2026-07-24
From Euler's formula to split geometry
Part 2. Split sine and split cosine of two-phase geometry
Trigonometric functions arose as a language for describing rotation. Cosine and sine allow one to represent the same motion through two mutually perpendicular projections, and Euler's formula combines these projections into a single complex exponential function. Thus, a single phase parameter simultaneously determines the position of a point on a circle, its real coordinate, and its imaginary coordinate.
However, this construction assumes that the entire geometric object is subject to a single, common phase. If an object consists of two independent components, each rotating according to its own law, a single phase coordinate is no longer sufficient. The challenge arises of constructing an analogue of Euler's formula that can describe two rotations simultaneously, without mixing them and preserving the familiar properties of sine and cosine.
Since each of the two rotations has its own phase, the usual sin(α) and cos(α) functions, which depend on a single argument, prove insufficient. This naturally leads to the introduction of their two-parameter generalization.
Below, it will be shown that such a construction can naturally be obtained in an idempotent basis. Its two mutually canceling components allow us to assign each phase its own plane of rotation, and then determine their common real and imaginary projections. These projections will be called the split-cosine and split-sine.
Classical Euler's formula
\[ e^{i\varphi}=\cos\varphi+i\sin\varphi \tag{1} \] shows that cosine and sine are the real and imaginary projections of a single rotation in the complex plane. Such a rotation is defined by a single phase \(\varphi\).
However, in two-phase geometry, a more general object arises
\[ J(a,b)=\j^{a}(-\j)^{b}, \tag{2} \] containing two independent phase coordinates. One of them refers to the \(\ep\) component, the other to the \(\em\) component. Therefore, it is no longer possible to represent the real and imaginary parts of operator (2) using just one ordinary sine and one ordinary cosine.
It is natural to introduce new functions split-cosine and split-sine to describe such a two-phase object. They do not replace the classical trigonometric functions, but rather extend them to the case of two independent rotations, while preserving the form of Euler's formula and the fundamental trigonometric identities.
1. Two independent rotations
Use an idempotent basis
\[ \ep^2=\ep,\qquad \em^2=\em,\qquad \ep\em=0,\qquad \ep+\em=1. \tag{3} \] Powers of the hyperbolic unit and its inverse are represented as
\[ \j^{a}=\ep+\em e^{i\pi a}, \tag{4} \] \[ (-\j)^{b}=\ep e^{i\pi b}+\em. \tag{5} \] Multiplying (4) and (5), we get
\[ \begin{aligned} J(a,b) &= \left(\ep+\em e^{i\pi a}\right) \left(\ep e^{i\pi b}+\em\right) \\[2mm] &=\ep e^{i\pi b}+\em e^{i\pi a}. \end{aligned} \tag{6} \] Mixed products disappear due to the condition \(\ep\em=0\). Therefore, each phase coordinate remains in its own idempotent component:
\[ \pi b\quad\text{in component}\quad\ep, \qquad \pi a\quad\text{in component}\quad\em. \tag{7} \] Thus, the operator \(J(a,b)\) combines two independent complex rotations into a single algebraic object.
2. Definition of split functions
Let's expand both exponents in formula (6):
\[ \begin{aligned} J(a,b) ={}&\ep\left[\cos(\pi b)+i\sin(\pi b)\right] \\ &+\em\left[\cos(\pi a)+i\sin(\pi a)\right]. \end{aligned} \tag{8} \] Collecting the real and imaginary parts separately, we introduce the definitions
\[ \boxed{ \Cos(a,b) =\ep\cos(\pi b)+\em\cos(\pi a) } \tag{9} \] and
\[ \boxed{ \Sin(a,b) =\ep\sin(\pi b)+\em\sin(\pi a). } \tag{10} \] Then operator (2) takes the form
\[ \boxed{ \j^{a}(-\j)^{b} =\Cos(a,b)+i\Sin(a,b). } \tag{11} \] Formula (11) is a two-phase analog of Euler's formula. The usual complex exponential describes a single rotation with a single phase, while the operator \(\j^{a}(-\j)^{b}\) describes two independent rotations with phases \(\pi a\) and \(\pi b\).
3. Connection with classical trigonometry
The most important property of the new functions manifests itself when two phase coordinates coincide:
\[ a=b. \tag{12} \] In this case
\[ \begin{aligned} \Cos(a,a) &=\ep\cos(\pi a)+\em\cos(\pi a) \\ &=(\ep+\em)\cos(\pi a) \\ &=\cos(\pi a), \end{aligned} \tag{13} \] and also
\[ \Sin(a,a)=\sin(\pi a). \tag{14} \] Investigatorbut,
\[ \boxed{ \Cos(a,a)=\cos(\pi a), \qquad \Sin(a,a)=\sin(\pi a). } \tag{15} \] Classical trigonometry turns out to be the diagonal case of two-phase geometry. When both components rotate synchronously, the difference between them disappears, and the split functions turn into ordinary sine and cosine.
For \(a\ne b\), the two components have different phases, so the operator \(J(a,b)\) in the general case cannot be replaced by a single ordinary exponential function \(e^{i\varphi}\). It is precisely this possibility of independently changing phases that constitutes the content of the new generalization.
4. The Basic Identity
Let's square functions (9) and (10). Due to the orthogonality of idempotents, the mixed terms disappear:
\[ \Cos^{2}(a,b) =\ep\cos^{2}(\pi b)+\em\cos^{2}(\pi a), \tag{16} \] \[ \Sin^{2}(a,b) =\ep\sin^{2}(\pi b)+\em\sin^{2}(\pi a). \tag{17} \] Adding these expressions and using the classical identity \(\cos^{2}x+\sin^{2}x=1\) separately in each component, we obtain
\[ \begin{aligned} \Cos^{2}(a,b)+\Sin^{2}(a,b) &=\ep+\em=1. \end{aligned} \tag{18} \] So,
\[ \boxed{ \Cos^{2}(a,b)+\Sin^{2}(a,b)=1 } \tag{19} \] Formula (19) completely preserves the structure of the classical identity.
\[ \cos^{2}x+\sin^{2}x=1. \tag{20} \] The only difference is that the classical formula pertains to a single phase, while the split identity combines two independent phases in a single normalized object.
5. Double-Argument Formulas
The connection between split functions and classical trigonometry is especially evident in double-argument formulas. For ordinary functions, the following identities hold:
\[ \cos^{2}x=\frac{1+\cos 2x}{2}, \qquad \sin^{2}x=\frac{1-\cos 2x}{2}, \tag{21} \] \[ \sin x\cos x=\frac{\sin 2x}{2}. \tag{22} \] Since each idempotent component obeys ordinary trigonometry, these formulas are transferred to split functions component-wise:
\[ \boxed{ \Cos^{2}(a,b) =\frac{1+\Cos(2a,2b)}{2} } \tag{23} \] \[ \boxed{ \Sin^{2}(a,b) =\frac{1-\Cos(2a,2b)}{2} } \tag{24} \] \[ \boxed{ \Sin(a,b)\Cos(a,b) =\frac{\Sin(2a,2b)}{2}. } \tag{25} \] Comparison of formulas (21) and (22) with formulas (23)–(25) shows that the structure of classical trigonometry remains unchanged. The only difference is the simultaneous doubling of two phase coordinates:
\[ (a,b)\longmapsto(2a,2b). \tag{26} \] This similarity is not a formal analogy. It follows directly from the expansion of split functions in idempotents and from the condition \(\ep\em=0\), which prevents the two phase planes from mixing.
6. Geometric meaning
The ordinary complex exponential \(e^{i\varphi}\) defines a rotation on a single unit circle. Its real and imaginary projections are \(\cos\varphi\) and \(\sin\varphi\).
Split operator
\[ J(a,b)=\ep e^{i\pi b}+\em e^{i\pi a} \tag{27} \] combines two unit circles, each with its own phase. The split cosine function defines the set of real projections of these rotations,
\[ \Cos(a,b) =\ep\cos(\pi b)+\em\cos(\pi a), \tag{28} \] and the split sine function defines the set of their imaginary projections:
\[ \Sin(a,b) =\ep\sin(\pi b)+\em\sin(\pi a). \tag{29} \] Therefore, the split sine and split cosine functions should be understood not as arbitrary combinations of ordinary functions, but as natural coordinate projections of a two-phase geometric object.
Conclusion
The introduction of the functions \(\Cos(a,b)\) and \(\Sin(a,b)\) allows us to preserve the Eulerian form of the representation for an operator containing two independent phase coordinates. The main result of the work can be written as follows:
\[ \j^{a}(-\j)^{b} = \Cos(a,b) + i\Sin(a,b) \]
In this case, the split cosine and split sine functions are defined as:
\[ \Cos(a,b) = \ep\cos(\pi b) + \em\cos(\pi a), \qquad \Sin(a,b) = \ep\sin(\pi b) + \em\sin(\pi a) \] Formula (30) shows that the single operator \(\j^{a}(-\j)^{b}\) combines two independent complex rotations. The phase \(\pi b\) belongs to the \(\ep\) component, and the phase \(\pi a\) belongs to the \(\em\) component. Due to the condition \(\ep\em=0\), these rotations do not mix but are included in a common normalized geometric object.
When the phases \(a=b\) coincide, the new functions transform into ordinary sine and cosine:
\[ \Cos(a,a)=\cos(\pi a), \qquad \Sin(a,a)=\sin(\pi a). \] At different phases \(a\ne b\), they describe a more general object consisting of two independent rotationsin orthogonal idempotent components.
Thus, classical trigonometry is not discarded, but rather incorporated into the new construction as its special diagonal case. Split trigonometry extends Euler's formula from one phase coordinate to two, preserving normalization and the most important classical identities.