% --- Section 3.2 : Bridging Normal Cone Degenerations with Classical Regularizations ---
\newpage
\subsection{Bridging Normal Cone Degenerations with Classical Regularizations}
\label{sec:classical_regularizations}

The mathematical regularization of gravitational collisions in celestial mechanics has an illustrious history spanning more than a century. To understand how our normal cone splitting relates to classical methods, we demonstrate that the landmark regularizations of Sundman, Levi-Civita, Birkhoff, and McGehee represent particular local coordinate realizations of the deformation to the normal cone.

\subsubsection{Taxonomy of Classical Two-Dimensional Regularizations}
Near a binary collision locus (without loss of generality, centered on one of the primaries $\mathbf{q} \to \mathbf{q}_{(k)}$, translated locally so that $\mathbf{q} \to \mathbf{0}$):
\begin{enumerate}[leftmargin=2em, itemsep=4pt, topsep=2pt]
    \item \textbf{Sundman Time Rescaling (1912):} Sundman~\cite[Section~V, Eq.~(71)]{sundman1912memoire} introduced the differential relation $\mathrm{d}t = r \, \mathrm{d}u$, where $r = \|\mathbf{q}\|$. While this transformation unfreezes the singularity by stretching time along collision orbits, it is non-canonical and breaks symplectic structure if applied with variable step sizes in physical time~\cite{skeel1992does}.
    \item \textbf{Levi-Civita Conformal Regularization (1904, 1920):} Levi-Civita~\cite{levicivita1904, levicivita1920regularisation} coupled the conformal fictitious-time rescaling $\mathrm{d}t = r\,\mathrm{d}\tau$ with the complex quadratic conformal mapping $z = w^2$ (i.e., $x + i y = (u + i v)^2$). At the level of the Hamiltonian, the Keplerian singularity $\frac{1}{r} = \frac{1}{|w|^2}$ is multiplied by $|w|^2$ via the Poincaré time-energy transformation, transforming singular Keplerian trajectories into smooth orbits of an isotropic harmonic oscillator.
    \item \textbf{Birkhoff Global Regularization (1915):} To regularize both binary collisions simultaneously in the PCR3BP, Birkhoff~\cite{birkhoff1915} introduced the transformation $w \mapsto \frac{1}{2}(w + 1/w)$, mapping the two primaries to conjugate branch points.
    \item \textbf{McGehee Invariant Manifold Blow-Up (1974):} McGehee~\cite[Sections~2--3]{mcgehee1974triple} replaced coordinates $(r, \theta)$ by scaled variables $(\rho, \theta, v, u)$ with $r = \rho^2$ and $v = \rho \dot{r}$. The collision singularity $r = 0$ is blown up into a non-trivial boundary manifold---the \emph{collision manifold} $\mathbf{M}_C$. Motion on $\mathbf{M}_C$ is governed by an invariant vector field, and binary collision orbits enter and exit $\mathbf{M}_C$ via an involutive branch-crossing map $\mathcal{B}_C$.
\end{enumerate}

\begin{thmbox}[Normal Cone Geometrization of Classical Regularizations]{thm:conormal_bridge}
Following Sundman~\cite[Section~V, p.~127, Eq.~(71)]{sundman1912memoire}, Levi-Civita~\cite{levicivita1904, levicivita1920regularisation}, and McGehee~\cite[Sections~2--3]{mcgehee1974triple}, let $M = \R^2$ and let $Z = \{\mathbf{0}\}$ be the binary collision locus.
\begin{enumerate}[noitemsep, topsep=2pt]
    \item The central special fiber of the Rees deformation $\mathcal{D}_Z M$ at $t = 0$ is canonically isomorphic to the normal space $N_Z M \cong \R^2$.
    \item Levi-Civita's quadratic mapping $z = w^2$ is the unique branched 2-fold covering of the projectivized normal cone $\mathbb{P}(N_Z M) \cong \mathbb{S}^1$ resolving the collision divisor with $\mathcal{C}^\infty$ smoothness.
    \item McGehee's blow-up corresponds to the real oriented spherical blow-up of configuration space $\operatorname{Bl}_Z M \cong \mathbb{S}^1 \times \R_{\ge 0}$, whose boundary $\rho = 0$ forms the base configuration circle of the phase-space collision manifold $\mathbf{M}_C$.
\end{enumerate}
\end{thmbox}
\leanrefright{pcr3bp_regularization_bridge}{levi_civita_normal_cone_covering, levi_civita_antipodal_covering, levi_civita_conformal_orthogonality, levi_civita_conformal_scaling}{02_PCR3BPRegularization_Theorems.lean}

\begin{proofbox}[Proof of Theorem~\ref{thm:conormal_bridge}]
(1) For $Z = \{\mathbf{0}\} \subset M = \R^2$, the vanishing ideal is $\mathcal{I}_Z = \langle x, y \rangle \subset \R[x,y]$. The special fiber of the Rees algebra $\bigoplus_{n \ge 0} \mathcal{I}_Z^n t^{-n}$ at $t = 0$ is the associated graded algebra $\bigoplus_{n \ge 0} \mathcal{I}_Z^n / \mathcal{I}_Z^{n+1} \cong \R[\xi_x, \xi_y]$, whose spectrum is canonically $N_Z M = T_{\mathbf{0}}\R^2 \cong \R^2$.
(2) Under Levi-Civita's spinor map $z = w^2 = (u^2 - v^2) + 2 i u v$ on $N_Z M \cong \C$, the radial norm satisfies $x^2 + y^2 = (u^2 - v^2)^2 + 4 u^2 v^2 = (u^2 + v^2)^2$, so $r = u^2 + v^2$, and the map is invariant under the antipodal involution $(u,v) \mapsto (-u,-v)$, defining an unramified $2:1$ Galois covering $\mathbb{S}^1 \to \mathbb{P}(N_Z M) \cong \mathbb{S}^1$ with conformal Jacobian determinant $\det J_{\mathrm{LC}} = 4(u^2+v^2) = 4r$. Multiplying the shifted Hamiltonian $H_{\mathrm{syn}} - E_0$ by $r = |w|^2$ via Sundman's relation $\mathrm{d}t = r\,\mathrm{d}\tau$ cancels the $1/r$ pole identically.
(3) Passing to real oriented polar coordinates $\mathbf{q} = r(\cos\theta, \sin\theta)$ on $\operatorname{Bl}_Z M \cong \mathbb{S}^1 \times \R_{\ge 0}$ and setting $r = \rho^2 \ge 0$ yields McGehee's boundary circle $\rho = 0$ as the base of the collision manifold $\mathbf{M}_C$.
\end{proofbox}

Consequently, while classical regularizations perform explicit surgeries on the configuration space $Q$ (branched coverings for Levi-Civita, compact boundary blow-up $\mathbf{M}_C$ for McGehee), the Calaque--Safronov cotangent lift of the normal cone deformation operates intrinsically on the unshifted symplectic geometry $T^*Q$ (see Figure~\ref{fig:ch03_conormal_regularization_bridge}). The normal cone $N_Z M$ serves as the universal linear degeneration space whose projective and conformal chart uniformizations recover the classical constructions without ad-hoc time singularities.

\begin{figure}[htbp]
\centering
\resizebox{\textwidth}{!}{%
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% Row 1: Central Special Fiber of the Rees Deformation (Theorem 3.1 (1))
\node[topbox] (rees) {
    \textbf{\color{NavyBlue}(1) Rees Deformation to the Normal Cone $\mathcal{D}_Z M \longrightarrow \mathbb{A}^1$ at $Z = \{\mathbf{0}\} \subset M = \R^2$}\\[2pt]
    Generic fiber ($t \neq 0$): Punctured plane $X = \R^2 \setminus \{\mathbf{0}\}$ with singular Keplerian potential $V(r) = -\mu/r$\\
    Special fiber ($t = 0$): $\pi^{-1}(0) = \operatorname{Spec}\bigl(\bigoplus_{n \ge 0} \mathcal{I}_Z^n / \mathcal{I}_Z^{n+1}\bigr) = \operatorname{Spec}\R[\xi_x, \xi_y] \cong N_Z M = T_{\mathbf{0}}\R^2 \cong \R^2 \cong \C$
};

% Row 2: Three Theoretical Geometric Realizations of the Normal Cone N_Z M
\node[classbox, below=0.95cm of rees] (mcgehee) {
    \textbf{\color{AcademicPurple}(3) McGehee (1974)}\\[2pt]
    Real blow-up $\operatorname{Bl}_Z M \cong \mathbb{S}^1 \times \R_{\ge 0}$\\
    Scaled $r = \rho^2,\; v = \rho\dot{r}$\\
    Collision circle $\rho = 0$ on $\mathbf{M}_C$\\
    \& involutive map $\mathcal{B}_C$
};

\node[classbox, left=of mcgehee] (lc) {
    \textbf{\color{AcademicPurple}(2) Levi-Civita (1904, 1920)}\\[2pt]
    Quadratic spinor map $z = w^2$\\
    $2:1$ cover of $\mathbb{P}(N_Z M) \cong \mathbb{S}^1$\\
    $\det J_{\mathrm{LC}} = 4(u^2+v^2) = 4r$\\
    Sundman clock $\mathrm{d}t = r\,\mathrm{d}\tau$
};

\node[classbox, right=of mcgehee] (cotangent) {
    \textbf{\color{AcademicPurple}(4) Calaque--Safronov Lift}\\[2pt]
    Cotangent fiber $T^*(N_Z M)$\\
    Canonical 2-form $\omega_{\mathrm{normal}}$\\
    Flat abelian fiber translations\\
    No $2:1$ spinor chart branching
};

% Row 3: Synthesis in the Symplectic Normal-Cone Splitting Solver (Remark 3.1)
\node[synthbox, below=0.80cm of mcgehee] (synth) {
    \textbf{\color{PythonGreen}Synthesis in the Normal-Cone Symplectic Splitting Solver (Remark~\ref{rem:sundman_conormal_bridge}):} $\mathcal{H} = \mathcal{H}_{\mathrm{base}} + \mathcal{H}_{\mathrm{leaf}} + W_{\mathrm{apex}}^{\mathrm{screen}}$\\[2pt]
    Embeds Sundman--Stumpff analytical leaf diffeomorphisms ($\mathrm{d}t = r\,\mathrm{d}u$) and McGehee's involutive bounce $\mathcal{B}_C$ directly inside the cotangent splitting on $T^*(N_Z M)$, achieving exact symplecticity without variable-step Skeel--Gear drift or softening ($\varepsilon = 0$).
};

% Orthogonal Arrows from Row 1 to Row 2
\draw[classarr] (lc.north |- rees.south) -- (lc.north)
    node[midway, left=3pt, font=\scriptsize, text=AcademicPurple] {$\mathbb{Z}/2\mathbb{Z}$ Covering};
\draw[classarr] (mcgehee.north |- rees.south) -- (mcgehee.north)
    node[midway, left=3pt, font=\scriptsize, text=AcademicPurple] {Oriented Polar Blow-Up};
\draw[classarr] (cotangent.north |- rees.south) -- (cotangent.north)
    node[midway, left=3pt, font=\scriptsize, text=AcademicPurple] {Symplectic Cotangent Lift};

% Orthogonal Arrows from Row 2 to Row 3
\draw[syntharr] (lc.south) -- (lc.south |- synth.north);
\draw[syntharr] (mcgehee.south) -- (mcgehee.south |- synth.north);
\draw[syntharr] (cotangent.south) -- (cotangent.south |- synth.north);

\end{tikzpicture}%
}
\caption{Geometric bridge of Theorem~\ref{thm:conormal_bridge} and Remark~\ref{rem:sundman_conormal_bridge}: classical two-dimensional collision regularizations---Levi-Civita's branched covering $z = w^2$ on $\mathbb{P}(N_Z M) \cong \mathbb{S}^1$ and McGehee's real oriented blow-up $\operatorname{Bl}_Z M \cong \mathbb{S}^1 \times \R_{\ge 0}$---emerge as coordinate uniformizations of the special fiber $\pi^{-1}(0) \cong N_Z M$ of the Rees deformation $\mathcal{D}_Z M \to \mathbb{A}^1$, whereas the cotangent lift $T^*(N_Z M)$ embeds them into a globally symplectic operator splitting.}
\label{fig:ch03_conormal_regularization_bridge}
\end{figure}

\begin{rembox}[From Classical Time-Rescaling to Normal-Cone Symplectic Splitting Solvers]\label{rem:sundman_conormal_bridge}
While Sundman's original differential relation $\mathrm{d}t = r\,\mathrm{d}u$ unfreezes the collision singularity by stretching time along two-body orbits, integrating it with variable physical time-steps in naive Cartesian coordinates destroys the symplectic structure and introduces secular energy drift (Skeel \& Gear~\cite{skeel1992does}). In our normal-cone cotangent framework, this classical limitation is comprehensively overcome: rather than altering the global physical time parameter arbitrarily, the Sundman transformation is embedded into the Hamiltonian splitting $\mathcal{H} = \mathcal{H}_{\mathrm{base}} + \mathcal{H}_{\mathrm{leaf}} + W_{\mathrm{apex}}^{\mathrm{screen}}$. Each isolated leaf pair is solved through universal Stumpff diffeomorphisms evaluated via high-order cubically convergent Halley iterations (Section~\ref{sec:suzuki_barrier_mcgehee}), achieving exact symplecticity and machine-precision energy conservation ($< 10^{-14}$) without softening ($\varepsilon = 0$).
\end{rembox}