[Paper Review] The toric sections: a simple introduction

Luca Moroni|arXiv (Cornell University)|Aug 1, 2017

Polynomial and algebraic computation1 references3 citations

TL;DR

This paper presents a didactic introduction to toric sections—the intersection curves of a torus and a plane—deriving their general quartic Cartesian equation and revealing an unexpected algebraic equivalence: toric sections can also be generated as the projection of a cone-cylinder intersection. The study emphasizes geometric intuition, compares toric sections to conic sections, and demonstrates interactive 3D visualization using GeoGebra.

ABSTRACT

We review, from a didactic point of view, the definition of a toric section and the different shapes it can take. We'll then discuss some properties of this curve, investigate its analogies and differences with the most renowned conic section and show how to build its general quartic equation. A curious and unexpected result was to find that, with some algebraic manipulation, a toric section can also be obtained as the intersection of a cylinder with a cone. Finally we'll show how it is possible to construct and represent toric sections in the 3D Graphics view of Geogebra. In the article only elementary algebra is used, and the requirements to follow it are just some notion of goniometry and of tridimensional analytic geometry.

Motivation & Objective

• To provide a didactic, accessible introduction to toric sections using elementary algebra and analytic geometry.
• To derive the general quartic Cartesian equation of a toric section from the intersection of a torus and a plane.
• To explore the geometric variety of toric sections, including Villarceau circles, Cassini ovals, Bernoulli lemniscates, and Hippopedes of Proclus.
• To establish an unexpected algebraic equivalence: toric sections can be obtained as the projection of a cone-cylinder intersection.
• To demonstrate interactive 3D visualization of toric sections using GeoGebra’s 3D Graphics view.

Proposed method

• Define the torus parametrically as the surface of revolution of a circle of radius $ r $ around an axis at distance $ R $, with $ R \geq r $.
• Model the intersecting plane using a normal vector and a point, deriving the implicit equation of the intersection curve in 3D space.
• Eliminate variables through algebraic manipulation to derive the general quartic equation of the toric section in Cartesian coordinates.
• Apply a change of coordinates and transformation to show that the same curve can be obtained as the projection onto the $ xy $-plane of the intersection between a cone and a cylinder.
• Construct the cone equation as $ x^2 = z^2 \cos^2\phi - (\rho \cos\phi - y \sin\phi)^2 $, which serves as a geometric bridge to the torus.
• Implement interactive 3D visualizations in GeoGebra, allowing dynamic exploration of toric sections by varying parameters $ R, r, \rho, \phi $.

Experimental results

Research questions

• RQ1 How can the general quartic equation of a toric section be derived from the intersection of a torus and a plane using elementary algebra?
• RQ2 What are the geometric and algebraic analogies and differences between toric sections and conic sections?
• RQ3 Can a toric section be equivalently generated through the intersection of a cone and a cylinder, and what is the geometric meaning of this equivalence?
• RQ4 How do specific toric sections—such as Villarceau’s circles, Cassini ovals, and Bernoulli’s lemniscate—arise from particular plane-torus configurations?
• RQ5 To what extent can interactive 3D visualization in GeoGebra facilitate the understanding of toric sections for educational purposes?

Key findings

• The general equation of a toric section is a quartic curve in Cartesian coordinates, derived through algebraic elimination of variables from the torus and plane equations.
• Villarceau’s circles arise when a plane passes through the center of the torus at a specific angle $ \arctan(\sqrt{R^2 - r^2}/r) $, producing two disjoint circles.
• Cassini’s ovals are defined as the locus of points where the product of distances to two foci is constant, and they emerge when the cutting plane is at distance $ \rho = r $ from the torus axis.
• Bernoulli’s lemniscate is a special case of a Cassini oval where $ R = 2r $, corresponding to a plane tangent to the inner equator of the torus.
• An unexpected result is that the same toric section curve can be generated as the projection of the intersection between a cone (with second-degree equation) and a cylinder (also second-degree), establishing a novel algebraic equivalence.
• The construction method using a cone and cylinder allows dynamic generation of toric sections in GeoGebra, with the cone acting as a geometric bridge between the torus and the final planar curve.