Revista de Matema´tica: Teor´ıa y Aplicaciones 1998 5(2) : 187–192
cimpa – ucr – ccss issn: 1409-2433
L3: the geometry of pseudoquaternions
Graciela Silvia Birman*
3 de junio de 2004
Received: February 10, 1998
Abstract
We introduce pseudoquaternions as an effective tool to describe the vector analysis
in L3, and we use them to characterize null curves and null cubics in S21 .
Keywords: pseudoquaternions, vector analysis, null curves.
Resumen
Introducimos los pseudocuaterniones como una herramienta efectiva para describir
el anlisis vectorial de L3, y los usamos para caracterizar curvas nulas y cbicas nulas
en S21 .
Palabras-clave: pseudocuaterniones, anlisis vectorial, curvas nulas.
AMS Subject Classification: 14H99
1. Introduction
Let L3 be the 3-dimensional Lorentzian space with inner product of signature −,+,+,
which will be denoted by dot.
In this paper we show that pseudoquaternions are an useful and natural tool to study
the elementary geometry of L3 and we have used them to characterize unitary null curves
in this space.
*CONICET, Depto. de Matema´tica, Fac.de Ciencias Exactas, Universidad Nacional del Centro de la
Provincia de Buenos Aires, Pinto 399, 7000 - Tandil - Argentina.
187
188 g. birman
2. Vector analysis in L3
As a generalization of complex numbers related with the system of quaternions we find
the pseudoquaternions [5], given by:
z = a+ bi+ ce+ df (1)
where a, b, c, d ∈ R and the complex units hold the following multiplication table:
— i e f
i — −1 f −e
e — −f 1 −i
f — e i 1
The conjugate pseudoquaternion of z, (1), will be
z∗ = a− bi− ce− df
and its norm or modulus will be
N(z) = a2 + b2 − c2 − d2
Trivially,
z−1 =
z∗
N(z)
when it is possible, and also, if x and y are two pseudoquaternions we get
(x · y)∗ = y∗ . x∗ and N(x . y) = N(x) .N(y).
We say that a pseudoquaternion z, (1), is pure if a = 0.
Pure pseudoquaternions verify z∗ = −z and N(z) = −z2.
The distance between two pure pseudoquaternions z1 = b1i + c1e + d1f , z2 = b2i +
c2e+ d2f is given by
d(z1, z2) =
√
−(b1 − b2)2 + (c1 − c2)2 + (d1 − d2)2
which coincides with the distance in L3.
The pseudoquaternions i,e,f are associated to the orthonormal vectors I,E,F .
If we note the inner product by dot, we have
I . I = −1, E .E = 1, F . F = 1
i.e., according to [3], I is timelike vector, E and F are spacelike vectors.
For all above we can identify the vectors of L3 with pure pseudoquaternions or equiv-
alently, with real linear combination of i, e, f .
We want to define an exterior product in L3 on the natural way, keeping in mind its
analogous in R3.
Let A = (a1, a2, a3), B = (b1, b2, b3) and C = (c1, c2, c3) be vectors in L3.
L3: the geometry of pseudoquaternions 189
Definition 1 The exterior product of A and B, A ∧ B, is the vector of L3 such that its
inner product with C is the determinant of the matrix a1 −a2 −a3−b1 b2 b3
c1 c2 c3
 .
Equivalently, we say
A ∧B = det
 i e fa1 −a2 −a3
−b1 b2 b3

= (a3b2 − a2b3)i− (a1b3 − a3b1)e+ (a1b2 − a2b1)f (2)
By straightforward computation we can verify
a) A ∧A = 0
b) A ∧B = −B ∧A
c) λA ∧B = A ∧ λB = λ(A ∧B) si λ > 0
d) A ∧B .B = A ∧A .A = 0
e) (A+B) ∧ C = A ∧ C +B ∧ C
f) (A ∧B) ∧ C = (A .C)B − (B .C)A
g) If A, B, C are vectors in L3 and a, b, c its corresponding pure pseudoquaternions,
it verifies
A ∧B .C = 1
2
(abc− cba)
h) Let A, B, C be future-pointing timelike vectors in L3, [1]; A, B, C are on line if and
only if
|(B −A) ∧ (C −A)| = 0
3. Unitary null curves
A curve q(s) verifying q′(s) . q′(s) = 0 is called a null curve and if in addition satisfy
q(s) . q(s) = 1 is called unitary null curve. A null frame in L3 is an ordered triple of vectors
(E1, E2, E3) such that
E1 .E1 = E2 .E2 = 0, E1 .E2 = −1, E3 .E3 = 1,
E1 .E3 = E2 .E3 = 0 and det
E1E2
E3
 = ±1 (3)
190 g. birman
Let (E1, E2, E3) be a null frame in L3. The orthonormal vectors I, E, F are the asso-
ciated orthonormal frame related to the null frame by
I =
1
2
(E1 +E2), E =
1
2
(E1 −E2), F = E3.
We take
E1 ∧E2 = −E3, E2 ∧E3 = E1 and E1 ∧E3 = −E2
and we obtain
I ∧E = F, E ∧ F = −(I +E)
2
, F ∧ I = (E − I)
2
and the others vanish.
A rotation in L3, around the origin, could be defined by the position of a null frame
(E1, E2, E3) respect to the initial basis I, E, F .
From the rotation defined by a pseudoquaternion q, the vectors Ei are associated to
the pseudoquaternions ei by
e1 = q∗ i q, e2 = q∗ e q, e3 = q∗ f q.
Explicity, if q = q0 + q1i+ q2e+ q3f we know that
q∗ = q0 − q1i− q2e− q3f eq = q0e− q1f + q2 − q3i
iq = q0i− q1 + q2f − q3e fq = q0f + q1e+ q2i+ q3
and we get
e1 = (q20 + q
2
1 + q
2
2 + q
2
3)i+ 2(q2q1 − q0q − 3)e + 2(q0q2 + q3q1)f
e2 = −2(q0q3 + q1q2)i+ (q20 − q21 − q22 + q23)e− 2(q0q1 + q2q3)f
e3 = 2(q0q2 − q3q1)i+ 2(q0q1 − q3q2)e+ (q20 − q21 + q22 − q23)f
These are the components of the pseudoquaternions ei as well as components of vectors
Ei, i : 1, 2, 3.
At every point of an unitary curve se associate the null frame (E1, E2, E3) and following
[3] we have the Frenet’s equations:
dE1
ds
= −k1(s)E1 + k2(s)E3
dE2
ds
= −k1(s)E2 + k3(s)E3 (4)
dE3
ds
= k3(s)E1 + k2(s)E2
The “curvatures.are
k1 =
−dE1
ds
.E2, k2 =
dE1
ds
.E3, k3 =
−dE3
ds
.E2
L3: the geometry of pseudoquaternions 191
and in terms of the pseudoquaternion q and its derivated
k1 = 2(−q′0q3 + q0q′3 − q2q′1 + q1q′2)
k2 = 2(q′3q1 − q0q′2 + q2q′0 − q3q′1) (5)
k3 = 2(−q3q′2 − q2q′3 − q0q′1 − q1q′0)
Also we find that (5) are the relative components (respect to the null frame (E1, E2, E3))
of the instant rotation vector, [4],
H = −k2E1 + k3E2 − k1E3
since dE
i
ds = H ∧Ei, i : 1, 2, 3.
The curve q = q(s) with s no proper time parameter, can be represented by the
pseudoquaternion q = q0(s) + q1(s)i + q2(s)e + q3(s)f , with the condition q . q = 1 and
q′ . q′ = 0 (q′ = dqds).
We will suposse that the qi(s) are C5, as [2].
At every point we can attach a null frame (Q1, Q2, Q3). Without loss of generality we
can choose Q1 as an scalar multiple of q′.
As Qi = Qi(s) we can write
dQi
ds
=
∑
j
wij Q
j
with w11 = w
2
2 = w
1
2 = w
2
1 = w
3
3 = 0, w
2
3 = −w31, w13 = −w32.
Now the Frenet’s equations are
dQ1
ds
= w13 Q
3
dQ2
ds
= w23 Q
3 (6)
dQ3
ds
= −w23 Q1 − w13 Q2
On the natural way, we can consider w13 as curvature and w
2
3 as torsion.
Comparing (4) and (6) we obtain k1(s) must be zero and from [2], theor. 6.1 the curve is
a null straight line. We also obtain w13 = k2 and w
2
3 = k3 and according to [3] (E
1, E2, E3)
become a Cartan frame and the curve is called a Cartan-framed curve.
In order to know about k2 and k3 we study the osculating sphere in L3, i.e., the sphere
passing through four consecutive points of a curve.
Keeping in mind that dot means the inner product of signature −,+,+, the equation
of this sphere is
(x− c) . (x− c)− r2 = 0
where x is a generic point of the sphere, c its center and r its radius.
192 g. birman
It is well known that necessary and sufficient condition that the surface f(s) has contact
of order n at the point P with the curve is that at P the relation hold:
f(s) = f ′(s) = ..... = f (n)(s) = 0 and f (n+1)(s) 6= 0.
In our case n = 3, f(s) = (x− c) . (x− c)− r2 and the relations becomes
(x− c) .Q1 = 0
k2(x− c) .Q3 = 0
(x− c) . (−k2k3Q1 − (k2)2Q2 + k′2Q3) = 0
We find (x− c) . (k2)2Q2 = 0 then k2 = 0.
The center is c = x+Q1 and the radius is zero.
For all above, we summarize in the following theorem.
Theorem 1 The curvatures (5) of a null curve is S21 are k1 = k2 = 0 or equivalently,
the null curves in S21 are null straight lines and there not exist osculating sphere of a null
spherical curve in L3.
At [2], pages 240 and 234, we find that a null cubics is a curve with k1 = 1 and
k2 = k3 = 0, thus
Corollary 1 There does not exist null cubics in S21 .
Referencias
[1] Birman, G.; Nomizu, K. (1984) “Trigonometry Lorentzian geometry”, Amer. Math.
Monthly, 91(9).
[2] Bonnor, W.B. (1969) “Null curves in a Minkowski space time”, Tensor N.S. 20: 229–
242.
[3] Graves, L.K. (1979) “Codimension one isomtric immersions between Lorentz spaces”,
Transactions of the Amer. Math. Society, 252: 367–392.
[4] Santalo´, L.A. (1974) “Curvas y cuaterniones”, Revista de la Unio´n Matema´tica Ar-
gentina 27(1): 41–52.
[5] Yaglom, I.M. (1968) Complex numbers in geometry. Academic Press, New York.