As it was in the case of complex lines, any hyperbolic line that passes through W0 = 0 and that is parallel to PZ0 (i.e., a hyperbolic line with the direction determined byZ0) is defined as the set
PZ0+W0:= μZ0+W0μ∈D .
We leave it to the reader to show that any hyperbolic line can be described as PZ0+W0for some bicomplex numbersZ0,W0.
We also leave it to the reader to prove the following results, which are anal- ogous to those given in the case of complex lines.
Proposition 4.2.2.1. Given the hyperbolic line PZ0 and given U ∈ PZ0 such that U = μ0Z0, with μ0 ∈/ S0 (hence U /∈ S0), then PU = PZ0. Reciprocally, if PU =PZ0, thenU ∈PZ0.
Another characterization of real two-dimensional planes passing through the origin and being a hyperbolic line is given in the following Proposition whose proof is left to the reader.
Proposition 4.2.2.2. A real two-dimensional planeT ⊂BCwith0∈T is a hyper- bolic line if and only ifU ∈T implieskU ∈T.
Corollary 4.2.2.3. A hyperbolic line P through the origin, regarded as a two- dimensional subspace ofR4∼=BC, is generated overRby anyU0∈P\S0 and by kU0, i.e., it is the real span of U0 andkU0.
Now, in analogy with complex lines, we describe the relation between the hyperbolic arguments of the points of a hyperbolic linePZ0.
Theorem 4.2.2.4. Consider the hyperbolic linePZ0 for a pointZ0∈BC\S0. Write Z0=|Z0|k
eiν01e+eiν20e† ,
with ΨZ0 = ν10e+ν20e† = argDZ0 ∈ D the principal hyperbolic argument of Z0. Then the setArgDU of any pointU ∈PZ0\S0 is one of the four sets:
ArgDU = ν10+ 2mπ e+
ν20+ 2nπ
e†m, n∈Z , ArgDU = ν10+ 2mπ
e+
ν20+ (2n+ 1)π
e†m, n∈Z , ArgDU = ν10+ (2m+ 1)π
e+
ν20+ 2nπ
e†m, n∈Z , or
ArgDU = ν10+ (2m+ 1)π e+
ν20+ (2n+ 1)π
e†m, n∈Z .
Proof. Since U ∈ PZ0 \S0, then U = μZ0 for some μ ∈ D\ S0. Since the trigonometric form ofμin hyperbolic terms is
μ=|μ|k
±e±e† , the statement follows from the fact that
U =μZ0=|μZ0|k
±eiν10e±eiν02e†
.
We are now ready to define the slope of hyperbolic lines; Theorem 4.2.2.4 ensures that the slope will not depend on the bicomplex numberZ0that determines its direction.
We now need to use a result about trigonometric functions of a hyperbolic variable, a topic that is studied in detail in Chapter 6. As we will show there, the tangent of a hyperbolic numberν =ν1e+ν2e† can be given a rigorous definition and it turns out that
tanν= tanν1e+ tanν2e†.
Definition 4.2.2.5. The slopeN of a hyperbolic linePZ0+W0, is defined by N := tan ΨZ0 = tanν10e+ tanν20e†,
whereΨZ0 ∈ArgD(Z0).
Example 4.2.2.6. (a) The hyperbolic lineDis generated by any non zero-divisor ae+be†whose hyperbolic argument is Ψae+be† = 0 = 0e+ 0e†. Therefore the slope ofDis tan 0 = tan 0e+ tan 0e† = 0. Thus, any hyperbolic lineD+W0
“parallel” toDhas the slope zero, and all of them together give the notion ofhyperbolic horizontality.
94 Chapter 4. Lines and curves inBC (b) Each real lineLinC(i) is exactly the intersection of a hyperbolic lineP and C(i) itself, and the slope of the hyperbolic line P coincides with the (usual real) slope ofL. Indeed, assume that 0∈L, then anyz∈Lis of the form
z=|z|(cosθ+isinθ) =|z|k
eiθe+eiθe†
; thus, the hyperbolic linePz=P which passes throughz has slope
N = tanθe+ tanθe†= tanθ.
(c) Consider the hyperbolic line of the typePie+β0
2e† or Pβ0
1e+ie†; since the hy- perbolic arguments are Ψie+β0
2e† = π
2e+ν20e† and Ψβ0
1e+ie† =ν10e+π 2e†, then the slopes are not well defined but can be symbolically represented as
tan Ψie+β0
2e†=∞e+ tanν20e†, tan Ψβ0
1e+ie†= tanν10e+∞e†.
We say in this case that these hyperbolic lines determine the notion ofhy-
perbolic verticality.
Recalling that a hyperbolic line is a real two-dimensional plane such that its projections onto BCe and BCe† are usual real lines, we can give a precise geometrical notion of the hyperbolic angle between two hyperbolic lines. Clearly it is enough to define the hyperbolic angle between hyperbolic lines that pass through the origin.
Definition 4.2.2.7. Take hyperbolic lines PZ0 and PW0; if argDZ0 = ν10e+ν20e† and argDW0 =μ01e+μ02e†, then the (trigonometric) hyperbolic angle α between PZ0 andPW0 is
α:= argDZ0−argDW0= (ν10−μ01)e+ (ν20−μ02)e†, (4.32) and the (geometric) hyperbolic angle between PZ0 andPW0 is
|α|k:=|argDZ0−argDW0|k
=|(ν10−μ01)e+ (ν20−μ02)e†|k=|ν10−μ01|e+|ν20−μ02|e†. (4.33) In analogy with the complex plane, if the angle α is a positive hyperbolic number, then we say that the angle between the lines is positively oriented; if the angleαis a hyperbolic negative number, then we say that the angle between the lines is negatively oriented; if the angle is neither negative nor positive, then no orientation is assigned.
Denoting byL1Z
0 and L1W
0 the real lines in BCe that are the projections of PZ0 and PW0 on BCe and by L2Z
0, L2W
0 the respective projections on BCe† we are not able to say which ofPZ0 or PW0 is the initial or the final hyperbolic line
that determines the orientation of the hyperbolic angle. The reason for this is that in the projections on BCe and BCe† the roles of L1Z
0, L1W
0 and L2Z
0, L2W do not coincide in general; just recall that D is not a totally ordered set but a0
partially ordered one. That is why we use not the difference of the arguments but its hyperbolic modulus.