Detour Extra Straight Lines in the Euclidean Plane

Using the theory of exploded numbers by the axiom-systems of real numbers and Euclidean geometry, we explode the Euclidean plane. Exploding the Euclidean straight lines we get super straight lines. The extra straight line is the window phenomenon of super straight line. In general, the extra straight lines are curves in Euclidean sense, but they have more similar properties to Euclidean straight lines. On the other hand, with respect of parallelism we find a surprising property: there are detour straight lines.


Short Introduction of the Theory of Exploded Numbers
The concept of exploded numbers had already been introduced in [1] and the explosion of k-dimensional space was discussed in the 4th part of [2]. In this article = 2 is used, only.
For the sake of a better understanding we repeat the main results of the theory of exploded numbers. Denoting by ℝ the set of real numbers, we use the axioms of real numbers: commutativity and associativity of both addition and multiplication, existence of unit elements for addition and multiplication (0 and 1, respectively) existence of additive inverse element (− ) where ∈ ℝ, and multiplicative inverse element of ∈ ℝ ( , where ≠ 0), and distributivity. Summarizing, we have the field (ℝ, +,• , =).

Requirement for explosion:
The set ℝ ∖ ℝ contains positive and negative elements, too.
After the latter requirement the following definitions are given: The exploded number is called positive if 0 < .

(By the Postulate of ordering it is fulfilled if and only if > 0.)
The exploded number is called negative if < 0.
(By the Postulate of ordering it is fulfilled if and only if < 0.)

Requirement of monotonity of super-addition: If
and ' are arbitrary exploded numbers and is smaller than ' , then, for any exploded number /, the super-sum ⨁/̌ is smaller than super-sum ⨁/.
Requirement of monotonity of super-multiplication: If and ' are arbitrary exploded numbers and is smaller than ' , then for any positive exploded number /̌ the superproduct ⨀/̌ is smaller than super-product '⨀/.

By isomorphism
⟷ , ∈ ℝ we can find that the set of exploded real number ℝ is an ordered field with respect to super-addition and super-multiplication. It is important to remark that super-operations ⨁ and ⨀ are not extensions of traditional operations + and ⋅, respectively. By the Postulate of extension the explosion formula (1.1) yields the compression formula
We denote by ℝ 9 the Euclidean plane with the Cartesian coordinate system with right angles. The point C = ( , ') of the plane is exploded by each coordinate to obtain the exploded plane ℝ 9 % ℝ 9 % = D E C -= ( , ' )FC ∈ ℝ 9 G. If C = (", H) ∈ ℝ 9 % ∖ ℝ 9 , then it is invisible on the Euclidean plane. The Euclidean plane has four invisible boundaries:

Extra-lines in the Euclidean Plane
Let C d = ( d , ' d ) ∈ ℝ 9 and e = #f = , f c $ ∈ ℝ 9 be a given point and vector, such that ‖e‖ = hf = 9 + f c 9 = 1. We consider the Euclidean straight line I i j ,k , given by the vector-equation where C W = ( W , ' W ) ∈ ℝ 9 . The straight line I i j ,k is described by the equation-system I i j ,k :  The appearance deludes. As C PXX@M = (0,1) and C MTUVW = (1.0) are not on the same side of the closed square ℝ 9 , so, by Property 2.8 , I i j , k @=WM is unambiguously determined.

Extra Straight Lines without Common Points in the Plane
Two straight Euclidean lines in one plane either intersect or not. The latter case is parallelism. The situation in extra geometry is more sophisticated. The concept of extra parallelism has already been introduced in [