Prompt-Aware Encoding

• The paper introduces prompt-aware encoding by mapping complex prompt parameters into affine-linear representations and deriving u,v-formulas for ellipse geometry.
• It establishes necessary and sufficient conditions for nondegenerate ellipses through discriminant analysis and explicit linear-ratio conditions.
• The study also details degenerate cases — reducing to segments or circles — and examines turning numbers and monotonicity in the geometric loci.

1. Poncelet‐Triangle Families: setup and parameters We recall two canonical “CAP” (Concentric, Axis‐Parallel) Poncelet families of triangles $T(\lambda)=A(\lambda)B(\lambda)C(\lambda)$ in the plane:

(a) Confocal (Elliptic‐Billiard) Pair Outer ellipse $\mathcal E$: $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,\qquad a>b>0,$$ Caustic (inner) also an ellipse, confocal with $\mathcal E$. Let $$c^2=a^2-b^2,\quad   \delta=\sqrt{\,a^4-a^2b^2+b^4\,},$$ then its semi-axes $a_c,b_c$ are $$a_c\;=\;\frac{a\bigl(\delta-b^2\bigr)}{c^2},   \quad   b_c\;=\;\frac{b\bigl(a^2-\delta\bigr)}{c^2}.$$ Under the affine map $(x,y)\mapsto(x/a,y/b)$ the outer ellipse becomes the unit circle $\,|z|=1$ in $\mathbb C$, and the inner caustic becomes a concentric ellipse with foci $$f=-c',\;g=+c',\quad   c'\;=\;\frac{1}{c}\,\sqrt{\,2\delta-(a^2+b^2)\,}.$$

(b) Incircle–Ellipse (Poristic) Family Outer ellipse as above, inner caustic a concentric circle of radius $r$. In this family the common circumradius $R$ of all Poncelet triangles is also fixed, so $\rho:=\frac{r}{R}   \quad\text{is constant.}$

2. Parametrization of an affine‐linear center locus Any triangle center of the form

$$X(\lambda)\;=\;\alpha\,X_2(\lambda)\;+\;\beta\,X_3(\lambda)\;+\;\gamma\,X_k(\lambda),$$

where $X_2$ is the barycenter, $X_3$ the circumcenter, and $X_k$ a center that is stationary over the family (e.g.\ $X_9$ for the confocal case or $X_1$ for the incircle case), can be written, after conjugating the affine map so the outer conic is $\{|z|=1\}$, in the form

$$X(\lambda) \;=\;u\,\lambda\;+\;v\,\lambda^{-1}\;+\;w, \qquad \lambda\in\mathbb C,\;|\lambda|=1,$$

for certain complex constants $u,v,w$ depending algebraically on $(a,b)$ (or on $(a,b,r)$) and on $\alpha,\beta,\gamma$. Equivalently, introducing the real parameter $t$ by $\lambda=e^{it}$,

$X(t)=u\,e^{it}+v\,e^{-it}+w.$

Lemma (Ellipse‐Parameter Lemma) The curve $t\mapsto X(t)$ is a (possibly degenerate) ellipse with: * center $w$, * semiaxes $$A=|\,u\,|+|\,v\,|,\quad B=\bigl\lvert\,|u|-|v|\,\bigr\rvert,$$ * rotated by angle $\tfrac12(\arg u+\arg v)$. In particular it is nondegenerate iff $u\neq0$, $v\neq0$ and $\,|u|\neq|v|\,$.

3. Nondegenerate ellipse: necessary and sufficient condition In full generality one obtains an ellipse (with two distinct positive semiaxes) precisely when

$$u\,v\;\neq\;0 \quad\text{and}\quad |u|\;\neq\;|v|.$$

Equivalently one may form the “discriminant”

$\Delta\;=\;|\,u\,|^2\;-\;|\,v\,|^2\;\neq\;0$

and require $u\neq0,\;v\neq0$. If the affine combination involves only $X_2,X_3$ (i.e.\ $\gamma=0$) then $u,v$ turn out real, and the ellipse is axis‐aligned in the affine circle‐model.

4. Degeneration to a segment (one axis zero) The semiminor axis $B$ vanishes exactly when

$$|u|\;=\;|v| \;\;\Longleftrightarrow\;\; \Delta\;=\;0.$$

For the confocal family, specializing to $$X=\alpha\,X_2+\beta\,X_3,\;\alpha,\beta\in\mathbb R,$$ one finds explicitly (by solving $u=\pm v$)

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

In the incircle family an entirely analogous pair of linear‐ratio conditions in $\alpha,\beta$ ensues, with $\delta$ replaced by $\sqrt{a^4-a^2b^2+b^4}$ and $b^2$ by $r^2$.

5. Degeneration to a circle (axes equal) A circle arises exactly when the ellipse has equal semiaxes, i.e.\

$$A\;=\;B \quad\Longleftrightarrow\quad |u|\,+|v|\;=\;\bigl\lvert\,|u|-|v|\,\bigr\rvert \;\Longleftrightarrow\; (u=0)\;\text{ or }\;(v=0).$$

Again for the confocal $\alpha X_2+\beta X_3$ case one solves $u=0$ or $v=0$ to obtain

$$\Bigl(\tfrac\alpha\beta\Bigr)_{\!\pm} \;=\; \frac{\delta-3\,a\,b\;\pm\;2\,(a^2+b^2)}{2\,a\,b}.$$

These two linear‐ratio values yield two distinct concentric circular loci (one “large,” one “small”).

6. Summary of behaviors Denote $\kappa=\alpha/\beta$. Then over the confocal family:

• Nondegenerate ellipse: $\displaystyle u,v\neq0$ and $\Delta=|u|^2-|v|^2\neq0$. Equivalently $$\kappa\neq\frac{2a^2-b^2+\delta}{2b^2},\;\frac{2b^2-a^2+\delta}{2a^2}$$ and $\kappa\neq(\kappa)_\pm$ below.
• Segment (degenerate ellipse): $\displaystyle\Delta=0$, i.e.\

  $\kappa=\frac{2a^2-b^2+\delta}{2b^2}$ or $\kappa=\frac{2b^2-a^2+\delta}{2a^2}.$

• Circle: $\displaystyle u=0$ or $v=0$, i.e.\ $\kappa=(\kappa)_\pm$ where (\displaystyle (\kappa)_\pm = \frac{\delta-3ab\pm2(a^2+b^2)}{2ab}. )

1. Turning number and monotonicity By Blaschke‐parametrization one shows:
   • As $\lambda=e^{it}$ runs once CCW around the unit circle, the Poncelet triangle family sweeps the outer ellipse exactly once in CCW order, and $\lambda\mapsto X(\lambda)$ has winding number $\pm3$ about its center $w$. – Except in the degenerate cases $|u|=|v|$, the speed $\bigl|\frac{dX}{dt}\bigr|$ never vanishes, hence $X(t)$ is traversed monotonically.

References: Helman–Laurain–Garcia–Reznik “Poncelet Triangles: a Theory for Locus Ellipticity,” which contains full derivations of the above $u,v$-formulas and the special ratio‐conditions in both the confocal and incircle families.