B -quadrea of A A A is A B (cid:0) A A A (cid:1) ≡ A ( Q , Q , Q ) . By the definition of the B -quadrance, this is also equal to A B ( v v v ) and to A B ( v v v ). So,the B -quadrea of a triangle is simply the B -quadrea of either of its two associated vector triangles.The following result extends the Quadrea theorem in [23] from the vector triangle setting to theaffine triangle setting. As the only variant to the result is the B -quadrea of the affine triangle, weomit the proof. Theorem 8 (Quadrea theorem)
For a triangle A A A in A with v ij ≡ −−−→ A i A j for any integer i and j between and , we have det B A B (cid:0) A A A (cid:1) = Q B ( v × B v ) = Q B ( v × B v ) = Q B ( v × B v )= Q B ( v × B v ) = Q B ( v × B v ) = Q B ( v × B v ) . The B - quadrume of a tetrahedron A A A A is V B (cid:0) A A A A (cid:1) ≡ B h −−−→ A A , −−−→ A A , −−−→ A A i B . By the linearity of the B -scalar triple product, this will be unchanged if we base the vectors at anotherpoint, for example h −−−→ A A , −−−→ A A , −−−→ A A i B = h −−−→ A A , −−−→ A A , −−−→ A A i B . The following result ensues.
Theorem 9 (Quadrume product theorem) If M ≡ −−−→ A A −−−→ A A −−−→ A A then V B (cid:0) A A A A (cid:1) = 4 det (cid:0) M BM T (cid:1) = 4 (det B ) (cid:0) det M M T (cid:1) = 4 (det B ) (det M ) . Proof.
The first expression is immediate from the Scalar triple product theorem, and the others arejust rewrites using the multiplicative property of the determinant.The B -quadrume is expressed in terms of the B -quadrances as follows. Theorem 10 (Quadrume theorem)
For a tetrahedron A A A A in A , define Q ij ≡ Q B ( A i , A j ) ,for integers i and j satisfying ≤ i < j ≤ . The B -quadrume of the tetrahedron A A A A satisfies V B (cid:0) A A A A (cid:1) = 12 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q Q + Q − Q Q + Q − Q Q + Q − Q Q Q + Q − Q Q + Q − Q Q + Q − Q Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . roof. From the Quadrume product theorem, V B (cid:0) A A A A (cid:1) = 4 det (cid:0) M BM T (cid:1) = 4 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v · B v v · B v v · B v v · B v v · B v v · B v v · B v v · B v v · B v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . By the definition of the B -quadratic form and the polarisation formula, this becomes V B (cid:0) A A A A (cid:1) = 12 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q Q + Q − Q Q + Q − Q Q + Q − Q Q Q + Q − Q Q + Q − Q Q + Q − Q Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) as required.The determinant present in the definition is called the Cayley-Menger determinant (see [4], [11,pp. 285-289] and [27, pp. 124-126]) and forms a general framework for calculating higher-dimensionaltrigonometric quantities of the ”distance” flavour. While named after Cayley and Menger, this formulawas known to Euler and dates back to work of Tartaglia.Given two lines l and l in A with respective direction vectors v and v , we define the B - spread between them as s B ( l , l ) ≡ − ( v · B v ) Q B ( v ) Q B ( v ) . Lagrange’s identity allows us to rewrite this as s B ( l , l ) = Q B ( v × B v )(det B ) Q B ( v ) Q B ( v ) . The following result, originally from [31, p. 82] and proven with B -vector products in [23], computesthe B -quadrea of a triangle in terms of its B -quadrances and B -spreads. We state it without proofhere. Theorem 11 (Quadrea spread theorem)
For a triangle A A A with B -quadrances Q ≡ Q B ( A , A ) Q ≡ Q B ( A , A ) and Q ≡ Q B ( A , A ) as well as B -spreads s ≡ s B ( A A , A A ) s ≡ s B ( A A , A A ) and s ≡ s B ( A A , A A ) and B -quadrea A ≡ A B (cid:0) A A A (cid:1) , we have that A = 4 Q Q s = 4 Q Q s = 4 Q Q s . Given two planes Π and Π in A with B -normal vectors n and n respectively, we define the10 - dihedral spread between them to be E B (Π , Π ) ≡ − ( n · B n ) Q B ( n ) Q B ( n ) . This is clearly independent of the rescaling of normal vectors. Note the similarities between thedefinition of the B -spread and the B -dihedral spread; this is a central theme in projective rationaltrigonometry (see [ ? ] and [32]). The B -dihedral spread can also be rewritten using Lagrange’s identityas E B (Π , Π ) = Q B ( n × B n )(det B ) Q B ( n ) Q B ( n ) . The B -dihedral spread satisfies the following property. Theorem 12 (Dihedral spread theorem)
Let Π be a plane with spanning vectors v and w , and Π be a plane with spanning vectors v and w , so that these two planes meet at a line with directionvector v . Then, E B (Π , Π ) = (det B ) [ v, w , w ] B Q B ( v ) Q B ( v × B w ) Q B ( v × B w ) . Proof.
We use the rearrangement of the definition of the B -dihedral spread using Lagrange’s identityto write E B (Π , Π ) = Q B (( v × B w ) × B ( v × B w ))(det B ) Q B ( v × B w ) Q B ( v × B w ) . By Corollary 6, E B (Π , Π ) = Q B (cid:16)h (det B ) (det M ) i v (cid:17) (det B ) Q B ( v × B w ) Q B ( v × B w )= (det B ) (det M ) Q B ( v ) Q B ( v × B w ) Q B ( v × B w )where M ≡ vw w . By the Quadrume product theorem, E B (Π , Π ) = (det B ) (det M ) Q B ( v ) Q B ( v × B w ) Q B ( v × B w )= (det B ) [ v, w , w ] B Q B ( v ) Q B ( v × B w ) Q B ( v × B w )as required.Take three concurrent lines l , l and l in A with respective direction vectors v , v and v . Wedefine the B - solid spread between them as S B ( l , l , l ) ≡ ([ v , v , v ] B ) (det B ) Q B ( v ) Q B ( v ) Q B ( v ) . The B -solid spread satisfies the following identity.11 heorem 13 (Solid spread theorem) Suppose three lines l , l and l in A meet at a single point O with respective direction vectors v , v and v . Furthermore, define the planes Π ≡ ( O, v , v ) , Π ≡ ( O, v , v ) and Π ≡ ( O, v , v ) . Then, S B ( l , l , l ) = E B (Π , Π ) s B ( l , l ) s B ( l , l )= E B (Π , Π ) s B ( l , l ) s B ( l , l )= E B (Π , Π ) s B ( l , l ) s B ( l , l ) . Proof.
By rewriting the definition of the B -spread using Lagrange’s identity, we have s B ( l , l ) = Q B ( v × B v )(det B ) Q B ( v ) Q B ( v ) and s B ( l , l ) = Q B ( v × B v )(det B ) Q B ( v ) Q B ( v ) . Given that E B (Π , Π ) = (det B ) [ v , v , v ] B Q B ( v ) Q B ( v × B v ) Q B ( v × B v )compute the product of the above three quantities to get our desired result. The other results followby symmetry.Given three concurrent lines l , l and l in A , we construct three lines k , k and k withrespective direction vectors n ≡ v × B v , n ≡ v × B v and n ≡ v × B v so that all six lines are concurrent and k is B -perpendicular to l and l , k is B -perpendicular to l and l , and k is B -perpendicular to l and l . We then define the B - dual solid spread betweenlines l , l and l to be D B ( l , l , l ) ≡ S B ( k , k , k ) . We now present an analog to the Solid spread theorem for B -dual solid spreads. Theorem 14 (Dual solid spread theorem)
Suppose three lines l , l and l in A meet at a singlepoint O with respective direction vectors v , v and v . Furthermore, define the planes Π ≡ ( O, v , v ) , Π ≡ ( O, v , v ) and Π ≡ ( O, v , v ) . Then, D B ( l , l , l ) = s B ( l , l ) E B (Π , Π ) E B (Π , Π )= s B ( l , l ) E B (Π , Π ) E B (Π , Π )= s B ( l , l ) E B (Π , Π ) E B (Π , Π ) . roof. First we construct three lines k , k and k with respective direction vectors n ≡ v × B v , n ≡ v × B v and n ≡ v × B v so that these three lines are concurrent to l , l and l . By the Scalar triple product of productstheorem, we know that [ n , n , n ] B = (det B ) ([ v , v , v ] B ) so that D B ( l , l , l ) = ([ n , n , n ] B ) (det B ) Q B ( n ) Q B ( n ) Q B ( n )= (det B ) [ v , v , v ] B Q B ( v × B v ) Q B ( v × B v ) Q B ( v × B v ) . Now, E B (Π , Π ) = (det B ) [ v , v , v ] B Q B ( v ) Q B ( v × B v ) Q B ( v × B v ) , E B (Π , Π ) = (det B ) [ v , v , v ] B Q B ( v ) Q B ( v × B v ) Q B ( v × B v )and, by Lagrange’s identity, s B ( l , l ) = Q B ( v × B v )(det B ) Q B ( v ) Q B ( v ) . Compute the product of the above three quantities to get our desired result. The other results followby symmetry.
In what follows, we consider a tetrahedron A A A A in A . For integers i and j satisfying 0 ≤ i 3, the B -quadrances between any two points A i and A j of A A A A will be denoted by Q ij ; the B -quadrea associated to the triangle A i A j A k of A A A A will be denoted by A ijk , for integers i , j and k satisfying 0 ≤ i < j < k ≤ 3; and its B -quadrume will be denoted by V .The B -spreads between two lines A i A j and A i A k of A A A A will be denoted by s i ; jk , for aninteger i satisfying 0 ≤ i ≤ j and k distinct from i satisfying 0 ≤ j < k ≤ 3, and the B -dihedral spreads between two planes A i A j A k and A i A j A l , for integers i and j satisfying 0 ≤ i < j ≤ k and l between 0 and 3 which are also distinct from i and j , will be denoted by E ij .Finally, the B -solid spreads and B -dual solid spreads between the three lines A i A j , A i A k and A i A l will be denoted respectively by S i and D i , for an integer i satisfying 0 ≤ i ≤ , and distinct integers j , k and l between 0 and 3 which are also distinct from i .The Quadrea theorem and Quadrume theorem gives us expressions for A , A , A , A and V in terms of the six B -quadrances of A A A A . In terms of the B -quadrances and B -spreads of13 A A A , we use the Quadrea spread theorem to express the B -quadreas as A = 4 Q Q s = 4 Q Q s = 4 Q Q s A = 4 Q Q s = 4 Q Q s = 4 Q Q s A = 4 Q Q s = 4 Q Q s = 4 Q Q s and A = 4 Q Q s = 4 Q Q s = 4 Q Q s . The following gives relations between face spreads of a tetrahedron. Theorem 15 (Alternating spreads theorem) For a tetrahedron A A A A with B -spreads s i ; jk ,for i , j and k distinct integers with ≤ j < k ≤ , we have s s s = s s s . Proof. From the Quadrea spread theorem, we know that A = 4 Q Q s = 4 Q Q s so that s s = Q Q . This is also a direct consequence of the Spread law in the triangle A A A . Similarly we have therelations s s = Q Q and s s = Q Q . The required result follows by taking the product of these three relations and cancelling all of thequadrances.Note that all the B -spreads in the formula involve the index 0 on the right hand side; so, includingthis relation, three other relations hold which will correspond to the other points of the tetrahedron. B -dihedral spreads The following result establishes a formula for the B -dihedral spread of a tetrahedron in terms of its B -quadrances, B -quadreas and B -quadrume. Theorem 16 (Tetrahedron dihedral spread formula) For a tetrahedron A A A A with B -quadrances Q ij for ≤ i < j ≤ , B -quadreas A , A , A and A , and B -quadrume V , the B -dihedralspread E can be expressed as E = 4 Q VA A . roof. Let v ≡ −−−→ A A , v ≡ −−−→ A A and v ≡ −−−→ A A . By the Dihedral spread theorem E = (det B ) [ v , v , v ] B Q B ( v ) Q B ( v × B v ) Q B ( v × B v ) . By the Quadrea theorem Q B ( v × B v ) = det B A B (cid:0) A A A (cid:1) = det B A and Q B ( v × B v ) = det B A B (cid:0) A A A (cid:1) = det B A . Given that V = 4det B [ v , v , v ] B we combine the above results to get E = 4 Q VA A as required.Similarly we have that E = 4 Q VA A , E = 4 Q VA A ,E = 4 Q VA A , E = 4 Q VA A and E = 4 Q VA A . The following result, a rational version of a result from [24], allows us to form a relationship betweenthe products of opposite B -dihedral spreads and the products of opposite B -quadrances. For somewhatmysterious reasons, the quantity R ≡ V A A A A is of significance in the study of the rational trigonometry of a tetrahedron. Unfortunately we do notcurrently have a good geometric interpretation of this quantity, although the two-dimensional analogis the quadratic curvature of the circumcircle of a triangle, and the following theorem does providea partial answer. The quantity R is the rational equivalent of a quantity denoted by h in [24], andhence we will call it the Richardson constant . Theorem 17 (Dihedral spread ratio theorem) For a tetrahedron A A A A with B -quadrances Q ij for ≤ i < j ≤ , B -quadreas A , A , A and A , B -quadrume V and B -dihedral spreads E ij for ≤ i < j ≤ , we have E E Q Q = E E Q Q = E E Q Q = R . roof. From the equivalent formulation of the Dihedral spread theorem for A A A A , we have E E = 4 Q VA A Q VA A = R Q Q and similarly E E = R Q Q and E E = R Q Q . Divide each result through by Q Q , Q Q and Q Q respectively to obtain our desired result. B -solid spreads We now present a formula for calculating the B -solid spreads of a tetrahedron in terms of its B -quadrances and B -quadrume. Theorem 18 (Tetrahedron solid spread formula) For a tetrahedron A A A A with B -quadrances Q ij for ≤ i < j ≤ and B -quadrume V , the B -solid spread S can be expressed as S = V Q Q Q . Proof. Defining v ≡ −−−→ A A , v ≡ −−−→ A A and v ≡ −−−→ A A , the definition of the B -solid spread S = ([ v , v , v ] B ) (det B ) Q B ( v ) Q B ( v ) Q B ( v ) . can be rewritten using the formula for the quadrume V as S = V Q Q Q which is our desired result.Similarly, we have S = V Q Q Q , S = V Q Q Q and S = V Q Q Q . We present an interesting result regarding the ratio of B -solid spreads. Theorem 19 (First solid spread ratio theorem) For a tetrahedron A A A A with B -quadrances Q ij for ≤ i < j ≤ , B -quadrume V , and B -solid spreads S k for ≤ k ≤ , we have S S = Q Q Q Q . Proof. This is an immediate consequence of the Tetrahedron solid spread formula, as S S = (cid:18) V Q Q Q (cid:19) ÷ (cid:18) V Q Q Q (cid:19) = Q Q Q Q . S S = Q Q Q Q and S S = Q Q Q Q etc . Theorem 20 (Second solid spread ratio theorem) For a tetrahedron A A A A with B -quadrances Q ij for ≤ i < j ≤ , B -quadrume V , and B -solid spreads S k for ≤ k ≤ we have S S S S = Q Q . Proof. This is an immediate consequence of the First solid spread ratio theorem, as S S S S = (cid:18) S S (cid:19) (cid:18) S S (cid:19) = (cid:18) Q Q Q Q (cid:19) (cid:18) Q Q Q Q (cid:19) = Q Q . Similarly, we will have S S S S = Q Q and S S S S = Q Q . We may also derive a result pertaining to the ratio of the product of three B -solid spreads to theproduct of three B -quadrances; this is a new result that is unique to this paper, which can only beunderstood by using the framework of rational trigonometry. Theorem 21 (Third solid spread ratio theorem) For a tetrahedron A A A A with B -quadrances A , A , A and A , B -quadrume V , B -solid spreads S , S , S and S , we have S S S Q Q Q = S S S Q Q Q = S S S Q Q Q = S S S Q Q Q = V Q Q Q Q Q Q . Proof. By the equivalent formulation of the Solid spread theorem for A A A A , we have S S S = V Q Q Q V Q Q Q V Q Q Q = (cid:18) V Q Q Q Q Q Q (cid:19) Q Q Q . Divide both sides by Q Q Q to get our desired result. The other results follow by symmetry. B -dual solid spreads We now present a formula for the B -dual solid spread of a tetrahedron in terms of its B -quadreas and B -quadrume. Theorem 22 (Tetrahedron dual solid spread formula) For a tetrahedron A A A A with B -quadreas A , A , A and A , and B -quadrume V , the B -dual solid spread D can be expressedas D = 4 V A A A . roof. Using the vectors v ≡ −−−→ A A , v ≡ −−−→ A A and v ≡ −−−→ A A , as we saw in the proof of the Dualsolid spread theorem, D = (det B ) [ v , v , v ] B Q B ( v × B v ) Q B ( v × B v ) Q B ( v × B v ) . But we know that V = V B (cid:0) A A A A (cid:1) ≡ B [ v , v , v ] B . and that Q B ( v × B v ) = det B A Q B ( v × B v ) = det B A Q B ( v × B v ) = det B A so that substituting we get D = (det B ) (cid:0) det B (cid:1) (cid:0) det B (cid:1) V A A A = 4 V A A A . Similarly we have D = 4 V A A A , D = 4 V A A A and D = 4 V A A A . The following result outlines the ratio of B -dual solid spreads to B -quadreas, one that is a rationalanalog of another classical result from [24]; it acts similarly to the Sine law in classical trigonometry,albeit in a very different context. The Richardson constant R will be present here as well. Theorem 23 (Dual solid spread and quadrea ratio theorem) For a tetrahedron A A A A with B -quadreas A , A , A and A , B -quadrume V , B -dual solid spreads D , D , D and D , andRichardson constant R , we have D A = D A = D A = D A = R . Proof. By the equivalent formulation of the Dual solid spread theorem for A A A A , we have D = 4 V A A A . Divide through by A to get D A = 4 V A A A A = R . The familiar formula for the projection of one vector onto another ([3, p. 206] and [28, p. 174]) holdsalso for more general bilinear forms; we define the B -projection of a vector v in the direction of thevector u as the vector (proj u v ) B ≡ u · B vQ B ( u ) u. For a tetrahedron A A A A with all the above quantities defined, Hilbert and Cohn-Vossen [16,pp. 13-17] established in the Euclidean case that the pairs of lines ( A A , A A ), ( A A , A A ) and( A A , A A ) of A A A A are skew, i.e. their meets do not exist. We now define n ≡ −−−→ A A × B −−−→ A A , n ≡ −−−→ A A × B −−−→ A A and n ≡ −−−→ A A × B −−−→ A A so that we define R ≡ Q B (cid:16) proj n −−−−→ P P (cid:17) , R ≡ Q B (cid:16) proj n −−−−→ P P (cid:17) and R ≡ Q B (cid:16) proj n −−−−→ P P (cid:17) to be the skew B - quadrances of A A A A associated to the respective pairs of opposing lines( A A , A A ), ( A A , A A ) and ( A A , A A ), where P ij is an arbitrary point on the line A i A j forintegers i and j satisfying 0 ≤ i < j ≤ 3. This quantity is independent of the selection of the P ij ’s,since if the two lines don’t meet, then the points on the line will lie on separate planes which areparallel.We establish a formula for the skew B -quadrances of a tetrahedron based on its B -quadrances and B -quadrume. We use [26] as inspiration to prove this result in our framework. Theorem 24 (Tetrahedron skew quadrance formula) For a tetrahedron A A A A with B -quadrances Q ij , B -quadrume V , and skew B -quadrances R , R and R , we have R = V Q Q − ( Q + Q − Q − Q ) R = V Q Q − ( Q + Q − Q − Q ) and R = V Q Q − ( Q + Q − Q − Q ) . Proof. For integers i satisfying 1 ≤ i ≤ 3, define vectors v i ≡ −−−→ A A i so that we may define n ≡ v × B ( v − v ). By the definition of skew B -quadrances, we have R = Q B (cid:16) proj n −−−→ A A (cid:17) = Q B (cid:18) n · B v Q B ( n ) n (cid:19) = [( v × B ( v − v )) · B v ] Q B ( v × B ( v − v )) = [ v , v − v , v ] B Q B ( v × B ( v − v )) . M ≡ v v v so we may use the bilinearity properties of the B -scalar and B -vector products, aswell as the Scalar triple product theorem, to rewrite the numerator as[ v , v − v , v ] B = ([ v , v , v ] B − [ v , v , v ] B ) = [ v , v , v ] B = (det M det B ) . Furthermore, the denominator becomes Q B ( v × B ( v − v )) = (det B ) h Q B ( v ) Q B ( v − v ) − ( v · B ( v − v )) i = (det B ) (cid:16) Q Q − [( v · B v ) − ( v · B v )] (cid:17) by Lagrange’s identity. Use the polarisation formula to obtain Q B ( v × B ( v − v )) = (det B ) (cid:16) Q Q − [( v · B v ) − ( v · B v )] (cid:17) = (det B ) Q Q − (cid:20) Q + Q − Q − Q + Q − Q (cid:21) ! = (det B ) Q Q − ( Q + Q − Q − Q ) ! = det B (cid:16) Q Q − ( Q + Q − Q − Q ) (cid:17) . Combine the results for the numerator and denominator with the Quadrume product theorem to get R = 4 (det M det B ) (det B ) (cid:16) Q Q − ( Q + Q − Q − Q ) (cid:17) = 4 (det B ) (det M ) Q Q − ( Q + Q − Q − Q ) = V Q Q − ( Q + Q − Q − Q ) as required. The other results follow by symmetry.It is curious to note that the denominator of the Tetrahedron skew quadrance formula is a rationalform of Bretschneider’s formula [6] for the quadrea of a general quadrangle (a collection of four coplanarpoints) in terms of the six quadrances between any two of its points (see [7], [10], [17, pp. 204-205]and [ ? ]). To finish we apply the framework devised in this paper to study a particularly fundamental type oftetrahedron, which is the analog of a right triangle in the three-dimensional setting. Just as manyproblems in metrical planar geometry can be resolved into right triangles, and in spherical or ellipticgeometry Napier’s rules highlight the importance of right spherical or elliptic triangles, so the tri-20ectangular tetrahedron plays a special role in three-dimensional geometry.We set A A A A to be a tetrahedron in A with all its trigonometric invariants denoted as above.Introducing v ≡ −−−→ A A , v ≡ −−−→ A A and v ≡ −−−→ A A we define A A A A to be B - tri-rectangular [1, pp. 91-94] at the point A precisely when v , v and v are all mutually B -perpendicular, that is when v · B v = v · B v = v · B v = 0 . While we can also similarly define a B -tri-rectangular tetrahedron at another point of A A A A ,we may suppose for the purposes of this study, and without loss of generality, that the tetrahedron A A A A is B -tri-rectangular at A .Then by the definition of the B -spread we have s = s = s = 1 . Furthermore, since v , v and v are all mutually B -perpendicular we deduce that E = E = E = 1 . By the Solid spread projective theorem, we then obtain S = 1.Because of this, it is natural to parametrize a B -tri-rectangular tetrahedron A A A A by thequadrances Q ≡ K , Q ≡ K and Q ≡ K . These quantities represent the B -quadrances of A A A A emanating from the point A . Doing this,we use Pythagoras’ theorem (see [15] and [23]) to obtain Q = K + K , Q = K + K and Q = K + K . By the Quadrume theorem, the B -quadrume of A A A A is V = 12 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K K 00 0 2 K (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 4 K K K . Then by the Quadrea spread theorem A = 4 Q Q = 4 K K A = 4 Q Q = 4 K K and A = 4 Q Q = 4 K K B -quadreas. To obtain A , we rely on the following generalization of a classicalresult from [8], which provides a parallel to Pythagoras’ theorem for B -quadreas of a B -tri-rectangulartetrahedron. Theorem 25 (de Gua’s theorem) For a B -tri-rectangular tetrahedron A A A A at A , we havethat A = A + A + A . Proof. By the definition of the B -quadrea, A = A ( Q , Q , Q )= (2 ( K + K + K )) − (cid:16) ( K + K ) + ( K + K ) + ( K + K ) (cid:17) = 4 (cid:0) K + K + K + 2 K K + 2 K K + 2 K K (cid:1) − (cid:0) K + K + K + K K + K K + K K (cid:1) = 4 K K + 4 K K + 4 K K = A + A + A as required.The result of the Quadrea spread theorem can be rearranged to obtain the remaining B -spreads,which are s = S K K K ( K + K ) , s = K K K ( K + K ) , s = K K + K K + K K ( K + K ) ( K + K ) ,s = K K K ( K + K ) , s = K K K ( K + K ) , s = K K + K K + K K ( K + K ) ( K + K ) ,s = K K K ( K + K ) , s = K K K ( K + K ) and s = K K + K K + K K ( K + K ) ( K + K ) . By the Tetrahedron dihedral spread formula, the remaining B -dihedral spreads are E = K ( K + K ) K K + K K + K K , E = K ( K + K ) K K + K K + K K and E = K ( K + K ) K K + K K + K K . The following elegant relation between B -dihedral spreads then becomes visible. Theorem 26 (Tri-rectangular dihedral spread theorem) For a B -tri-rectangular tetrahedron A A A A at A , we have that E + E + E = 2 . Proof. Use the above quantities to immediately obtain our result, as follows: E + E + E = K ( K + K ) K K + K K + K K + K ( K + K ) K K + K K + K K + K ( K + K ) K K + K K + K K = 2 . By the Tetrahedron solid spread formula, the remaining B -solid spreads are S = K K ( K + K ) ( K + K ) , S = K K ( K + K ) ( K + K ) and S = K K ( K + K ) ( K + K ) . A A A has a right angleat A then s B ( A A , A A ) + s B ( A A , A A ) = 1 . Here is a three-dimensional extension of this involving solid spreads. Theorem 27 (Tri-rectangular solid spread theorem) For a B -tri-rectangular tetrahedron A A A A at A , we have that (1 − S − S − S ) = 4 S S S . Proof. With the values of S , S and S above, we have1 − S + S + S = 1 − K K ( K + K ) ( K + K ) − K K ( K + K ) ( K + K ) − K K ( K + K ) ( K + K )= 2 K K K ( K + K ) ( K + K ) ( K + K )so that (1 − S − S − S ) = 4 K K K ( K + K ) ( K + K ) ( K + K ) = 4 S S S as required.It appears interesting to ask if this result extends in some fashion to more general tetrahedra.As for the dual solid spreads, the value at A is D = 1because the normals to the lines meeting there are the lines themselves. Note that this is consistentwith D = 4 V A A A = 4 (4 K K K ) (4 K K ) (4 K K ) (4 K K ) = 1which uses the Tetrahedron dual solid spread formula. Using this same formula, we also get D = 4 V A A A = 4 (4 K K K ) (4 K K ) (4 K K ) (4 K K + 4 K K + 4 K K )= K K K K + K K + K K and similarly D = K K K K + K K + K K and D = K K K K + K K + K K . The following result pertaining to D , D and D then follows. Theorem 28 (Tri-rectangular dual solid spread) For a B -tri-rectangular tetrahedron A A A A at A , we have that D + D + D = 1 . roof. With the values of D , D and D above, we compute that D + D + D = K K K K + K K + K K + K K K K + K K + K K + K K K K + K K + K K = 1 . There is clearly a big step in going from the two-dimensional to the three-dimensional situation intrigonometry. One of the reasons is simply that the number of objects can increased considerably;instead of just three points, three lines and a triangle, we have four points, six lines, four faces,and a tetrahedron, and so the range of metrical notions must also expand to include the variousconfigurations that are possible when we combine these in various ways.So when we contemplate higher dimensional trigonometry, the situation will become much moreinvolved even when we restrict to the case of the simplex, and will also require the addition of higherdimensional invariants. 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