Distance, Normals and Double Normals for Real Plane Curves with Singularrities

Technical Summary: Distance, Normals, and Double Normals for Real Plane Curves with Singularities

Problem Statement
This paper investigates the relationship between normals, double normals, and the critical points of the squared Euclidean distance function ($D_u$) for real algebraic plane curves that possess singularities. While the complex setting offers a detailed description via Puiseux expansions and the ED-degree (counting normals on the regular part), the real setting presents distinct challenges. Specifically, the authors address how algebraic singularities—points where the gradient vanishes but the curve may still be locally smooth (e.g., $C^1$ but not $C^2$)—affect the topological structure of the distance function. The study aims to characterize the topological ED-discriminant ($\Delta_{tED}$), which separates stable points from those where the local topological type of the distance function changes, and to establish counting formulas for normals and double normals that account for these singularities.

Methodology
The authors employ a combination of algebraic geometry, singularity theory, and topological Morse theory.

1. Local Parametrization: Curves are analyzed via local parametrizations of the form $x_1(t) = t^p, x_2(t) = c t^q + \sigma(t)$, distinguishing between smooth branches ($p$ odd) and geometric singularities/cusps ($p$ even).
2. Topological Criticality: Instead of relying solely on differential calculus (which fails at non-$C^2$ points), the paper defines critical points based on topological regularity. A point is topologically critical if the distance function is not locally homeomorphic to a linear function.
3. Competition Factor ($\rho$): A key analytical tool introduced is the competition factor $\rho = q/2p$, which describes the contact order between the squared distance function and the curve at a singularity. This factor determines the existence and location of focal points.
4. Bifurcation Analysis: The authors perform explicit computations on the dominant terms of the distance function to determine bifurcations of critical points (minima, maxima, saddles) as the observation point $u$ moves relative to the curve.
5. Double Normals: The study extends to the squared distance function $B(a,b) = |a-b|^2$ between two curves (or a curve with itself), analyzing critical points on the product space of normalizations.

Key Contributions and Results

1. The Topological ED-Discriminant ($\Delta_{tED}$)
The paper characterizes $\Delta_{tED}$ as the complement of stable points. Unlike the smooth $C^2$ case where the discriminant is solely the evolute (caustic), the presence of singularities adds specific components:

• The Caustic: The envelope of normals, extended to include focal points at geometric singularities.
• Distinguished Normal Lines: The discriminant includes all normal lines at geometric singular points (cusps).
• Strict $C^1$-Points: For smooth points that are only $C^1$ embedded (where $p$ is odd and $q < 2p$), the normal line may also belong to the discriminant, a phenomenon absent in standard $C^2$ theory.

2. Focal Points and Bifurcations
The behavior of focal points on normal lines is classified by $\rho$:

• $\rho > 1$ ($q > 2p$): No focal points exist on the normal line; the distance function remains a minimum everywhere on the line.
• $\rho = 1$ ($q = 2p$): A finite focal point exists (center of curvature), where the type of critical point changes.
• $\rho < 1$ ($q < 2p$): The focal point coincides with the singular point itself ($u=x$).
  The paper provides a comprehensive table (Table 1) detailing 10 distinct cases of discriminant types and bifurcations based on the parity of $p$ and $q$ and the value of $\rho$.

3. Counting Formulas for Normals
The authors establish lower bounds for the number of normals ($N(u)$) passing through a generic point $u$:

• General Bound: $N(u) \ge 2$.
• Contribution of Cusps: Every cuspidal point contributes at least one normal (either a minimum or maximum). Specifically, $N(u) \ge N_{\text{cusps}}$.
• Morse Counting: The total number of normals is the sum of maxima and minima. For a compact curve, $m(u) = M(u) \ge 1$.

4. Double Normals
The study extends to double normals (segments connecting two points on the curve such that the segment is normal to the curve at both ends).

• Types: The authors classify double normals into $rr$ (regular-regular), $rs$ (regular-singular), $ss$ (singular-singular), and intersection points.
• Counting Results:
  • The number of double normals is at least 2.
  • Every cuspidal point contributes to at least one double normal.
  • Every pair of cuspidal points contributes to a double normal connecting them.
  • Self-intersections act as absolute minima.
• Morse Theory Application: Using the topology of the configuration space (Moebius band for a single curve, torus for two), the paper derives Morse inequalities relating the number of maxima, minima, and saddles. For a single curve, the number of double normals is at least 2, with $\#(\text{Max}) + \#(\text{Min}) = \#(\text{Saddles})$.

5. Comparison with Complex Theory
The paper contrasts the real case with the complex ED-discriminant. In the complex setting, the discriminant includes components related to points at infinity (atypical discriminant) and flex points with isotropic tangents (iflex discriminant). These components do not appear in the real case. The real discriminant is shown to be a subset of the complex one, specifically $\Delta_{tED} = \Delta_{\text{focal}} \cup \Delta_{\text{sing}}$, where $\Delta_{\text{sing}}$ may contain fewer normal lines than its complex counterpart.

Significance and Claims
The paper claims that including singular points is essential for a complete understanding of maximal and minimal distances in the real setting, a requirement often overlooked in approaches focusing solely on the ED-degree of the regular part. By introducing the concept of topological critical points and the competition factor $\rho$, the authors provide a framework that:

• Unifies the treatment of smooth and singular points under topological Morse theory.
• Identifies new phenomena, such as the inclusion of normal lines at strict $C^1$ points in the discriminant.
• Provides explicit counting formulas that relate the geometry of singularities (cusps) directly to the number of normals and double normals.
• Offers a detailed local description of bifurcations and focal sets that extends classical differential geometry to algebraic curves with singularities.

The work is presented as a rigorous extension of classical results (caustics, evolutes) to the singular real algebraic context, providing necessary tools for computational geometry and the study of distance functions on non-smooth objects.