# UAV Trajectory Optimization for Time Constrained Applications

Emmanouil Fountoulakis, Georgios S. Paschos, and Nikolaos Pappas
E. Fountoulakis and N. Pappas are with the Department of Science and
Technology, Linköping University, Norrköping, Sweden. Emails:
{emmanouil.fountoulakis<EMAIL_ADDRESS>G. S. Paschos is with
Amazom.com, Luxembourg city, Luxembourg. Email<EMAIL_ADDRESS>The work
of G. S. Paschos was done before joining Amazon.com. The views and opinions
expressed in this article are those of the authors and do not necessarily
reflect the official policy or position of Amazon.com.

###### Abstract

In this paper, we consider a UAV flying over multiple locations and serves as
many users as possible within a given time duration. We study the problem of
optimal trajectory design, which we formulate as a mixed-integer linear
program. For large instances of the problem where the options for trajectories
become prohibitively many, we establish a connection to the orienteering
problem, and propose a corresponding greedy algorithm. Simulation results show
that the proposed algorithm is fast and yields solutions close to the optimal
ones. The proposed algorithm can be used for trajectory planning in content
caching or tactical field operations.

##  I Introduction

The growing popularity of mobile devices and applications that require
bandwidth hungry services has fueled an increase in mobile traffic which in
many cases, limits the ability of systems to offer high quality
communications. For example, in peak demand hours, or during a popular event
in specific areas, communication systems become congested and service quality
is critically impaired. In addition, applications that require operations in
areas without infrastructure, called tactical field operations, raise the need
for fast and reliable reaction from the communication systems. In such
situations, it is desirable to have helpers to offload traffic from congested
networks or areas without infrastructure [1], [2]. Unmanned Aerial Vehicles
(UAVs) can fly and serve congested network areas or specific areas that
require urgently specific information. However, UAVs have limited flight time
duration. Therefore, a UAV may not have the energy resources to visit all the
areas. In this paper, our goal is to design an optimal trajectory in order to
serve areas with higher emergency within a certain time.

Recently, research on UAVs that act as small base stations or caching helpers,
has attracted a lot of interest [1, 2, 3]. The authors in [4] consider a UAV
that flies from one location to another and accomplishes a certain amount of
computation tasks. UAVs are used as small cells with caches in [5]. Caching
and UAV placement strategies are provided while considering limited UAV
battery budget constraints. Authors in [6] consider that a UAV transmitting
files to a group of ground terminals (GTs). Based on device-to-device (D2D)
communications, GTs can share the files, received by the UAV, to their
adjacent GTs when they are requested. In [7], a proactive caching technique is
considered. The authors propose a solution for UAVs deployment and caching
content placement in order to maximize the quality-of-service (QoS). In [8],
the authors consider the UAV deployment for data delivery in vehicular
networks. _To the best of our knowledge, there is no work that considers a UAV
that flies over multiple areas with high importance and serve as many as
possible within a certain time._ The importance of each area is expressed with
a score. We formulate an optimization problem whose solution provides a
trajectory for the UAV for which the collected score is maximized. By drawing
an analogy from the orienteering problem [9], we prove that the problem is NP-
hard. We provide a greedy algorithm that finds an approximate solution to the
optimization problem in scalable manner.

##  II System model and Problem Formulation

We consider a UAV that flies over multiple geographical areas and collects the
corresponding scores. Location $i$ has a score which is denoted by
$\lambda_{i}$ 111E.g. the reward can play the role of user demand in that
location.. Our goal is to design a trajectory that maximizes the scores
collected by the UAV.

Trajectory selection. We consider that each location $i\in\mathcal{I}$ may be
the barycenter (or centroid) of that area, or some central hotspot point. Let
$x_{i}$ denote its coordinates on the plane. The UAV forms a _trajectory_ by
visiting a subset of locations in a specified order.

For two locations $i,j\in\mathcal{I}$, let
$d_{ij}=d_{ji}\propto\|x_{i}-x_{j}\|^{2}$ be a _distance_ that measures the
amount of time it takes the UAV to move from one location to the other (for
example the Euclidean distance of $x_{i},x_{j}$ divided by the maximum
velocity of the UAV). Consider an undirectional complete graph
$G=(\mathcal{I},E,\boldsymbol{d})$, where
$E=\left\\{\\{i,j\\}:i,j\in\mathcal{I},i\neq j\right\\}$ is the set of links
connecting the locations, and for each link $\\{i,j\\}$ we have an associated
distance $d_{ij}$ (or $d_{ji}$). A trajectory $\mathcal{T}$ is a tour on graph
$G$, i.e., an ordered set of nodes
$\mathcal{T}\triangleq(i_{1},i_{2}\dots,i_{k},i_{1})$, such that the UAV
visits the nodes in the described order and all nodes are visited once, except
$i_{1}$. An example of our system model is shown in Fig. 1, where
$\mathcal{T}=(6,4,2,3,6)$.

Figure 1: Illustration of our system model with six areas.

For each link $(i,j)\in E$, we introduce flow variables $f_{ij}\in\\{0,1\\}$,
where $f_{ij}=1$ if and only if nodes $i,j$ appear in the trajectory
$\mathcal{T}$. The flow that enters node $i$ must be equal with the one that
goes out:

$\displaystyle\sum_{j}f_{ji}=\sum_{j}f_{ij}=\left\\{\begin{array}[]{ll}1\text{,}&\text{if
}i\in\mathcal{T}\\\ 0\text{,}&\text{if
}i\notin\mathcal{T}\end{array}\right.,\quad\forall i\in\mathcal{I}.$ (3)

However, including only the constraints in (3), we may produce tours that are
not connected. In order to create solutions that do not contain disconnected
tours, we introduce the following _subtour elimination constraints_ :

$\displaystyle\sum_{(i,j)\in\mathcal{I}\text{, }i\neq
j}f_{ij}\leq|\mathcal{S}|-1\text{,
}\forall\mathcal{S}\subset\mathcal{I}\text{.}$ (4)

For each nonempty subset $\mathcal{S}$ of the set of nodes $\mathcal{I}$, (4)
ensures that the number of edges of $\mathcal{S}$ must be at most
$|\mathcal{S}|-1$. Hence, (4) eliminates solutions with two or more
disconnected subtours. Note, that any non-zero integer flow $\boldsymbol{f}$
that satisfies (3) and (4) is a tour, i.e., a path on graph $G$ that starts
and ends at the same node. In case we need a trajectory that starts and ends
at a specific node $s\in\mathcal{I}$, we can include $\sum_{j}f_{sj}=1$ as a
constraint.

The total time of the trajectory $\mathcal{T}$ is denoted by $D$ and is equal
to the sum of all traversed distances, $D=\sum_{(i,j)}d_{ij}f_{ij}$. A
trajectory is called _feasible_ if its total time is no larger than a
specified limit $D_{\max}$:

$\displaystyle D=\sum_{(i,j)}d_{ij}f_{ij}\leq D_{\max}.$ (5)

Our target is to compute a UAV trajectory that passes through the areas with
highest scores in time less than $D_{\text{max}}$. To this end, we formulate
the UAV Trajectory Design (UTD) problem as:

$\displaystyle\max_{\boldsymbol{f}}$
$\displaystyle\sum_{(i,j)}f_{ij}\lambda_{i}$ (6a) s. t. $\displaystyle\text{
}(\ref{eq: flow}),(\ref{cons: subtour}),(\ref{cons: deadline}),$ (6b)
$\displaystyle\text{ }\sum_{j}f_{ij}\leq 1\text{, }\forall
i\in\mathcal{I}\text{,}$ (6c) $\displaystyle\text{
}\boldsymbol{f}\in\\{0,1\\}^{|\mathcal{I}|\times|\mathcal{I}|}\text{.}$ (6d)

Constraint (6c) ensures that up to one flow can enter and exit node $i$.

###  II-A Strategic Content Placement Use Case

In this subsection, we discuss the potential application of the trajectory
design to the content caching problem. We assume that each area contains users
that request popular file content. Furthermore, we assume that the UAV carries
a cache in which we can cache popular file content and deliver the files to
the users. However, the capacity of the cache is limited and the time flight
of the UAV is limited as well. Therefore, we should jointly decide the
trajectory and the file placement. Then, in problem (6), the reward would be
the product of demand times the popularities of the files at each visited
location.

##  III Algorithm for UAV Trajectory Design

###  III-A Orienteering

First, we characterize the complexity of this problem by drawing an analogy
from the _Orienteering Problem_ (OP) [9], a sport in which starting and ending
points are specified in a forest along with other locations (checkpoints) with
associated scores for visiting. Boyscouts must travel from the starting to the
ending point before a certain deadline expires, and on their way they seek to
visit a subset of the locations that maximizes the total collected score.
Consider, in the OP problem, that $\lambda_{i}$ are the boyscout rewards
collecting from each checkpoint $i$, $d_{ij}$ is the travel distance between
the checkpoints $i$ and $j$, and $D_{\text{max}}$ is the total time frame
available for waypoint collection. Then, there is an 1-1 mapping between the
UTD and OP problem. Since the OP problem is NP-hard and particularly APX-hard,
we get the following result.

Corollary 1. The UTD problem in (6) is NP-hard, and particularly, APX-hard.
222 It is APX-hard because it has a 2+e poly-time algorithm, but it has no
Polynomial-time Approximation Scheme (PTAS), i.e., it cannot be approximated
to within any constant larger than 1 [10].

Remark. The authors in [9] first defined the Orienteering problem, and show
that is NP-hard with a reduction from the _Travelling Salesman Problem_. The
work in [11] shows that the Orienteering is APX-hard, i.e., any polynomial
time algorithm will fail to approximate the optimal within $\frac{1481}{1480}$
(unless P=NP). Also, it was provided a 4–approximation using dynamic
programming to compute min-excess paths, i.e., paths that achieve a targeted
prize by introducing a minimum amount of excess cost. [12] provides a
3–approximation of the rooted Orienteering problem, based on Linear
Programming relaxation and rounding. Improved guarrantees are also given in
[13, 10] where a 2–approximation guarantee is provided using $k$-TSP
techniques.

###  III-B Subtour elimination: lazy constraints approach

Note that if the number of nodes is of size $n$, then, there are $2^{n}-2$
subsets of $\mathcal{S}$ of $\mathcal{I}$, excluding $\mathcal{S}=\mathcal{I}$
and $\mathcal{S}=\emptyset$. In order to avoid constructing an exponential
number of constraints for each scenario resulting in a formulation that is
complex even to state, we include the constraints in (4) in a lazy fashion.
More specifically, we relax all subtour elimination constraints (SECs) (4) and
solve the remaining Integer Linear Program (ILP) by using Gurobi solver333Note
that in order to obtain an optimal solution from the solver, we do not
restrict its runtime.. When the solver finds a feasible solution that
satisfies the other constraints, we check the number of edges and determine
whether the found solution has disconnected subtours or not. If the number of
edges of the shortest tour is equal to the number of visited nodes, the found
solution has no subtours, hence it satisfies the subtour elimination
constraints (even if we did not require them) and the optimization problem is
solved. Otherwise, we add the corresponding subtour elimination constraint
that is violated and solve the problem again. We repeat until the found
solution has no subtours.

Theorem 1. Lazy constraints approach is optimal.

###### Proof.

At each iteration, we find a solution of minimum cost of the relaxed problem.
We denote the relaxed minimum cost by $c_{\text{min}}$. $c_{\text{min}}$ is a
lower bound to the optimal cost of the original problem $c_{\text{opt}}$.
After the last iteration of SEC approach, we find an optimal solution to the
relaxed problem that is actually feasible in the original problem. Hence, it
must be $c_{\text{min}}\geq c_{\text{opt}}$. Therefore we conclude that
$c_{\text{min}}=c_{\text{opt}}$. ∎

We note that this approach provides no guarantees that we will not have to
eventually add all subtour elimination constraints (and hence it requires
exponentially many steps), however, experience shows that it can be quite
efficient in some problems [14].

###  III-C UAV trajectory algorithm

1 Input: Graph$=(\mathcal{I},E,{\boldsymbol{d}})$, budget tour
$D_{\text{max}}$, start/end node $s$

2 Output: trajectory $\mathcal{T}$

3 $t_{d}\leftarrow 0$ //traversed time

4 next_node $\leftarrow\infty$

5 $\mathcal{P}\leftarrow\left\\{s,s\right\\}$ //we start and end at the node
$s$

6 next_segment $\leftarrow\infty$

7 do

8 if  _$(\text{next\\_node} <I+1)$_ then

9 $\ell\leftarrow|\mathcal{P}|$

10 $m\leftarrow$ next_segment

11 $t_{d}\leftarrow
t_{d}+d_{m,\text{next\\_node}}+d_{\text{next\\_node},m+1}-d_{m,m+1}$

12 $\mathcal{P}_{m+2:\ell+1}\leftarrow\mathcal{P}_{m+1:\ell}$

13 $\mathcal{P}_{m+1}\leftarrow\text{next\\_node}$

14 for  _$\forall j\in|\mathcal{T}|-1$_ do

15 $\ell_{c}\leftarrow\emptyset$, $\ell_{e}\leftarrow\emptyset$

16 for  _$\forall i\in\mathcal{I}$_ do

17 if  _( $i\notin\mathcal{T}$)_ then

18 $i_{1}\leftarrow\mathcal{T}_{j}$

19 $i_{2}\leftarrow i$

20 $i_{3}\leftarrow\mathcal{T}_{j+1}$

21 if  _$(t_{d}+d_{j,i}+d_{i,j+1}-d_{j,j+1}\leq D_{\text{max}})$_ then

22 $\ell_{c}\leftarrow\ell_{c}\cup i$ //local candidates

23
$\ell_{e}\leftarrow\ell_{e}\cup\left\\{t_{d}+d_{j,i}+d_{i,j+1}-d_{j,j+1}\right\\}$
//add extra distance

24

25

26 if  _$(\ell_{c}=\emptyset)$_ then

27 $g_{c_{j,:}}\leftarrow\left\\{I+2,0\right\\}$ // we didn’t find any
candidate

28 else

29 $u_{k}\leftarrow\frac{\lambda_{k}}{\ell_{e_{k}}}\text{, }\forall
k\in\ell_{c}$ // Score

30 $\ell_{c_{max}}\leftarrow\text{find the node with the maximum utility
value}$

31 $g_{c}\leftarrow\ell_{c_{max}}\cup max(u)$ //assign the maximum utility
value and the corresponding node

32

33 next_node $\leftarrow\emptyset$, next_segment $\leftarrow\emptyset$

34 if  _$(max(g_{c_{:,2}} >0))$_ then

35 //we found some candidates

36 $u\leftarrow g_{c_{:,2}}$//assign all rows of the second column

37 next_segment $\leftarrow$ find node with the maximum utility

38 next_node $\leftarrow g_{c_{\text{next\\_segment,1}}}$

39

40while  _next_node $\neq\emptyset$_;

return $\mathcal{T}$

Algorithm 1 UAV trajectory design algorithm TABLE I: The case with $80$ nodes. Execution time. Optimal solution vs greedy algorithm. $D_{\text{max}}$ (min) | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16
---|---|---|---|---|---|---|---|---
Solver | 103.22 sec | 377.95 sec |  $>2$h |  $>2$h |  $>2$h |  $>2$h |  $>2$h | 41 sec
Greedy | 0.35 sec | 0.27 sec | 0.25 sec | 0.25 sec | 0.25 sec | 0.39 sec | 0.31 sec | 0.31 sec

|
---|---

(a) Collected score by the UAV.

(b) Execution time.

(c) Number of visited nodes.

(d) Total journey time.

Figure 2: The case with $60$ nodes. Optimal solutions vs greedy algorithm.

Even with the lazy constraints approach, the OP problem is APX-hard, and as
the instance of the problem increases, the solver will take too long to return
a solution (if ever). In order to have a solution in reasonable time, we
propose a heuristic that is described in Algorithm 1. Our approach is inspired
from a recent study that proposes touristic itineraries on Google maps [15].
Although our algorithm provides no guarantees, it follows the ideas of the
knapsack relaxation, i.e., greedily adding waypoints that maximize the
efficiency ratio $\frac{\text{reward}}{\text{added time}}$.

The algorithm builds a trajectory by progressively adding waypoints
considering: (i) the feasibility of the tour (step 21), (ii) the cost
efficiency of a waypoint addition by means of ratio of reward/added travel
time (step 23). Specifically, we begin with the origin and add the waypoint
$a$ that maximizes the ratio reward/distance (step 28). At this point the
trajectory is simply origin $\rightarrow$ $a$ $\rightarrow$ origin. Next, for
every hop in the trajectory, we find the maximal node that if added in the
hop, it will maximize the ratio reward/added distance (step 34). Hence it
could be $o\rightarrow b\rightarrow a\rightarrow o$ or $o\rightarrow
a\rightarrow b\rightarrow o$. Specifically for the second step, there is
symmetry and both solutions will be equal. But for the following steps, every
hop results in a possibly different maximal waypoint, and we must select the
best. At every step of the way, a waypoint can be added only if the new total
travel time does not exceed our constraint. When no such node can be found
(step 36), our heuristic has converged.

##  IV Simulation Results

We consider that the velocity of the UAV is equal to $70$ km/h444
https://www.drone-world.com/dji-phantom-4-specs/. and $100$ different
topologies with $50$ nodes each. The location of each node is randomly
generated according to a normal distribution that takes values in $[-1,1]$.
For each topology, we generate score $\lambda_{i}$ for each node. Each score
takes values according to a normal distribution in $[0,10]$. We consider that
the UAV always starts from $(0,0)$ location points. Optimal and suboptimal
trajectories are designed by the solver and algorithm, respectively, for each
topology. Then, we take the average of the collected score, execution time,
number of visited nodes, and total journey over the topologies. We repeat for
different values of $D_{\text{max}}$. We obtain the optimal solution by using
Gurobi software.

In Fig. 2(a), we compare the collected score for the solution provided by the
solver and algorithm. We observe that the score collected by greedy algorithm
solution is very close to the optimal one. The algorithm utilizes the
available budget in an efficient way, as shown in Fig. 2(c) and Fig. 2(d). The
algorithm needs less than $1$ sec to provide an approximate solution, as shown
in Fig. 2(b), that is important when we have a large system to solve or need
to run the routine multiple times within another algorithm. On the other hand,
the solver needs more than $1$ min to provide an optimal solution, and as
$D_{\text{max}}$ increases its runtime increases dramatically. However, we
observe that the runtime of the solver decreases after a certain point, as
shown in Fig. 2(b). The constraint that affects the number of the trajectory
options is (5), i.e., the flight time budget of the UAV. As the time flight
budget increases, the number of nodes that can be visited without violating
the constraint, approaches the number of the nodes that cannot be visited as
shown in Fig. 2(c). Therefore, the trajectory options increase and the solver
needs more time to provide the optimal solution. However, when
$D_{\text{max}}$ is greater than $6$ min, we observe that the number of
visited nodes is greater than the number of not visited ones. For example, for
$D_{\text{max}}=8$ min, the UAV visits $35$ nodes out of total $60$ nodes, as
shown in Fig. 2(c). Therefore, it is easier now for the solver to find an
optimal solution. To give a better intuition on this, consider that the flight
time budget is infinite. Then, the solution is trivial; visit all the nodes
without taking into account the order. The order does not affect the value of
the objective function.

Additional results are provided in Table I, for a larger topology with $80$
nodes. We see that for some cases, the solver needs more than $2$ h to provide
the solution555We set up the program to stop after $2$h of waiting period.. On
the other hand, our proposed algorithm can provide an approximate solution in
reasonable time for arbitrary number of nodes.

##  V Conclusions

In this paper, we study the trajectory design problem of a UAV that flies over
multiple areas and collects the corresponding scores. We formulate an
optimization problem in order to maximize the collected score over multiple
geographical locations. We show that the problem is equivalent to the
Orienteering Problem from operation research, and therefore it is APX-hard. We
then provide a fast heuristic algorithm, and a simplified MIP approach and
compare their performance. Simulation results show that the algorithm performs
well and provides solutions for the cases where the solver collapses. The
proposed UAV trajectory design problem can be applied for tactical network and
strategic content caching applications.

## References

  * [1] N. Zhao, F. R. Yu, L. Fan, Y. Chen, J. Tang, A. Nallanathan, and V. C. Leung, “Caching unmanned aerial vehicle-enabled small-cell networks: Employing energy-efficient methods that store and retrieve popular content,” _IEEE Veh. Technol. Magazine_ , vol. 14, no. 1, pp. 71–79, 2019.
  * [2] H. Baek and J. Lim, “Design of future UAV-relay tactical data link for reliable UAV control and situational awareness,” _IEEE Commun. Magazine_ , vol. 56, no. 10, pp. 144–150, 2018.
  * [3] Y. Zeng, R. Zhang, and T. J. Lim, “Wireless communications with unmanned aerial vehicles: Opportunities and challenges,” _IEEE Commun. Magazine_ , vol. 54, no. 5, pp. 36–42, 2016.
  * [4] X. Cao, J. Xu, and R. Zhangt, “Mobile edge computing for cellular-connected UAV: Computation offloading and trajectory optimization,” _in Proc. IEEE SPAWC_ , pp. 1–5, Jun. 2018.
  * [5] E. Lakiotakis, P. Sermpezis, and X. Dimitropoulos, “Joint optimization of UAV placement and caching under battery constraints in UAV-aided small-cell networks,” _in Proc. ACM SIGCOMM MAGESys_ , pp. 8–14, Aug. 2019\.
  * [6] X. Xu, Y. Zeng, Y. L. Guan, and R. Zhang, “Overcoming endurance issue: UAV-enabled communications with proactive caching,” _IEEE J. Sel. Areas. Commun._ , vol. 36, no. 6, pp. 1231–1244, 2018.
  * [7] M. Chen, M. Mozaffari, W. Saad, C. Yin, M. Debbah, and C. S. Hong, “Caching in the sky: Proactive deployment of cache-enabled unmanned aerial vehicles for optimized quality-of-experience,” _IEEE J. Sel. Areas. Commun._ , vol. 35, no. 5, pp. 1046–1061, 2017.
  * [8] M. Samir, S. Sharafeddine, C. Assi, T. M. Nguyen, and A. Ghrayeb, “Trajectory planning and resource allocation of multiple UAVs for data delivery in vehicular networks,” _IEEE Networking Letters_ , vol. 1, no. 3, pp. 107–110, 2019.
  * [9] B. L. Golden, L. Levy, and R. Vohra, “The orienteering problem,” _Naval research logistics_ , vol. 34, no. 3, pp. 307–318, 1987.
  * [10] C. Chekuri, N. Korula, and M. Pál, “Improved algorithms for orienteering and related problems,” _ACM TALG_ , vol. 8, no. 3, p. 23, 2012.
  * [11] A. Blum, S. Chawla, D. R. Karger, T. Lane, A. Meyerson, and M. Minkoff, “Approximation algorithms for orienteering and discounted-reward TSP,” _SIAM J. Comput._ , vol. 37, no. 2, pp. 653–670, 2007.
  * [12] Z. Friggstad and C. Swamy, “Compact, provably-good LPs for orienteering and regret-bounded vehicle routing,” _in Springer International Conference on Integer Programming and Combinatorial Optimization_ , pp. 199–211, Jun. 2017\.
  * [13] C. Chekuri and A. Kumar, “Maximum coverage problem with group budget constraints and applications,” _Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. Springer_ , pp. 72–83, 2004\.
  * [14] U. Pferschy and R. Staněk, “Generating subtour elimination constraints for the TSP from pure integer solutions,” _Central European journal of operations research_ , vol. 25, no. 1, pp. 231–260, 2017.
  * [15] Z. Friggstad, S. Gollapudi, K. Kollias, T. Sarlos, C. Swamy, and A. Tomkins, “Orienteering algorithms for generating travel itineraries,” _in Proc. Eleventh ACM WSDM_ , pp. 180–188, 2018.


Extended geometry and kinematics induced by
biquaternionic and twistor structures

Vladimir V. Kassandrova,1 and Nina V. Markovab,2

a

Institute of Gravitation and Cosmology, Peoples’ Friendship University of
Russia, Moscow, Russia

b

Department of Applied Mathematics, Peoples’ Friendship University of Russia,
Moscow, Russia

###### Abstract

The algebra of biquaternions possess a manifestly Lorentz invariant form and
induces an extended space-time geometry. We consider the links between this
complex pre-geometry and real geometry of the Minkowski space-time. Twistor
structures naturally arise in the framework of biquaternionic analysis. Both
together, algebraic and twistor structures impose rigid restriction on the
transport of singular points of biquaternion-valued fields identified with
particle-like formations.

##  1 Biquaternions as the algebra of extended space-time

The algebra of complex quaternions, biquaternions, $\mathbb{B}$ is the most
appropriate candidate for the role of a space-time algebra (the notion
introduced firstly by D. Hestenes in [1]). Indeed, its symmetry (automorphism)
group $SO(3,\mathbb{C})$ is isomorphic to the spinor Lorentz group
$SL(2,\mathbb{C})$, and its multiplication table can be represented in a
manifestly Lorentz invariant form [2, 3]. However, vector space of
$\mathbb{B}$ has four complex dimensions and does not correspond therefore to
the structure of real Minkowski space M. Physical sense of four “superfluous”
coordinates (if any) remains indefinite, and we shall concentrate on this
problem below.

Nonetheless, biquaternions and $\mathbb{B}$-valued functions, as well as the
structure of extended complex space-time $\mathbb{C}\bf M$ itself, have been
exploited in a number of interesting approaches in the field theory and
general relativity. Particularly, in GTR complex space-time serves as a source
of null shear-free congruences (NSFC) on its “real cut” which are known to
determine the structure of Robinson-Trautman metrics. Moreover, consideration
of a pointlike source “moving” in a complex extension $\mathbb{C}\bf M$
results in a number of interesting conclusions about its characteristics and
characteristics of particlelike formations – caustics of corresponding NSFC
arising on the real cut [4, 5, 6]. A well-known example of this situation is
the anomalous Dirac value [7] of the gyromagnetic ratio of the Kerr singular
ring, the caustic of corresponding NSFC. The latter property has been used in
a number of models of electron, in particular the “Dirac – Kerr-Newman
electron” elaborated by A. Burinskii [8].

As for the biquaternions themselves, there exists a wide variety of works
devoted to various applications of them in field theory and physical geometry
(for a review see, e.g., [9, 10]). In particular, the quaternionic version of
Special Relativity, based on the properties of $\mathbb{B}$ and the concept of
3-dimensional time, has been elaborated in the works of A. Yefremov [11, 12].
There is also a lot of attempts (see [9, 10]) in which physical fields are
regarded as $\mathbb{B}$-valued functions subject to a kind of
differentiablity conditions. The latter, usually constructed in a full analogy
with the Cauchy-Riemann (CR) conditions from complex analysis, play the role
of the primary field equations.

However, all these generalizations reproduce the linear CR-structure, do not
take in account the non-commutativity of $\mathbb{B}$ and, moreover, are not
mathematically justified (value of a derivative usually depends on the
direction of infinitesimal increment, contrary to the complex case).

On the other hand, in our version of non-commutative analysis (see, e.g., [13,
14, 15] and references therein) non-linearity of the differentiability
conditions for (bi)quaternionic functions follow directly from the property of
$\mathbb{B}$ non-commutativity, so that $\mathbb{B}$-differentiable functions
can be actually treated as (self-)interacting physical fields.

Moreover, the system of $\mathbb{B}$-differentiability equations possess
natural 2-spinor and twistor structures, and its general solution represents
in fact an invariant version of the celebrated Kerr-Penrose theorem [16, 17].
The latter gives a complete algebraic description of the NSFC on the Minkowski
$\bf M$ or Kerr-Shild spaces [16] via twistor generating functions. In the
turn, the NSFC define a set of fundamental (both gauge and spinor) fields for
which the corresponding vacuum equations are identically satisfied [18].
Singularities of these fields (point-, string- or membrane-like) can be
treated as particlelike formations participating in a self-consistent
collective dynamics [19]. We obtain thus a non-trivial set of interacting
fields and their singular sources whose properties are determined solely by
the $\mathbb{B}$-differentiability conditions; this approach has been called
“algebrodynamics” [20, 13].

However, the complex space-time geometry defined by the structure of
$\mathbb{B}$-algebra embarrass the physical applications of biquaternions and,
particularly, the interpretation of results obtained in the framework of the
algebrodynamical approach. In our previous works [21, 22] a number of
constructions has been presented which, rather naturally, relate complex pre-
geometry with real geometry of the Minkowski space or its extensions. These
constructions based on $\mathbb{B}$-invariant bilinear maps lead to a peculiar
kinematics of the particlelike formations; they are described in Section 2.

In Section 3 we impose some additional restrictions on the particles’
kinematics motivated by the properties of the primary $\mathbb{B}$-field and
its twistor partner. These properties define the transport of twistor field
and, in addition, the transport of singulartities, caustic-like structures. In
result, one obtains a remarkable classification of the properties of
singularities-particles themselves. Some concluding remarks are made in the
last Section 4.

##  2 Real geometry determined by the algebra of biquaternions

Biquaternion algebra $\mathbb{B}$ is isomorphic to the full $2\times 2$ matrix
algebra over $\mathbb{C}$, $Mat(2,\mathbb{C})$, so that for any
$Z\in\mathbb{B}$ one has, say, the following representation:

$Z={\bf I}z_{0}+{\bf\Sigma}\cdot{\bf
z}=\left(\begin{array}[]{cc}z_{0}+z_{3}&z_{1}-\imath z_{2}\\\ z_{1}+\imath
z_{2}&z_{0}-z_{3}\end{array}\right),$ (1)

where $\\{z_{0},\bf z\\}\in\mathbb{C}$, I is the unit matrix and
${\bf\Sigma}=\\{\sigma_{a}\\},a=1,2,3$ is the triplet of Pauli matrices.

Consider now a Hermitian ($M=M^{+}$) matrix,

$M:=ZZ^{+}=\left(\begin{array}[]{cc}T+R_{3}&R_{1}-\imath R_{2}\\\ R_{1}+\imath
R_{2}&T-R_{3}\end{array}\right)=T+\bf\Sigma\cdot\bf R,$ (2)

in which quantities $\\{T,\bf R\\}$ depend on the complex coordinates and
their conjugates as follows:

$\begin{array}[]{l}T:=z_{0}\bar{z}_{0}+\bf z\cdot\bf\bar{z},~{}~{}{\bf
R}:={\bf P}+{\bf Q},\\\ {\bf P}=z_{0}\bf\bar{z}+\bar{z}_{0}\bf z,~{}~{}{\bf
Q}=\imath~{}\bf z\times\bf\bar{z}\end{array}$ (3)

Bilinear in $Z$ quantities $\\{T,\bf R\\}$ could represent the real time and
space coordinates, respectively; however, more justified interpretation will
be discussed below.

One observes that spatial coordinates are defined as the sum of two different
parts, and the two 3D subspaces $\bf P$ and $\bf Q$ are easily seen to be
orthogonal,

${\bf P}\cdot{\bf Q}=0.$ (4)

Making now use of the well-known properties of determinants,
$\det(ZZ^{+})=(\det Z)(\det Z^{+})=|\det Z|^{2}$, one obtains the following
remarkable identity [22]:

$T^{2}-{\bf R}^{2}=S^{2},~{}~{}~{}S:=|\det Z|=|z_{0}^{2}-{\bf z}^{2}|,$ (5)

which, in account of (3) and the orthogonality relation (4), takes the form

$T^{2}-{\bf P}^{2}-{\bf Q}^{2}=S^{2}.$ (6)

It is worthy to note that, since $S^{2}=\Re(S)^{2}+\Im(S^{2})$, equation (6)
represents a remarkable nine squares identity closely related to the structure
of $\mathbb{B}$-algebra. Of course, identity (6) can be immediately written
out in an explicit coordinate form.

Consider now a particular case when the component $z_{0}$ is zero. Then
equation (5) reduces to the identity of six squares for $\bf z$ of the form

$\begin{array}[]{l}t^{2}-{\bf q}^{2}=s^{2},\\\ t={\bf z}\cdot\bar{\bf
z},~{}~{}q=\imath~{}{\bf z}\times\bar{\bf z},~{}~{}s=|{\bf
z}^{2}|,\end{array}$ (7)

which has been previously obtained and discussed in [21]. Importantly, the
identity (7) remain valid, together with (5), if even $z_{0}\neq 0$.

Under transformations from the group $SO(3,\mathbb{C})$ which preserve the
multiplication law of $\mathbb{B}$-algebra, both ${\bf z}^{2}$ and $z_{0}$ are
evidently invariant, so that the quantity $S$ (and the reduced one $s$) can be
identified with the Minkowski interval. Under real 3D-rotations, the
quantities $T$ and $\bf R$ evidently form a scalar and vector, respectively.
On the other hand, when the angle of rotation is imaginary, these quantities
transform in a way different from the Lorentz transformations. For a reduced
case such transformations have been studied in [21].

However, if we consider instead the left shift transformation $Z\mapsto
AZ,~{}A\in SL(2,\mathbb{C})$, then the corresponding Hermitian matrix
$M=ZZ^{+}$ transforms according to the canonical law, $M\mapsto AMA^{+}$, so
that the quantities $\\{T,\bf R\\}$ undergo standard Lorentz transformations.
Note that the dual mapping $Z\mapsto Z^{+}Z$ and related right shift
transformation $Z\mapsto ZA$ define in fact the same Lorentz 4-vector
$\\{T,\bf R\\}$ but for the point in $\mathbb{C\bf M}^{3}$ corresponding to
complex conjugate coordinates $\bf z\rightarrow\bar{z}$.

One should, however, bear in mind that the introduced quantities $\\{T,\bf
R\\}$, being bilinear in $Z$, are not translationary invariant. That is, for
two points $Z_{1}$ and $Z_{2}$ and corresponding real coordinates
$T_{1},T_{2}$ and ${\bf R}_{1},{\bf R}_{2}$ the holonomy conditions $\Delta
T=T_{2}-T_{1},~{}\Delta{\bf R}={\bf R}_{2}-{\bf R_{1}}$ are no longer valid.
Here $\Delta T$ and $\Delta{\bf R}$ are the quantities expressed through the
differences of complex coordinates $\Delta Z$ and their conjugates $\Delta
Z^{+}$ making use of the same definitions (3).

In view of this fact, one should identify the 4-vector $\\{T,\bf R\\}$ with
some other quantity attributed to a point of $\mathbb{C}{\bf M}^{4}$,
precisely with a set of characteristics of a material pointlike source. We
shall return to this problem below. As for the coordinates themselves, one can
define only their increments $\\{\delta T,\delta\bf R\\}$ using corresponding
definitions (3), that is:

$\begin{array}[]{l}\delta T:=dz_{0}d\bar{z}_{0}+d{\bf z}\cdot d\bar{\bf
z},~{}~{}\delta{\bf R}:=\delta{\bf P}+\delta{\bf Q},\\\ \delta{\bf
P}:=dz_{0}d\bar{\bf z}+d\bar{z}_{0}d{\bf z},~{}~{}\delta{\bf
Q}:=\imath~{}\delta{\bf z}\times\delta\bar{\bf z}\\\ \delta
S:=|dz_{0}^{2}-d{\bf z}^{2}|.\end{array}$ (8)

However, these increments do not constitute full differentials, so that
corresponding real coordinates turn our to be unholonomic. This property,
though seeming rather exotic, leads up to a number of interesting concepts and
conclusions (“random time”, geometric origin of time irreversibility,
effective curvature, etc.) some of which have been discussed in [15].

As the quantities $\\{T,\bf R\\}$ themselves, the increments of real
coordinates satisfy the identities similar to (5) and (6), that is

$\delta T^{2}-\delta{\bf R}^{2}=\delta S^{2},$ (9) $\delta T^{2}-\delta{\bf
P}^{2}-\delta{\bf Q}^{2}=\delta S^{2},$ (10)

as well as the orthogonality condition

$\delta{\bf P}\cdot\delta{\bf Q}=0.$ (11)

Note that infinitesimal Minkowski interval $\delta S$ in (8) is necessarily
non-negative, so that one deals here with only the causal part of the
Minkowski-like space-time. On the other hand, the time increment $\delta T$,
according to the definition (8), is positive definite, $\delta T>0$, so that
one observes here a sort of time irreversibility: any displacement in the
primary complex space results in an increase of time in the induced real
space.

Consider now a pointlike source moving in the primary complex space
$\mathbb{C}{\bf M}$. Then, for the case of nonzero interval, one can define a
real 4-velocity vector ($\mu=0,1,2,3$)

$V^{\mu}:=\frac{\delta X^{\mu}}{\delta S},~{}\delta X^{\mu}:=\\{\delta
T,\delta{\bf R}\\},$ (12)

which, in account of (9), is identically unit,

$V_{\mu}V^{\mu}=\left(\frac{\delta T}{\delta
S}\right)^{2}-\left(\frac{\delta{\bf R}}{\delta S}\right)^{2}=1,$ (13)

while the 3-velocity ${\bf V}:=\frac{\delta{\bf R}}{\delta T}$ is always
subluminar,

$\delta T(1-V^{2})=\delta S>0,\rightarrow V^{2}<1.$ (14)

Note finally that, in view of the identity (10) and orthogonality condition
(11) the full 3-velocity is represented by “partial” velocities

${\bf u}:=\frac{\delta{\bf P}}{\delta T},~{}~{}{\bf w}:=\frac{\delta{\bf
Q}}{\delta T}$ (15)

corresponding to the motions in the two real 3-subspaces $\bf P$ and $\bf Q$
and mutually orthogonal,

$V^{2}=u^{2}+w^{2},~{}~{}~{}{\bf u}\cdot{\bf w}=0.$ (16)

In the second case of zero interval, $\delta S=0$, when the complex cone $\det
Z=\delta z_{0}^{2}-\delta{\bf z}^{2}=0$ is mapped into the future part of the
real light cone $\delta T^{2}-\delta{\bf R}^{2}=0,~{}~{}\delta T>0$, the full
3-velocity is equal to the speed of light, $V=1$, and for partial velocities
we have instead of (16):

$u^{2}+w^{2}=1,~{}~{}~{}{\bf u}\cdot{\bf w}=0.$ (17)

Remarkably, in this case our $\mathbb{B}$-geometry leads up to the geometry
proposed earlier by I. Urusovskii [23, 24, 25] in the framework of his “3+3
treatment of the Special Relativity”. From his viewpoint, the speed of light
is not only invariant but universal, i.e. all material points have one and the
same velocity in the full 6D space.

We have seen thus that the $\mathbb{B}$-induced real geometry not only
reproduces the principal properties of the Minkowski space-time but impose, in
addition, some restrictions on admissible kinematics of pointlike particles.
In the next section we shall consider other kinematical relations which arise
on the base of $\mathbb{B}$-analysis and corresponding twistor structures.

##  3 $\mathbb{B}$-analysis, twistors and kinematics in $\mathbb{B}$-induced
real space

In the algebrodynamical program [20, 13, 14, 15] primary physical fields are
$\mathbb{B}$-differentiable functions. Conditions of
$\mathbb{B}$-differentiability, a non-commutative generalization of the
Cauchy-Riemann conditions from complex analysis, are considered as the primary
field equations; in the principal case they reduce to the overdetermined
matrix system of equations for differentials,

$d\xi=\Phi dZ\xi$ (18)

for a 2-spinor $\xi=\\{\xi_{A}(Z)\\}$ and complex 4-vector
$\Phi=\\{\Phi_{AB}(Z)\\}$ fields ($A,B=0,1$, and $Z$ is the
$\mathbb{B}$-coordinate matrix). The structure of general solution of (18) can
be easily obtained by the following its transformation

$d\xi=\Phi d(Z\xi)-(\Phi dZ)\xi,\rightarrow\xi=\xi(\tau),$ (19)

where $\tau:=Z\xi$ together with $\xi$ constitute the (projective) twistor of
complex space $\mathbb{C}{\bf M}$. Therefore, in an invariant form general
solution of (18) is represented by a system of two algebraic equations [18]

$\Pi^{(C)}(\xi,Z\xi)=0,~{}~{}C=1,2,$ (20)

with $\Pi^{(C)}$ being two arbitrary analytical functions of four twistor
variables. Making use of the projective invariant structure of the incidence
relation $\tau=Z\xi$, one can reduce (20) to a simpler form of one equation
with one homogeneous generating twistor function $\Pi$,

$\Pi(\xi,Z\xi)=0,$ (21)

from which only the ratio $G$ of two spinor components $\xi_{A}$ can be
determined at each fixed point of $\mathbb{C}{\bf M}$.

On the Minkowski subspace $\bf M$ of ${\mathbb{C}}\bf M$ represented by
Hermitian matrices of coordinates $Z\mapsto X=X^{+}$, solution of (18) in the
form (21) reproduces the celebrated Kerr-Penrose theorem [16, 17] completely
describing the structure of null shear-free congruences (NSFC) on $\bf M$
mentioned in Section 1. Thus, $\mathbb{B}$-differentiable functions explicitly
give rise to a complex extension of the Minkowski twistors and, geometrically,
to a complex extention of NSFC.

Caustics of the rays of NSFC corresponding to multiple roots of the equation
(21) and defined, therefore, by the condition

$\frac{d\Pi}{dG}=0$ (22)

Structure and dynamics of caustics correspond to those of singularities of
fundamental fields defined by NSFC which can be treated as particlelike
formations. For more details we refer the reader, say, to [26, 18] and
references therein.

Let us consider now the (infinitesimal) transformations of complex coordinates
which leave invariant the primary twistor field and the caustic structure.
Specifically, from the equation (21) as the condition of twistor preservation
one obtains $(dZ)\xi=0$, which in components leads up to the following two
real relations for differentials $dZ$:

$dy_{0}={\bf n}\cdot d{\bf y},~{}~{}~{}d{\bf x}={\bf n}~{}dx_{0}+{\bf n}\times
d{\bf y},$ (23)

where $dx_{\mu}$ and $dy_{\mu}$ are the real and imaginary parts of the
increments of four complex coordinates $dz_{\mu}=dx_{\mu}+\imath dy_{\mu}$.
The (projective invariant) unit director 3-vector $\bf n$ is formed from the
components of the principal spinor $\xi$ as follows:

${\bf n}:=\frac{\xi^{+}{\bf\Sigma}\xi}{\xi^{+}\xi},~{}~{}{\bf n}^{2}\equiv 1.$
(24)

Of course, conditions (23) can be easily inverted,

$dx_{0}={\bf n}\cdot d{\bf x},~{}~{}~{}d{\bf y}={\bf n}~{}dy_{0}-{\bf n}\times
d{\bf x},$ (25)

and the increments identically satisfy the complex null cone condition:

$dz_{0}^{2}-d{\bf z}^{2}=0.$ (26)

It is worthy to note that under transition to the Minkowski space-time, when
$Z\mapsto X=X^{+}$, conditions (23) reduce to $d{\bf x}={\bf n}dt$, where
$dt:=dx_{0}$ is the real Minkowski time. The latter condition corresponds to
the well-known transport of twistor field along a 3-direction (defined by the
initial value of the principal spinor) with velocity of light.

The complex cone condition can be equivalently written out in the form of two
real conditions:

$\begin{array}[]{l}dx_{0}^{2}+d{\bf y}^{2}=dy_{0}^{2}+d{\bf
x}^{2}=\frac{1}{2}\delta T,\\\ d{\bf x}\cdot d{\bf
y}=dx_{0}dy_{0}.\end{array}$ (27)

The last equality in the first equation follows from comparison of the latter
with the previous definition of the time increment in (8). Note that the
topology of the twistor preserving 6D increment space (actualy, that of the
complex null cone (26)), in a consequence of (27), is $S^{3}\times S^{2}\times
R_{+}$).

From three equations (27) one can, in particular, express the invariant
$|dz_{0}|^{2}=dx_{0}^{2}+dy_{0}^{2}$ through the invariants of spacial
increments,

$dx_{0}^{2}+dy_{0}^{2}=\sqrt{(d{\bf x}^{2}-d{\bf y}^{2})^{2}+(2d{\bf x}\cdot
d{\bf y})^{2}},$ (28)

or, after a simple transformation, obtain another important relation:

$(\delta{\bf x}^{2}+\delta{\bf y}^{2})^{2}-|2d{\bf x}\times d{\bf
y}|^{2}=(dx_{0}^{2}+dy_{0}^{2})^{2}.$ (29)

Thus, despite the fact that the previously defined complex and real Minkowski
intervals in the twistor preserving geometry are both zero, one can, in
account of (29), consider as infinitesimal Minkowski interval $ds$ the
invariant

$ds:=|dz_{0}|^{2}=dx_{0}^{2}+dy_{0}^{2}\geq 0,$ (30)

while for the redefined time $\delta t$ and space $\delta\bf r$ increments of
the coordinates take the quantities

$\delta t:=\delta{\bf x}^{2}+\delta{\bf y}^{2}\geq 0,~{}~{}\delta{\bf
r}:=2d{\bf x}\times d{\bf y}.$ (31)

By this identifications, the principal relation (29) of the twistor preserving
geometry takes the Minkowski-like form (again with non-negatively definite
increments $\delta t$ and $\delta s$):

$\delta t^{2}-|\delta{\bf r}|^{2}=\delta s^{2}$ (32)

(compare with the reduced identity (7) for corresponding increments in account
of the complex null cone condition $dz_{0}^{2}=d{\bf z}^{2}$).

On the other hand, for the full increments of the coordinates (8) one obtains:

$\begin{array}[]{l}\delta T=\sqrt{(d{\bf x}^{2}-d{\bf y}^{2})^{2}+(2d{\bf
x}\cdot d{\bf y})^{2}}+d{\bf x}^{2}+d{\bf y}^{2},\\\ \delta{\bf P}=2({\bf
n}\cdot d{\bf x})d{\bf x}+({\bf n}\cdot d{\bf y})d{\bf y},~{}~{}\delta{\bf
Q}=2d{\bf x}\times d{\bf y},\end{array}$ (33)

and the relations $\delta T^{2}-\delta{\bf P}^{2}-\delta{\bf
Q}^{2}=0,~{}\delta{\bf P}\cdot\delta{\bf Q}=0$ are valid for any $d{\bf
x},d{\bf y}$.

##  4 Caustic preserving geometry and effective kinematics in real space

Let us now impose, in addition to the twistor preserving condition, the
analogous requirement on the caustic structure. Specifically, let us consider
the increments of complex coordinates which transfer any caustic point into
another one. From the defining equation (22) of the caustic locus, assuming
the constancy of twistor field, one obtains the preservation condition for the
matrix components:

$dZ_{A}^{B}=\Pi_{A}\xi^{B}d\lambda,~{}~{}A,B=0,1,$ (34)

where $\Pi_{A}=d\Pi/d\tau^{A}$ are the derivatives of generating function
$\Pi$ w.r.t. two of its twistor arguments, and $d\lambda$ is one sole
infinitesimal complex parameter.

From (34) it follows that, since the twistor-valued $\Pi_{A}$ and $\xi_{A}$
remain constant, the combined twistor-caustic preserving transport takes place
along a complex null straight line. If $dz_{0}\neq 0$, condition (34) can be
equivalently rewritten in a vector form,

$d{\bf z}={\bf m}~{}dz_{0},~{}~{}{\bf m}^{2}=1,$ (35)

where ${\bf m}$ is a unit complex 3-vector,

${\bf m}:={\bf a}+\imath{\bf b};~{}\rightarrow{\bf a}^{2}-{\bf
b}^{2}=1,~{}{\bf a}\cdot{\bf b}=0.$ (36)

In the other case, when $dz_{0}=0$, condition (34) gives:

$d{\bf z}={\bf l}~{}d\tau,~{}~{}{\bf l}^{2}=0,$ (37)

with a null complex 3-vector,

${\bf l}={\bf c}+\imath{\bf d};\rightarrow{\bf c}^{2}={\bf d}^{2},~{}~{}{\bf
c}\cdot{\bf d}=0,$ (38)

and an infinitesimal complex parameter $d\tau$.

Calculating then the increments of real temporal and spacial coordinates (8),
in the first case (35), one obtains

$\delta T={\bf a}^{2}~{}ds,~{}~{}\delta{\bf P}={\bf a}~{}ds,~{}~{}\delta{\bf
Q}=({\bf a}\times{\bf b})ds,$ (39)

while the increment of proper time parameter turns to be null, $\delta S=0$!
The increment of real parameter $ds$ is defined by the square of modulus of
the zeroth complex coordinate, $ds=2|dz_{0}|^{2}$. Thus, in account of (10),
in the induced real space-time singularity moves with the full velocity equal
to the speed of light. The latter consists of the two components, ${\bf
u}=\delta{\bf P}/\delta T<1$ and ${\bf w}=\delta{\bf Q}/\delta T<1$ which both
are smaller than $c$ and mutually orthogonal.

In the second case (35), when $dz_{0}=0$, one has, respectively:

$\delta T={\bf c}^{2}~{}ds,~{}~{}\delta{\bf P}\equiv 0,~{}~{}\delta{\bf
Q}=({\bf c}\times{\bf d})ds,$ (40)

where $ds=2|d\tau|^{2}$, and again the increment of proper time parameter
$\delta S=0$. One observes that singularity again moves with the the speed of
light but in this case only one component of velocity is nonzero.

We have already mentioned that the quantities $T,\bf R$ as well as their
increments $\delta T,\delta\bf R$ are not invariant under translations and
unholonomic. Therefore, these (bilinear in complex coordinates $z_{0},\bf z$)
quantities should be interpreted not as coordinates but rather as energy and
momentum of a corresponding particlelike formation.

##  5 Conclusion

Existence of complex pre-geometry seems to be unavoidable in algebrodynamical
approach. Indeed, in the framework of the latter, the whole physical geometry
and dynamics should be derived solely from the properties of a space-time
algebra. However, the only candidate for this role is the algebra of
biquaternions with its $SO(3,\mathbb{C})\sim SL(2,\mathbb{C}$) automorphism
group and 4D complex vector space.

Thus, to construct the unified biquaternionic algebrodynamics, one should
discover the links between the complex geometry $\mathbb{C}\bf M$ and real
Minkowski space-time $\bf M$. In the most natural way one can do it via
bilinear mapping $Z\mapsto X:=ZZ^{+}$ used in the paper. Then the induced
kinematics of particles-singularities turns to be Lorentz invariant. However,
quantities $X$ are not translation invariant and correspond rather to the
energy-momentum 4-vector of a particle. As for the real coordinates
themselves, they can be identified with real parts of complex coordinates
while imaginary parts constitute the 4D timelike evolution parameter, see
(23). Note that multidimensional time can be the origin of real time
irreversibility, of quantum uncertainty etc. [15].

In the paper, we have also studied the effective kinematics of a singular
point defined by the requirements of preservation of the primary twistor field
and the singularity itself. These conditions arise in the framework of
$\mathbb{B}$-analysis and lead to two different types of the transport of a
singular point along a straight line in both of which, however, the full real
space velocity of transport is equal to the speed of light. Note, however,
that for the points on a focal line (unique Worldline) [19] the effective
kinematics (dynamics) can be much more complicated since the primary spinor
field of the congruence $G$ is indefinite on the worldline itself. Complete
investigation of this case will be presented elsewhere.

## References

  * [1] Hestenes D 1966 Space-time algebra (New York: Gordon and Breach)
  * [2] Grgin E Phys. Lett. B431 15
  * [3] Kassandrov V V and Rizcallah J A 2016 Relativistic algebra of space-time and algebrodynamics Grav. Cosm. 22 230 (Preprint 1612.02455)
  * [4] Kozameh C N and Newman E T 2005 Electromagnetic dipole radiation fields, shear-free congruences and complex center of charge worldlines Class. Quant. Grav. 22 4667 (Preprint gr-qc/0504093)
  * [5] Newman E T 2002 On a classical geometric origin of magnetic moments, spin-angular momentum amd the Dirac gyromagnetic ratio Phys. Rev. D 65 104005 (Preprint gr-qc/0201055)
  * [6] Burinskii A Ya 2003 Complex Kerr geometry and nonstationary Kerr solutions Phys. Rev. D 67 124024 (Preprint gr-qc/0212048)
  * [7] Carter B 1968 Phys. Rev. 174 1559
  * [8] Burinskii A Ya 2008 The Dirac – Kerr-Newman electron Grav. Cosm. 14 109 (Preprint hep-th/0507109)
  * [9] Gsponer A and Hurni J P 2005 Quaternions in mathematical physics (1): alphabetical bibliography (Preprint math-ph/0510059)
  * [10] Gsponer A 2005 Quaternions in mathematical physics (2): analytical bibliography (Preprint math-ph/0511092)
  * [11] Yefremov A P 1996 Grav. & Cosm. 2 77; 2 355
  * [12] Yefremov A P 2005 Quaternionic Spaces, reference frames and fields (Moscow: Peoples’ Friendship Univ. Press, in Russian)
  * [13] Kassandrov V V 1995 Biquaternionic electrodynamics and Weyl-Cartan geometry of space-time Grav. & Cosm. 1 216 (Preprint gr-qc/0007027)
  * [14] Kassandrov V V 2007 Quaternionic alalysis and the algebrodynamics Space-time structure: algebra and geometry eds. Pavlov D G et al. ( Moscow: Lilia-Print) p. 441 (Preprint 0710.2895)
  * [15] Kassandrov V V 2009 Algebrodynamics over complex space and phase extension of the Minkowski geometry Phys. Atom. Nuclei 72 813 (Preprint 0907.5425)
  * [16] Debney G C, Kerr R P and Schild A 1969 J. Math. Phys. 10, 1812
  * [17] Penrose R and Rindler W 1986 Spinors and space-time, vol.2 (Cambridge: Cambridge Univ. Press)
  * [18] Kassandrov V V and Rizcallah J A 2017 Maxwell, Yang-Mills, Weyl and eikonal fields defined by any shear-free congruence Int. J. Geom. Meth. Mod. Phys. 1750031 (Preprint 1612.06718)
  * [19] Kassandrov V V, Khasanov I Sh and Markova N V 2015 Collective Lorentz invariant dynamics on a single “polynomial” worldline J. Phys. A: Math. Theor. 48 395204 (Preprint 1501.01606)
  * [20] Kassandrov V V 1992 Algebraic structure of space-time and algebrodynamics (Moscow: Peoples’ Friendship Univ. Press, in Russian)
  * [21] Kassandrov V V 2005 Algebrodynamics in complex space-time and the complex-quaternionic origin of Minkowski geometry Grav. Cosm. 11 354 (Preprint gr-qc/0602088)
  * [22] Kassandrov V V 2012 Urusovskii’s geometry, algebrodynamics and universal quantum-like kinematics in complex space Proc. Int. Sci. Meet. on Physical Interpretation of Relativity Theory (PIRT-2011), eds. M. C. Duffy et al. ( Moscow: Bauman Tech. Univ. Press) p. 131 (Preprint 1209.2377)
  * [23] Urusovskii I A 1996 Zarubezhnaya Radioelectronika No. 3, 359
  * [24] Urusovskii I A 2000 Zarubezhnaya Radioelectronika No. 6, 66
  * [25] Urusovskii I A 2009 Proc. Int. Sci. Meet. on Physical Interpretation of Relativity Theory (PIRT-2009) eds. M. C. Duffy et al. (Moscow: Bauman Tech. Univ. Press) p. 173
  * [26] Kassandrov V V 2004 Singular sources of Maxwell fields with self-quantized electric charge Has the last word been said on classical electrodynamics? eds. A. Chubykalo et al. ( Rinton Press) p. 42 (Preprint physics/0308045)