Totally Degenerate Locus

• Totally Degenerate Locus is defined as a configuration where the typical elliptical path of a triangle center collapses into a segment or circle when |u| equals |v|.
• It arises from explicit algebraic constraints on the linear combination of triangle centers and confocal parameters, with specific ratios such as α/β playing a key role.
• This framework provides a closed-form criterion for classifying degenerate cases, impacting the turning number and monotonicity of the locus in Poncelet families.

A totally degenerate locus, in the context of loci traced by triangle centers over Poncelet triangle families, refers to a degeneration of the locus—normally a nondegenerate ellipse—into a geometrically simpler form, specifically a segment (flattened ellipse) or a circle. Such degeneracy is determined by explicit algebraic conditions on the linear combination of triangle centers and the confocal parameters, and has precise implications for the structure and traversal of the locus (Helman et al., 2021).

1. Poncelet Triangles and 3-Periodic Families

A classical setting involves Poncelet 3-periodic families, specifically triangles interscribed between an outer ellipse $\mathcal{E}: x^2/a^2 + y^2/b^2 = 1$ (with $a > b > 0$) and an inner confocal caustic ellipse derived from the parameters

$$c = \sqrt{a^2 - b^2}, \quad \delta = \sqrt{a^4 - a^2 b^2 + b^4},$$

$$a_c = \frac{a(\delta - b^2)}{c^2},\quad b_c = \frac{b(a^2 - \delta)}{c^2}$$

where $a_c$, $b_c$ are the semi-axes of the inner caustic. The family of 3-periodic triangles generated in this configuration sweeps the outer ellipse $\mathcal{E}$ monotonically as the complex parameter $\lambda = e^{it}$ traverses the unit circle exactly once.

2. Parametrization of Triangle-Center Loci

Given centers $X_2$ (barycenter), $X_3$ (circumcenter), and a stationary center $X_k$ (e.g., $X_9$ in the confocal case), any triangle center of the affine-linear form

$$X = \alpha X_2 + \beta X_3 + \gamma X_k, \quad \alpha, \beta, \gamma \in \mathbb{R}$$

traces a locus parametrized by

$$X(t) = u\, e^{it} + v\, e^{-it} + w, \quad t \in [0, 2\pi).$$

The complex constants $u, v, w$ depend affinely on $\alpha, \beta$ (not on $\gamma$) and on the confocal parameters $a$, $b$.

The locus $X(t)$ is an ellipse centered at $w$ with semi-axes

$$A = |u| + |v|, \qquad B = \bigl|\,|u| - |v|\,\bigr|.$$

The nondegeneracy condition for a true ellipse (i.e., not a segment or a circle) is $|u| \neq |v|$.

3. Conditions for Total Degeneracy

Degeneracy arises precisely when $|u| = |v|$. This can be analyzed in two principal cases:

• Degeneration to a Segment: $B = 0$ (i.e., $|u| = |v|$, $A > 0$), yielding a flattened segment. For $\alpha, \beta \in \mathbb{R}$ (hence $u, v \in \mathbb{R}$), this occurs if and only if

$$\frac{\alpha}{\beta} = \frac{2a^2 - b^2 + \delta}{2b^2} \quad \text{or} \quad \frac{\alpha}{\beta} = \frac{2b^2 - a^2 + \delta}{2a^2}.$$

Alternatively, these ratios may be written in terms of the inradius-to-circumradius ratio $\rho = r/R$ (invariant across the confocal family) as

$$\frac{\alpha}{\beta} = \frac{3}{2} \frac{1 \pm \sqrt{1-2\rho}}{\rho \pm \sqrt{1-2\rho}}.$$

• Degeneration to a Circle: $A = B$ (i.e., $|u| + |v| = \bigl|\,|u| - |v|\,\bigr|$), which in the real $u,v$ case occurs if $u = 0$ or $v = 0$. The locus is then a (nontrivial) circle of radius $A = B > 0$. This corresponds to the special ratios:

$$\left(\frac{\alpha}{\beta}\right)_\pm = \frac{\delta - 3ab \pm 2(a^2 + b^2)}{2ab}.$$

4. Tabulation of Elliptic and Totally Degenerate Cases

Letting $\mu = \alpha/\beta$, the full set of behaviors can be summarized as follows:

Behavior	Condition on $\mu$
Nondegenerate ellipse	$\mu \neq$ any of the four degenerate values below
Segment (flattened)	$$\displaystyle \mu = \frac{2a^2 - b^2 + \delta}{2b^2}$$ or $$\displaystyle \mu = \frac{2b^2 - a^2 + \delta}{2a^2}$$
Circle	$$\displaystyle \mu = \frac{\delta - 3ab \pm 2(a^2 + b^2)}{2ab}$$

This algebraic criterion establishes a complete, closed-form method for determining the fully degenerate configurations of a triangle-center locus in the classical confocal Poncelet setting.

5. Turning Number and Monotonicity

As $\lambda$ winds once counterclockwise around the unit circle, the locus $X(t)$ winds three times around its center $w$. The total turning number is: $$\operatorname{wind}(X) = 3 \times \operatorname{sign}(|u|^2 - |v|^2) = \pm 3,$$ where the sign reverses exactly at the two segment-degeneracy ratios where $|u| = |v|$.

For $|u|\neq|v|$, the instantaneous speed $|\tfrac{d}{dt}X(t)|$ never vanishes, ensuring $X(t)$ traverses its nondegenerate ellipse monotonically and without backtracking. In the degenerate cases ($|u|=|v|$ or $uv=0$), the velocity vanishes at some point, reflecting the collapse to a segment or a circle.

6. Connections Beyond the Confocal Pair

All statements and formulae for the confocal Poncelet family extend, mutatis mutandis, to the concentric-circle (incircle) Poncelet family via substitutions such as $a_c = b_c = r$ and $c'^2 = a^2b^2 - (a^2 - b^2)^2 / c^2$. Thus, the framework for identifying totally degenerate loci is applicable to broader classes of 3-periodic triangle center loci generated by Poncelet families.

7. Implications and Algebraic Decidability

This framework enables a comprehensive and easily checkable algebraic determination of when a locus of an affine-linear combination of triangle centers—two moving, one stationary—collapses from a nondegenerate ellipse to a segment or circle, for all Poncelet 3-periodic families considered. The conditions are reducible to explicit constraints on the parameters $(\alpha, \beta, a, b)$, yielding a direct classification of ordinary and totally degenerate loci (Helman et al., 2021).