2026-11-01
Radius vector in split-geometry. Spherical and cylindrical forms
Two-phase representation of three-dimensional coordinates
A point in space can be specified by three coordinates. Often, however, it is more convenient to give its distance from the origin and two angles: the inclination of its direction from the vertical and its orientation in the horizontal plane. This is the idea behind spherical coordinates. In this article, we combine the two angles in a single operator expressed in an idempotent basis to obtain a compact representation of a position vector. We then show how to convert this representation into spherical, cylindrical, and Cartesian coordinates, recover the original parameters, and calculate distances.
This construction follows “From Euler’s Formula to Split Geometry” and “Split Sine and Split Cosine of Two-Phase Geometry”. The problem considered here is purely mathematical: both phase parameters are independent and are not assigned any physical meaning.
1. Two Phases in One Representation
We use two complementary idempotents:
\[\tag{1} \ep^2=\ep,\qquad \em^2=\em,\qquad \ep\em=0,\qquad \ep+\em=1. \] The hyperbolic unit and the operator of two independent phases are given by
\[\tag{2} \j=\ep-\em,\qquad \j^2=1, \] \[\tag{3} J(a,b)=\j^a(-\j)^b =\ep e^{i\pi b}+\em e^{i\pi a}. \] Equation (3) uses principal values of the powers, with the phase of minus one chosen as \( \pi \). To avoid repeating the factor \( \pi \) in the geometric formulas, we introduce angles in radians:
\[\tag{4} \tilde a=\pi a,\qquad \tilde b=\pi b. \] The phase representation then becomes
\[\tag{5} J(a,b)=\ep e^{i\tilde b}+\em e^{i\tilde a}. \] The two angles are stored in separate idempotent components. We introduce a nonnegative radial scale:
\[\tag{6} \boxed{Q=r J(a,b),\qquad r\ge0.} \] To cover all spatial directions, we choose the ranges
\[\tag{7} 0\le\tilde a\le\pi,\qquad 0\le\tilde b<2\pi, \qquad\text{that is}\qquad 0\le a\le1,\quad 0\le b<2. \] The expression \( Q \) is an algebraic representation of the parameters. To make it represent a point in three-dimensional space, we must specify a mapping to spatial coordinates. Below, this is provided by the spherical mapping. The orthogonality of the idempotent components alone does not determine this mapping.
2. Spherical Form
Let \( \tilde a \) be the polar angle measured from the positive \( z \)-axis, and let \( \tilde b \) be the azimuth measured in the \( xy \)-plane from the positive \( x \)-axis toward the positive \( y \)-axis. The spherical coordinates can be read directly from the parameters:
\[\tag{8} (r,\theta,\varphi)=(r,\tilde a,\tilde b). \] First consider the unit sphere. The height of a point is \( \cos\tilde a \), and the radius of its horizontal cross-section is \( \sin\tilde a \). The azimuth determines the projections of this horizontal radius onto the \( x \)- and \( y \)-axes. Multiplying by \( r \), we obtain the spatial mapping:
\[\tag{9} \boxed{ \mathcal P[rJ(a,b)]=\mathbf r =r\begin{pmatrix} \sin\tilde a\cos\tilde b\\ \sin\tilde a\sin\tilde b\\ \cos\tilde a \end{pmatrix}. } \] Thus, the Cartesian coordinates are
\[\tag{10} x=r\sin\tilde a\cos\tilde b,\qquad y=r\sin\tilde a\sin\tilde b,\qquad z=r\cos\tilde a. \] We can verify the length of the position vector by adding the squares of the horizontal coordinates and then the square of the height:
\[\tag{11} x^2+y^2 =r^2\sin^2\tilde a \bigl(\cos^2\tilde b+\sin^2\tilde b\bigr) =r^2\sin^2\tilde a, \] \[\tag{12} x^2+y^2+z^2 =r^2\bigl(\sin^2\tilde a+\cos^2\tilde a\bigr)=r^2. \] The parameter \( r \) is therefore the Euclidean length of the spatial position vector. Equation (9) determines this length through the mapping \( \mathcal P \); there is no need to identify it with an arbitrary algebraic norm of \( Q \).
3. Cylindrical Form
In cylindrical coordinates, a point is specified by its distance \( \rho \) from the \( z \)-axis, its azimuth \( \varphi \), and its height \( z \). The same geometry of a horizontal cross-section gives
\[\tag{13} \boxed{ \rho=r\sin\tilde a,\qquad \varphi=\tilde b,\qquad z=r\cos\tilde a. } \] The horizontal coordinates are then those of a point on a circle of radius \( \rho \):
\[\tag{14} x=\rho\cos\varphi,\qquad y=\rho\sin\varphi. \] They can be combined using an ordinary complex exponential:
\[\tag{15} x+iy=\rho e^{i\varphi} =r\sin\tilde a\,e^{i\tilde b}. \] The length of the spatial position vector follows from the Pythagorean theorem:
\[\tag{16} r^2=\rho^2+z^2. \] The phase \( \tilde b \) determines the position on the horizontal circle, while the phase \( \tilde a \) determines its radius and height. The spherical and cylindrical forms describe the same point and differ only in the choice of intermediate coordinates.
4. Connection with Split Sine and Split Cosine
In two-phase geometry, the real and imaginary parts of the operator define split cosine and split sine:
\[\tag{17} \Cos(a,b)=\ep\cos\tilde b+\em\cos\tilde a, \] \[\tag{18} \Sin(a,b)=\ep\sin\tilde b+\em\sin\tilde a, \] \[\tag{19} J(a,b)=\Cos(a,b)+i\Sin(a,b). \] The \( \ep \) component contains the azimuthal projections \( \cos\tilde b \) and \( \sin\tilde b \). The \( \em \) component contains \( \cos\tilde a \) and \( \sin\tilde a \), which specify the height and the radius of the horizontal cross-section as fractions of the total length. Thus, all angular data needed for equations (10) and (13) are already contained in the split functions.
For example, to calculate \( x \), we extract the numerical coefficient \( \sin\tilde a \) from the \( \em \) component of split sine and the coefficient \( \cos\tilde b \) from the \( \ep \) component of split cosine. We multiply their ordinary numerical product by \( r \). The coordinate \( y \) is obtained similarly, while \( z \) requires only the coefficient \( \cos\tilde a \).
Extracting numerical coefficients is essential: directly multiplying the idempotent components produces no mixed terms, because \( \ep\em=0 \). The spherical mapping combines the two phases through a separate geometric rule.
The split cosine of the phase difference between two operators can be calculated directly from the operators themselves. Let \( J_1=J(a_1,b_1) \) and \( J_2=J(a_2,b_2) \), and let the overbar denote complex conjugation with respect to \( i \), leaving the idempotents unchanged. Since \( J\overline J=1 \), the conjugate operator equals the inverse. The composition law gives
\[\tag{20} J_1\overline{J_2} =J(a_1-a_2,b_1-b_2) =\ep e^{i(\tilde b_1-\tilde b_2)} +\em e^{i(\tilde a_1-\tilde a_2)}. \] The conjugate product has the opposite phases:
\[\tag{21} \overline{J_1}J_2 =J(a_2-a_1,b_2-b_1) =\ep e^{-i(\tilde b_1-\tilde b_2)} +\em e^{-i(\tilde a_1-\tilde a_2)}. \] Adding expressions (20) and (21) and dividing by two cancels the imaginary parts in each component. The half-sum of the exponentials gives the ordinary cosine:
\[\tag{22} \frac{J_1\overline{J_2}+\overline{J_1}J_2}{2} =\ep\cos(\tilde b_1-\tilde b_2) +\em\cos(\tilde a_1-\tilde a_2). \] The right-hand side is precisely the definition of split cosine of the phase difference. Therefore,
\[\tag{23} \boxed{ \Cos(a_1-a_2,b_1-b_2) =\frac{J_1\overline{J_2}+\overline{J_1}J_2}{2}. } \] Equation (23) gives split cosine through multiplication, conjugation, and addition of operators, without separately recovering the angles. The result retains two cosines of phase differences, one in each idempotent component. The ordinary cosine of the spatial angle between position vectors is a scalar; for the chosen spherical mapping, it is calculated using equation (28).
5. Inverse Transformation
Suppose the Cartesian coordinates of a point are known. First, we find the total length of its position vector and its distance from the \( z \)-axis:
\[\tag{24} r=\sqrt{x^2+y^2+z^2},\qquad \rho=\sqrt{x^2+y^2}. \] For \( r>0 \), the angles are recovered as follows:
\[\tag{25} \tilde a=\operatorname{atan2}(\rho,z),\qquad \tilde b=\operatorname{atan2}(y,x)\pmod{2\pi}. \] The function \( \operatorname{atan2}(u,v) \) accounts for the signs of both coordinates: it returns the argument of the complex number \( v+iu \). For the azimuth, we choose the representative in \( [0,2\pi) \). The original operator parameters are
\[\tag{26} a=\frac{\tilde a}{\pi},\qquad b=\frac{\tilde b}{\pi}. \] If cylindrical coordinates are known, the formulas are even simpler:
\[\tag{27} r=\sqrt{\rho^2+z^2},\qquad \tilde a=\operatorname{atan2}(\rho,z),\qquad \tilde b=\varphi\pmod{2\pi}. \] On the positive \( z \)-axis, the polar angle is zero; on the negative axis, it is \( \pi \). In both cases, the azimuth is arbitrary. At the origin, both angles are arbitrary because \( Q=0 \) for any \( J \). The correspondence therefore becomes one-to-one only after conventions have been chosen for these special cases.
6. Angle, Dot Product, and Distance
Consider two representations, \( Q_1=r_1J(a_1,b_1) \) and \( Q_2=r_2J(a_2,b_2) \). For nonzero position vectors, the angle \( \gamma \) between their spatial directions is determined by
\[\tag{28} \cos\gamma =\cos\tilde a_1\cos\tilde a_2 +\sin\tilde a_1\sin\tilde a_2 \cos(\tilde b_1-\tilde b_2). \] This follows by substituting coordinates (10) into the ordinary dot product. Next,
\[\tag{29} \mathbf r_1\cdot\mathbf r_2=r_1r_2\cos\gamma, \] \[\tag{30} d=\lvert\mathbf r_1-\mathbf r_2\rvert =\sqrt{r_1^2+r_2^2-2r_1r_2\cos\gamma}. \] On a sphere of radius \( r \), the straight-line distance and the shortest arc length are different:
\[\tag{31} d=2r\sin\frac{\gamma}{2},\qquad \ell=r\gamma,\qquad 0\le\gamma\le\pi. \] These calculations therefore do not require all Cartesian coordinates to be written out first. The radii and phase parameters suffice. If one vector is zero, the angle is undefined, but the dot product is zero and the distance equals the length of the other vector.
7. Numerical Example
Choose the radius and angles
\[\tag{32} r=2,\qquad a=\frac13,\qquad b=\frac14, \qquad \tilde a=\frac{\pi}{3},\quad \tilde b=\frac{\pi}{4}. \] The compact representation of the point is
\[\tag{33} Q=2J\!\left(\frac13,\frac14\right) =2\left(\ep e^{i\pi/4}+\em e^{i\pi/3}\right). \] In cylindrical form, we obtain
\[\tag{34} \rho=\sqrt3,\qquad \varphi=\frac{\pi}{4},\qquad z=1. \] The Cartesian coordinates of the same point are
\[\tag{35} (x,y,z)=\left(\frac{\sqrt6}{2},\frac{\sqrt6}{2},1\right), \qquad x^2+y^2+z^2=\frac32+\frac32+1=4. \] The inverse transformation gives \[\tag{36} r=2, \quad \tilde a=\operatorname{atan2}(\sqrt3,1)=\pi/3, \quad \tilde b=\operatorname{atan2}(\sqrt6/2,\sqrt6/2)=\pi/4.\] We recover the original parameters \( a=1/3 \) and \( b=1/4 \).
8. What This Representation Provides
The expression \( Q=rJ(a,b) \) combines a radial scale and two independent phases. Its spatial meaning is defined by equation (9). Standard spherical geometry is preserved, while the idempotent basis allows both phases to be stored in a single algebraic object and their projections to be expressed through split sine and split cosine.
Algebraic operations on \( Q \) must be distinguished from operations on spatial vectors. For example, the product \( J(a_1,b_1)J(a_2,b_2)=J(a_1+a_2,b_1+b_2) \) adds the phases. By itself, it is neither a dot product nor a cross product, and it does not provide a general composition law for three-dimensional rotations. To add spatial vectors, we can convert them to Cartesian coordinates, add the coordinates, and recover the parameters using equations (24)–(26).
We have obtained an alternative two-phase representation of a position vector, with explicit conversions to standard coordinate systems. It combines the compact notation \( rJ \), the spherical mapping, and ordinary Euclidean calculation rules.