## Abstract

An original mathematical instrument matching two different operational procedures aimed to change orientation and velocity of a spacecraft is suggested and described in detail. The tool’s basements, quaternion algebra with its square-root (pregeometric) image, and fractal surface are represented in a parenthetical but in a sufficient format, indicating their principle properties providing solution to the operational task. A supplementary notion of vector-quaternion version of relativity theory is introduced since the spacecraft-observer mechanical system appears congenitally relativistic. The new tool is shown to have a simple pregeometric image of a fractal pyramid whose tilt and distortion evoke needed changes in the spacecraft’s motion parameters, and the respective math procedures proved to be simplified compared with the traditionally used math methods.

### Keywords

- spacecraft motion
- operation
- quaternion
- fractal surface

## 1. Introduction

In classical mechanics, rotation of a rigid body (in particular, a spacecraft) and its translational motion are normally regarded as drastically different actions leading to changes in its position and are respectively described by different groups. Relativistic mechanics, in its turn, deals with these two types of motions “more homogeneously” since rotation and linear motion are described in this case by 4 × 4 matrices from the Lorentz group

In this study, we suggest an essential development of the last (single rotation) method leading, first, to noticeable simplification of computations, and second, to possibility of introduction of additional parameters responsible for the spacecraft acceleration. This development is fully based on fundamental properties of subgeometric dyad forming the fractal space in a way underlying the 3D physical space. Moreover, we suggest subgeometric images (fractal joystick and fractal pyramid) of the math tools realizing the spacecraft’s reorientation and acceleration tasks. As well, we give a brief comparative analysis of simplicity (or complexity) of conventional and new methods.

The study is composed as following. In Sections 2–4 we offer a detailed mathematical introduction. In Section 2, we renew our knowledge of quaternion algebra giving traditional (Hamiltonian) and more compact (tensor) notions and correlations. In Section 3, we briefly reproduce the quaternion version of the relativity theory. In Section 4, we consider main notions and properties of the 2D fractal space and show how to build a 3D frame out of a dyad element.

Sections 5–7 are devoted to new math methods making operations of a spacecraft simpler and more functional. Section 5 is devoted to presentation of three methods to reorient a spacecraft with accent on convenience of the single rotation method involving a fractal joystick model. In Section 6, we suggest a very simple way to introduce (apart from space rotation) an acceleration of the spacecraft and demonstrate a subgeometric image of the respective math tool having a shape of fractal pyramid. Finally, in Section 7, we give a sketch of a technological map previewing necessary steps to simultaneously reorient and accelerate the spacecraft followed by a series of relevant pictures.

## 2. Basic notions and relations of quaternion algebra

Quaternion (Q-) numbers were discovered by Hamilton in 1843 [3]. A quaternion is a math object of the type

Q-numbers and the multiplication law (1) can be more compactly rewritten in the vector (and tensor) notations

Summation in repeated indices is implied, and

Quaternions admit the same operations as real and complex numbers. Comparison of Q-numbers is reduced to their equality: two Q-numbers are equal if coefficients at respective units are equal. Commutative addition (subtraction) of Q-numbers is made by components. Q-numbers are multiplied as polynomials; the rules (1, 2) state that multiplication is noncommutative (left and right products are defined), but still associative. A quaternion

Written in components, Eq. (3) becomes the famous identity of four squares

Identities of the type (4) exist only in four algebras: of real numbers (trivial identity), of complex numbers (two squares), of quaternions (four squares), and of octonions (the last exclusive algebra with one real and seven imaginary units admits identity of eight squares; multiplication in this algebra is no more associative).

Geometrically, the imaginary Q-units are associated with three unit vectors initiating a Cartesian coordinate system (Q-triad, Q-frame). This image, in particular, follows from the fact that, according to Eq. (2), each imaginary unit appears as ordered product of the two others:

We form the product of the two units and demand that its trace vanishes that is given as

then Eq. (6) gives expression for the third imaginary Q-unit * i*:

and the imaginary Q-triad given as Eq. (7) describes a constant Q-vector frame.

However, a Q-frame may be variable, rotating, and moving. There are two types of transformations changing the frame but retaining the form of the multiplication law (2). The first is rotational-type transformation

where

Superposition of any number (N) of real rotations (product of relevant matrices) gives a (nonplane) real rotation

Product of multiple hyperbolic rotations is physically sensible if accompanied by real rotations in the framework of vector version of theory of relativity (see Section 3); so in general, the matrices of the type

are used in applications.

The second type of transformations is performed by an operator

It is evident that the transformation (12) keeps the form of the basic law (2). The operators

As well, in formulation of quaternion relativity (see Section 3), we shall need notion of a biquaternion (BQ-) number. Such a number has the form

whose real and imaginary parts are mutually orthogonal

There are evidently zero dividers in Eq. (14), hence division is not well defined, but the subset (13 and 14) comprises basic formulas describing relative motion of arbitrary accelerated frames of reference.

## 3. Vector-quaternion version of the relativity theory

According to Eqs. (13) and (14), the interval of Einstein’s relativity theory^{1}

admits a BQ-square root

where displacement of observed object

then the orthogonality condition is fulfilled automatically

The interval (15) is invariant under Lorentz transformations of coordinate system ^{2} [5, 6]

Eqs. (17) and (18) in particular mean that within the group

The matrix (19) may describe a series of simple rotations, but real rotations should be always performed about vector

Main idea of Q-version of relativity is to replace line element of Einstein’s relativity (15) and its invariance under Lorentz group by adequate BQ-vector (16) invariant under rotational group represented by matrices

Physical measurements in the Q-model are made with the help of three spatial rulers

Now, the principal statement of the Q-version of relativity follows: all physically sustainable frames of reference are interconnected by “rotational equations”

The sustainability means form-invariance of BQ-vector (16) or (20) under transformations (21). Kinematic effects of special relativity are straightforwardly found in the Q-version; here, we demonstrate only one effect important for fractal pyramid technology accelerating a spacecraft (see Section 6).

* Boost*. Σ-observer always can align one of his spatial vectors (e.g.,

Let the frame

with the matrix

yields familiar coordinate transformations

with respective effects of length and time segments contraction. If observed particle is the body of reference of the frame Σ′, then

Specific features of the Q-vector version of relativity will be effectively used below in the fractal-pyramid math method to operate a spacecraft. Now, we turn to notions of a fractal space.

## 4. Fractal space underlying physical space

In this section, we show that a 3D space (e.g., physical space) may be endowed with a pregeometry [7] mathematically described by a complex-numbered surface, a 2D fractal space, each its vector having dimensionality half compared to that of the 3D space. We start with 2D space and construct out of its basic elements a basis of 3D space.

Let there exist a smooth 2D space (surface) endowed with a metric

the surface may be curved, so covariant and contravariant metric components differ. In a point, we choose a couple of unit orthogonal vectors

A domain of the surface in vicinity of the dyad’s initial point (together with respective part of tangent plane having the metric

Considering direct (tensor) products of the dyad vectors with mixed components [8], we can construct only four such products (

and two nilpotent matrices

Next, we built sum and difference of the idempotent matrices

and sum and difference of the nilpotent matrices

If the units Eqs. (31b) and (31c) are slightly corrected so that their product is the third unit (31d), then we obtain the basis of quaternion (and biquaternion) numbers

Now, we recall the spectral theorem (of the matrix theory) stating that any invertible matrix with distinct eigenvalues can be represented as a sum of idempotent projectors with the eigenvalues as coefficients, the projectors being direct products of vectors of a biorthogonal basis. The unit

Right and left eigenfunctions of

As mentioned above, the similarity transformation of the units

preserves the form of algebras’ multiplication law (2). Therefore, vector units from Eq. (32) can be obtained from a single unit, say,

Hereinafter, we introduce shorter 2D-index-free matrix notations for the dyad: a vector is a column, a co-vector is a row, and a parity indicator

this helps to rewrite the above expressions more compactly. The dyad orthonormality conditions (28, 29) acquire the form

the idempotent projectors are denoted as

and the units (32) are expressed through the single dyad vectors (co-vectors) as

Eq. (37) obviously demonstrates that the dyad elements are in a way “square roots” from 3D vector units. So, if we put dimensionality of any 3D line to be a unity, then dimensionality of a line on the 2D space (e.g., dimensionality of a dyad vector) must be ½; hence from the viewpoint of the 3D space, the surface determined by a dyad is fractal. The next important observation concerns transformations. The transformation (34) clearly results from the

So, apart from vector-type (8) and spinor-type (12) transformations of a Q-triad (an element of 3D space), there exists a possibility to deal with more fundamental math elements, vectors, and covectors describing “pregeometric” 2D cell of a fractal surface. These simpler math objects are subject to evidently simpler mapping (38); moreover, in the following sections, we will show that the operators of the transformations, being themselves BQ-numbers, suggest simpler and less numerous equations to solve, thus reducing degree of math load and probability of mistakes.

## 5. Three methods to reorient a spacecraft and fractal joystick

The orientation tasks are relevant with computations over 3D flat space modeling a local domain of the physical space. Two types of the orientation problem solutions are traditional: (i) a series of subsequent several angles rotation and (ii) a one-angle rotation about an instant axis. Mixed variants exist, but are less productive, and they are not normally considered.

If magnitudes involved in calculations are generically measured in real numbers, then both techniques (i) and (ii) are based on the vector rotation group

Quaternions are widely known to fit better than real numbers for the orientation tasks mostly due to the fact that three vector units represent models of three mutually orthogonal gyroscope axes. As well, use of the Q-algebra formalism essentially simplifies calculations, especially for the technique (ii), since both the vector rotation group

### 5.1. Quaternion SO(3,R) approach to the reorientation problem: Technique (i)

Orientation of a spacecraft in 3D space is determined by three angles between axes of some global coordinate system and unit vectors of a frame attached to the moving body taking into account its physical symmetry. The global coordinates, e.g., are represented by a spherical system, and its local initiating vectors pointing:

Outlined above technique (i) demands that the matrix

Direct reorientation problem, i.e., reaching object’s assigned orientation, can be solved by a sequence of plane rotations mathematically described by a sequent multiplication of matrices [see Eq. (10)]. This problem has no unique solution since the group

Even with more difficult, we meet trying to use matrices from the group

### 5.2. Reorientation by a single rotation of the quaternion frame: Technique (ii)

Consider a

One readily demonstrates that the matrix

where

and

The unit vector (44) represented through the constant basis (7) has the form;

therefore,

with

Eq. (47) in fact interlinks the

Multiplied by

Introducing now two artificial unit vectors with complex number components

Eq. (49) is just an explicit formulation of the spectral theorem applied on a 3D orthogonal matrix. Since its determinant differs from zero, this matrix is nonsingular, all its eigenvalues

Here,

The value of the single rotation angle follows from computation of the trace of the matrix (49)

antisymmetric part of the matrix yields the components of unit vector directing the rotation axis

Eqs. (51) and (52) represent parameters of the single rotations, the angle Φ. and components

### 5.3. Reorientation as transformation of a fractal surface, technique (iii)

In Section 4, we demonstrated that each vector of any Q-triad

Further on, we use for the dyad the eigenvectors

Normalization and orthogonality conditions are identically satisfied. The matrix

Therefore, Eq. (53) takes the form

Eq. (56) shows that the nonlinear problem formulated within the technique (ii), on the fractal surface level, is reduced to a linear task of the 2D basis rotation.

To get technological formulas convenient for fast numerical computation, we denote the final values of the new 2D basis as

Then, we notice that only one new dyad vector is to be computed,

The second vector

This helps to represent the 3D reorientation processes “subgeometrically”, on the 2D fractal level, as a displacement of a “joystick” tool (see [12] and Figure 1).

2D complex-numbered space can be imaged as a pyramid (with no base) consisting of one real, one imaginary, and two mixed real-imaginary joined surfaces. The joystick has one of its end matched with the pyramid’s top by a hinge; a certain shift of the stick gives components of a new dyad vectors and co-vectors. From these fractal elements, a new Q-frame providing the assigned reorientation of the spacecraft is straightforwardly built.

All reorientation parameters providing operations in the fractal space are in fact the components of the matrix

Eqs. (59), (52), (56), and (37) suggest a very simple algorithm for computation of all parameters of a single rotation and resulting matrices of a reoriented Q-triad describing new orientation of a spacecraft.

The technological scheme of the reorientation procedure can be briefly outlined as the following steps:

A spacecraft reorientation is assigned by a series of simple rotations [Eq. (40)].

Components of the rotation axis vector are computed [Eq. (52)].

The angle of fractal rotation is computed [Eq. (59)].

The dyad and resulting Q-triad are computed [Eqs. (56), (37), much simpler than in Eq. (49)].

If the computed and assigned frames match, then the rotation parameters are sent to the operational systems realizing the reorientation.

The study suggested in Section 5 gives detailed analysis of math mechanisms linking two different approaches to solution of an object’s reorientation task, a consequent 3D rotations described by matrices and a single rotation about an instant axis described by matrices. We like to emphasize importance (and original form) of Eqs. (48) and (49) explicitly demonstrating the projector-eigenvalue decomposition of any

However, thorough analysis of the Q-math reveals its additional, and important, option quite helpful in operational tasks. Namely, extension of the groups

## 6. Hyperbolic rotations and a fractal pyramid

In this section, we essentially extend the methods briefly described above. The crucial point of the extension is introduction of an imaginary parameter of rotation, thus involving hyperbolic functions. We assume that this action will result in possibility to control not only orientation, but as well dynamics of the spacecraft. We will prove the assumption within extended formulation of the technique (iii).

But at first, to make the picture more clear, we show it in framework of 3D serial rotations [technique (i)], and for simplicity, we implement just one supplement plane hyperbolic rotation about one axis

so that hyperbolic functions are introduced. Then, complete rotational operator is

We rewrite the operator (61) in the spinor-type form where the tilde denotes some initial basis

and the components of the instant rotation axis vector given by Eq. (52). It is important to note that in the computation procedure, we have to deal with vectors belonging to the same frame. Therefore, we express

This expression is again a quaternion and we denote it as

where

parameter

Expression for the vector-directing axis of the single rotation is found from Eqs. (65) and (66)

Eq. (61) represents an operator performing the serial rotation, and Eqs. (65), (68) give parameters of a single rotation. Physical content of this rotation is easily revealed when the mapping is made in the fractal surface format, and then returned into 3D space. Despite seeming complexity of the given expressions, the final calculation is shown to be very simple.

So, following the ideology of geometrization of the algebraic actions, we plunge into the fractal medium, and we consider the technique (iii). We rewrite fractal mapping with the operator (62) in the form

where the intermediate dyad is a result of the real rotation (similar with the covectors)

We also stress that all dyad elements used in the computations are always the eigenvectors (eigencovectors) of the quaternion unit

hence, Eq. (69) produces a new fractal basis simply multiplying the intermediate dyad by an exponent

By other words, one dyad vector and one co-vector (here

This primitive mapping has clear physical sense concerning kinematic of a spacecraft. To reveal it, we, using Eq. (75), build an “imaginary constituent” of the 3D frame vector

However from Eqs. (37b, c), we find

substitution of the Eq. (74) into Eq. (73) yields

Eq. (75a) rewritten in terms of the Pauli-type matrices [as in Eqs. (20), (22)]

Using results of Section 3, we associate the hyperbolic functions with the time ratio

(linking time

Then, Eq. (75b) takes the form of “vector interval” of quaternion version of relativity theory (23)

when squared, it gives the spacecraft’s special relativistic space-time interval linked with the frame at rest by the Lorentz (hyperbolic) transformation

describing kinematics of a frame moving along

besides, the velocity modulus may be variable in time; hence, the spacecraft is accelerated.

So, introducing imaginary rotation angles, we obtain a possibility to control an arbitrary space reorientation of a spacecraft with variation of its velocity in the direction that can be as well changing with time (In this sample, the vector

This math tool has two important properties. First, a spacecraft endowed by the tool with a velocity is initially described as a relativistic system; one comes to the classical mechanics considering the hyperbolic parameter small. Second, the tool accelerates the spacecraft always in the direction of the frame vector appointed to indicate “yaw”; if this vector rotates, changing the yaw, the acceleration arrow changes with it; i.e., the spacecraft is accelerated along a curve line. These properties can be useful in real motion control.

On the 2D fractal level, the spacecraft’s more complex 3D motion comprising reorientation and acceleration is accompanied by respective rotation and deformation of the mentioned above fractal pyramid. Here, this subgeometric image of the math instrument necessarily enriches a simpler model of the joystick, and moreover, to make the picture symmetric, we show positive and negative directions of the pyramid (see Figure 2).

Computations providing the spacecraft’s reorientation and acceleration are performed on the fractal level by Eq. (58) with the functions

with hyperbolic conjugation (

where vector

and rest of the dyad elements is found by primitive math actions

Eqs. (82), (37) immediately give expressions of all spacecraft’s frame vectors, thus solving the reorientation and acceleration problem in explicit form.

One straightforwardly finds that use of the fractal technique (iii) essentially simplifies computation procedures. In paper [13], we compare math difficulty of the discussed three techniques in solution of the simple problem of the spacecraft’s one-plane space rotation and acceleration. It is demonstrated there that the techniques (i) and (ii) demand solution of at least seven equations, among them are matrix equations, while the fractal technique (iii) suggests solution of only four relatively simple algebraic equations.

## 7. Technological scheme and concluding remarks

A sketch of technological scheme aimed to realize mixed rotation-acceleration maneuver of a spacecraft can be suggested as the following consequence of actions fit for any mentioned above approach.

The initial and final parameters of reorientation and acceleration are assigned and memorized.

Parameters as functions of time must be determined and input.

Time intervals are divided into standard steps (quantized), the standard input.

Process of computation of quantum steps starts resulting in obtaining of a series of related parameter values describing the orientation and velocity of the spacecraft’s frame.

The data of each step are transmitted to the systems changing the spacecraft orientation and velocity until the assigned values are achieved.

And we emphasize two most important results of this study.

First, we succeeded to show that an extrarotation by an imaginary angle entails endowing a spacecraft with a (relativistic) velocity, hence in addition to reorientation, to accelerate it. This math observation seems to be a novel one since no similar information is met in related literature.

Second, we show that the most mathematically economical way to compute operational parameters needed for realization of the maneuver is to utilize the “fractal pyramid” technique (definitely a new tool) comprising minimal number of math actions, where major of them are simple algorithms, other approaches having no such advantages.

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## Notes

- Standard interval of special relativity is regarded for simplicity; similarly, interval of general relativity can be considered in tangent space ds2=θ02−θkθk with θα=gαλdyλ being basic one-form and Greek indices in brackets enumerating tangent space tetrad, and those without brackets are related to curved manifold holonomic coordinatesηαβ=diag1−1−1−1
- ds2=θ02−θkθkθα=gαλdyλ: four-dimensional indices are raised and lowered by Minkowski metric ηαβ=diag1−1−1−1.