Tools for analyzing the intersection curve between a torus and a quadric through projection and lifting

TL;DR

The paper tackles the problem of analyzing the real intersection curve between a centered torus and an irreducible quadric by studying its projection (the cutcurve) onto the plane $z=0$ and then lifting points back to 3D. It develops a systematic framework based on resultant and subresultant polynomials, defining the cutcurve via the zero set of $\widehat{{\bf S}_0}$ constrained by silhouette inequalities, and provides explicit lifting rules that depend on $sres_1$ and its companion, enabling reconstruction of the 3D intersection from the 2D projection. The authors establish that ${\bf S}_0=\widehat{{\bf S}_0}/\gcd(\widehat{{\bf S}_0},p_2+q_1^2-2q_0)$ generates the ideal of the cutcurve under generic conditions, relate the cutcurve to silhouette curves, and derive practical computational tools (identities linking $\widetilde{{\bf S}_0}$ with silhouette polynomials) to locate intersection points efficiently. They also distinguish special cases, notably $q_1\equiv0$ vs $q_1\not\equiv0$, addressing tangency, multiple components, and singularities, and provide a suite of examples to illustrate lifting, projection singularities, and the geometry of the intersection. Overall, the work provides a robust algebraic pipeline for projecting, lifting, and characterizing torus–quadric intersections with explicit computational paths and geometric interpretations, useful for CAD/GD applications and qualitative analysis of space curves.

Abstract

This article introduces efficient and user-friendly tools for analyzing the intersection curve between a ringed torus and an irreducible quadric surface. Without loose of generality, it is assumed that the torus is centered at the origin, and its axis of revolution coincides with the $z$-axis. The paper primarily focuses on examining the curve's projection onto the plane $z=0$, referred to as the cutcurve, which is essential for ensuring accurate lifting procedures. Additionally, we provide a detailed characterization of the singularities in both the projection and the intersection curve, as well as the existence of double tangents. A key tool for the analysis is the theory of resultant and subresultant polynomials.

Figures (14)

• Figure 1: Left: $\Delta_{\cal T}$ (blue), $\Delta_{\cal Q}$ (purple) and the cutcurve (green). Right: The intersection curve defined by two parallel circles (red).
• Figure 2: Left: $\Delta_{\cal T}$ (blue) and the cutcurve (green). Right: The intersection curve defined by three parallel circles (red).
• Figure 3: Left: $\Delta_{\cal T}$ (blue), $\Delta_{\cal Q}$ (purple) and the curve defined by ${\bf S}_0(x,y)$ (green) for Example \ref{['curvareal']}. Right: The intersection curve (red).
• Figure 4: Left: $\Delta_{\cal T}$ (blue), $\Delta_{\cal Q}$ (purple) and the cutcurve (green). Right: The intersection curve (red). For example \ref{['t9-elipse2']}
• Figure 5: Left: $\Delta_{\cal T}$ (blue), $\Delta_{\cal Q}$ (purple), the cutcurve (green) and the line $q_1=0$ (black) for Example \ref{['toro-8']}. Right: The intersection curve (red).

Theorems & Definitions (69)

• Definition 1.1
• Theorem 2.1
• proof
• Definition 2.2
• Proposition 2.3
• proof
• Lemma 2.4
• proof
• Proposition 2.5
• proof