Paper 2-2: Dual Rotation Structure: Π Transformation Rules for Torus

Author: Suhang Zhang (Heluo Mathematical School)

Abstract

A torus is a 3D solid formed by rotating a circle (generatrix circle) of radius a around a coplanar non-intersecting axis. It involves two independent radii: the radius a of the generatrix circle and the revolution radius R (the distance from the rotation axis to the center of the generatrix circle). This paper incorporates the torus into the framework of the \Pi operator, defines a dual-radius parameter system, derives operator expressions for the volume and surface area of the torus, establishes the dimension-raising mapping from the 2D generatrix circle to the 3D torus as well as its inverse transformation, and analyzes extreme degeneration cases: the torus degenerates into a circular loop as a \to 0, and into a sphere as R \to 0. This research further verifies the applicability of the \Pi operator to geometric bodies with multiple rotational degrees of freedom.

Keywords: \Pi operator; torus; dual rotation; generatrix circle; meridian cross-section

1. Introduction

A cylinder features a single rotational radius, generated by rotating a rectangle around an axis. By contrast, a torus involves compound rotation: first construct a 2D generatrix circle with radius a, then rotate this generatrix circle around an external axis whose distance to the circle center is R (R > a). Such a "rotation of rotation" generates a toroidal surface. In classical geometry, the volume and surface area of a torus are given by:
V = 2\pi^2 R a^2, \quad S = 4\pi^2 R a

Within the \Pi operator system, the torus can be regarded as a compound transformation: the \Pi operator first generates the generatrix circle from a point or a small circle, and the second \Pi operator rotates the generatrix circle around another axis. For conciseness, this paper takes the 2D generatrix circle as the original figure and implements one-time dimension raising via rotation about an external axis to obtain the torus. This approach is adopted hereinafter, and the manifestations of the dual radii in the operator system are discussed.
Section 2 defines the generatrix circle and rotational parameters; Section 3 presents the \Pi transformation rules and derives formulas for volume and surface area; Section 4 explores meridian cross-sections and inverse transformations; Section 5 discusses extreme degeneration; Section 6 draws conclusions.

2. Generatrix Circle and Dual-Radius Parameters

2.1 2D Original Figure: Generatrix Circle

Define a circle C \subset \mathbb{R}^2 on a 2D plane, with its center at (R, 0) and radius a, satisfying R > a > 0. The parametric equations of the generatrix circle are written as:

C:
\begin{cases}
x = R + a\cos\phi \\
y = a\sin\phi
\end{cases}, \quad \phi \in [0, 2\pi)

where \phi denotes the angular parameter on the generatrix circle. The y-axis (i.e., the line x=0) is set as the rotation axis. Notably, the distance from the center of the generatrix circle to the rotation axis is exactly R.

2.2 Origin of the "Dual Rotation" Property

The generatrix circle itself is a rotational structure generated with radius a. Rotating it around the y-axis for one full revolution means every point on the generatrix circle sweeps out a circular trajectory, eventually forming a 3D torus with the topological structure S^1 \times S^1.

3. Π Operator Transformation Rules

3.1 Dimension-Raising Mapping

Adopt the \Pi operator (geometric rotation channel) to rotate the generatrix circle C around the y-axis by 2\pi. For an arbitrary point (x, y) on C where x = R + a\cos\phi,\ y = a\sin\phi, rotating around the y-axis yields a circle in 3D space with radius x at height y. The union of all such point sets constitutes the torus \mathcal{T}.

Under cylindrical coordinates (r, \theta, y) (where r is the radial distance to the y-axis and \theta is the azimuth angle around the y-axis), the torus is expressed as:

\mathcal{T} = \left\{ (r, \theta, y) \,\bigg|\,
\begin{array}{l}
y = a\sin\phi,\quad \phi\in[0,2\pi) \\
r = R + a\cos\phi,\quad \theta\in[0,2\pi)
\end{array}
\right\}

Its equivalent parametric equations in Cartesian coordinates are:

\begin{cases}
x = (R + a\cos\phi)\cos\theta \\
z = (R + a\cos\phi)\sin\theta \\
y = a\sin\phi
\end{cases}, \quad \phi,\theta \in [0,2\pi)

3.2 Volume Transformation Formula

Generalizing the Pappus's Centroid Theorem: the volume of a solid of revolution generated by rotating a planar region around a coplanar non-intersecting axis equals the product of the area of the region and the travel length of its centroid.

The area of the generatrix circle is:
A_C = \pi a^2
The centroid of the generatrix circle coincides with its geometric center, whose trajectory around the y-axis has a perimeter of 2\pi R. Thus the volume of the torus is:
V = A_C \cdot 2\pi R = \pi a^2 \cdot 2\pi R = 2\pi^2 R a^2

In the \Pi operator framework, define a coefficient k such that V = A_C \cdot k, then:
k = \frac{2\pi^2 R a^2}{\pi a^2} = 2\pi R
Since 2\pi R is the perimeter of the centroid trajectory, the volume-oriented \Pi transformation is formulated as:
\mathcal{\Pi}^{(I)}(C) = A_C \cdot 2\pi R

3.3 Surface Area Transformation Formula

The surface area of a torus is the product of the circumference of the generatrix circle and the perimeter of the centroid trajectory:

- Circumference of the generatrix circle: 2\pi a
- Perimeter of the centroid trajectory: 2\pi R

Hence the surface area:
S = 2\pi a \cdot 2\pi R = 4\pi^2 R a

Geometrically, a toroidal surface can be interpreted as a circular tube of radius a bent into a ring of radius R. The surface area formula can also be derived by applying the \Pi operator to the circumference of the generatrix circle. The circumference-oriented \Pi transformation is defined as:
\mathcal{\Pi}_{\text{circumference}}(C) = 2\pi a \cdot 2\pi R = 4\pi^2 R a

4. Inverse Transformation and Meridian Cross-Section

4.1 Definition of Meridian Cross-Section

A meridian cross-section refers to the intersection between the torus and a plane passing through the rotation axis (the y-axis), e.g., the half-plane z=0,\ x \ge 0.

Set \theta = 0 and \theta = \pi in the parametric equations of the torus to solve for intersection curves:

- When \theta = 0: x = R + a\cos\phi,\ z=0,\ y = a\sin\phi, which corresponds to the circle (x-R)^2 + y^2 = a^2 located in the region x \ge R-a > 0.
- When \theta = \pi: x = -(R + a\cos\phi),\ z=0,\ y = a\sin\phi, which corresponds to the circle (x+R)^2 + y^2 = a^2 outside the half-plane x \ge 0.

In general, a plane passing through the rotation axis cuts the torus into two disjoint circles with radius a, whose centers are at (R,0) and (-R,0) respectively. To recover the 2D original generatrix circle from the 3D torus, we select one circle from the meridian cross-section. The two circles are mirror images of each other, and either can be taken as the generatrix circle.

4.2 Inverse Transformation Operator

The dimension-reduction operation \mathcal{\Pi}^{-1}(\mathcal{T}) yields one circle from the meridian cross-section:
\mathcal{\Pi}^{-1}_{I}(\mathcal{T}) = C \quad \text{or} \quad C'
where C denotes the original generatrix circle centered at (R,0) with radius a, and C' denotes its mirror circle centered at (-R,0) with the same radius. In accordance with the curl conservation axiom of the \Pi operator, the inverse transformation uniquely determines the original figure once the rotation axis and initial orientation are specified. In this system, we adopt the circle on the side x \ge 0 as the standard original figure.

5. Analysis of Extreme Degeneration

5.1 Case: a \to 0

As a \to 0, the generatrix circle shrinks to a single point (R,0). Rotating this point around the y-axis produces a closed circular loop of radius R. Meanwhile:
V = 2\pi^2 R a^2 \to 0,\quad S = 4\pi^2 R a \to 0
The volume and surface area both approach zero, which is geometrically consistent.

5.2 Case: R \to 0

As R \to 0, the rotation axis passes through the center of the generatrix circle. Rotating the generatrix circle around its own center generates a sphere of radius a.

It can be seen that the original torus volume formula V = 2\pi^2 R a^2 approaches zero as R \to 0, which conflicts with the standard sphere volume \displaystyle \frac{4}{3}\pi a^3. The discrepancy arises because Pappus's Centroid Theorem is no longer applicable when the rotation axis intersects the generatrix circle. When R=0, the generatrix circle rotates about its own diameter to form a sphere, which requires a separate formula.

In the \Pi operator system, the transformation rules switch from "rotating a circle around an external axis" to "rotating a circle about its diameter" when the rotation axis intersects the generatrix circle. The torus formulas are only valid for standard tori with R > a. For 0 < R < a, the surface becomes a self-intersecting spindle torus with more complex volume formulas, which will not be discussed in this paper.

5.3 Degeneration to Cylinder

Replacing the generatrix circle with a rectangle yields a cylinder, while a torus cannot directly degenerate into a cylinder. When R \to \infty, the local part of the torus approximates a cylindrical tube.

6. Conclusions

This paper successfully integrates the torus into the \Pi operator system, with major achievements summarized as follows:

1. A dual-radius parameter system is established, clarifying the geometric relationship between the generatrix circle radius a, the revolution radius R and the rotation axis.
2. Operator expressions for volume and surface area are derived:
\mathcal{\Pi}^{(I)}(C) \ (\text{Volume}) = \pi a^2 \cdot 2\pi R = 2\pi^2 R a^2
\mathcal{\Pi}_{\text{circumference}}(C) \ (\text{Surface Area}) = 2\pi a \cdot 2\pi R = 4\pi^2 R a
3. The inverse transformation via meridian cross-section is verified: a plane through the rotation axis cuts the torus into two circles of radius a, one of which serves as the 2D original generatrix circle.
4. Extreme degeneration behaviors are analyzed: the torus degenerates into a circular loop as a \to 0; the case R=0 corresponds to a sphere and will be discussed in follow-up Paper 2-4; self-intersecting tori for R < a are reserved for future research.

As a typical dual-rotation geometric structure, the torus validates the capability of the \Pi operator in describing compound rotational geometries. The subsequent Paper 2-3 will focus on periodic micro-element channels and implement the \Pi transformation for helical curves and helical surfaces.

References: Omitted

Author's Statement

The content of this paper is original research based on the \Pi operator system proposed by the Heluo Mathematical School.

 

Next Paper 2-3: Periodic Micro-element Channel: Operator Implementation of Helical Curves and Helical Surfaces