--- title: Collapsed Coordinate Transformation url: https://www.emergentmind.com/topics/collapsed-coordinate-transformation type: topic --- # Collapsed Coordinate Transformation Searching arXiv for relevant papers on "collapsed coordinate transformation" and adjacent usages. arXiv search query: "collapsed coordinate transformation" Collapsed coordinate transformation denotes, in the cited literature, a family of technically distinct coordinate changes that concentrate the effective dynamics, geometry, or observable content into a reduced or canonical representation. In unbalanced power systems, it is a Geometric Algebra construction that identifies the two-dimensional signal locus in $\mathbb{R}^n$ from two samples and rotates it into the canonical $\sigma_{12}$-plane [2506.10835]. In nonlinear diffusion with fast-decay mobility, it is a diffeomorphic change of variables that maps a bounded interval to $\mathbb{R}$ and converts a diffusion with degenerate space-dependent mobility into one with linear mobility [1902.02764]. Related collapse mechanisms also appear in the longitudinal part of Lorentz boosts [1611.08018], in the $0\to R_1$ radial transformation of cylindrical transformation optics [1707.02500], and in singular or regular transformations from an $SO(4)$-invariant metric to a Schwarzschild-like form in the collapsed phase of Euclidean dynamical triangulations [2305.16011]. ## 1. Comparative scope of the term The phrase does not refer to a single universal formalism. The cited papers use it for different operations: dimensional reduction of a signal locus, absorption of a mobility into a Jacobian, longitudinal contraction under a boost, collapse of a virtual radial interval to an inner boundary, and transformation of an $SO(4)$-invariant metric into an $SO(3)$-invariant form. This suggests a common structural motif: a coordinate map absorbs nontrivial spatial or geometric dependence into transformed variables so that the resulting problem becomes lower-dimensional, canonical, or more directly interpretable. | Domain | Transformation | Principal outcome | |---|---|---| | Unbalanced power systems | Rotor $\mathbf{R}=\mathbf{R}_2\mathbf{R}_1$ acting by $\mathbf{v}'=\mathbf{R}\mathbf{v}\mathbf{R}^\dagger$ | Collapse of an $n$-phase signal to two coordinates [2506.10835] | | Nonlinear diffusion | $y=\Phi(x)$ with $\Phi(x)=\int_{x_0}^x \frac{dz}{g(z)}$ and $\rho=\Phi'(x)u$ | Linear mobility in the transformed PDE [1902.02764] | | Special relativity | Coordinate-free boost projected onto axes aligned with $\hat{\mathbf v}$ | Longitudinal coordinate collapse by $1/\gamma$ [1611.08018] | | Cylindrical transformation optics | $\rho_q(\rho)=R_1+Q(\rho)$ | Exact pull-back of fields, but no cloak [1707.02500] | | Collapsed-phase EDT | Singular or regular map from $(\rho,\chi)$ to $(r,t)$ | Schwarzschild-like $t=0$ metric with interior [2305.16011] | A recurring distinction is between formal equivalence and physical effect. In some settings the collapse is analytically productive, as in diffusion and power systems. In others it exposes limitations: Li et al. argue that collapsing $0\le \rho\le R_2$ into $R_1\le \rho_q\le R_2$ leaves a nonzero inner-boundary field and therefore cannot produce cylindrical cloaking [1707.02500]. ## 2. Geometric Algebra collapsed frame in unbalanced power systems The most explicit use of the expression appears in the analysis of unbalanced $n$-phase, $(n+1)$-wire sinusoidal systems. An orthonormal basis $\{\sigma_1,\sigma_2,\dots,\sigma_n\}$ is chosen, and an instantaneous electrical quantity is represented by \[ \mathbf{v}(t)=\sum_{k=1}^n v_k(t)\,\sigma_k \in \mathbb{R}^n. \] A bivector $\mathbf{B}=\mathbf{a}\wedge\mathbf{b}$ represents an oriented plane, and a rotor $\mathbf{R}$ with $\mathbf{R}\mathbf{R}^\dagger=1$ rotates multivectors by the sandwich product $\mathbf{M}'=\mathbf{R}\mathbf{M}\mathbf{R}^\dagger$ [2506.10835]. For an unbalanced sinusoidal $n$-phase system, the trajectory $\mathbf{v}(t)$ lies on a $2$D plane in $\mathbb{R}^n$. The method identifies that plane from two samples, \[ \mathbf{v}_1=\mathbf{v}(t_1), \qquad \mathbf{v}_2=\mathbf{v}(t_2), \] with $t_2-t_1\le T$ and $\mathbf{v}(t_2)\not\parallel\mathbf{v}(t_1)$. The plane bivector is \[ \mathbf{B}=\mathbf{v}_1\wedge \mathbf{v}_2 =\sum_{1\le i3$ dimensions two planes need not intersect, the construction uses a two-step rotor. First, \[ \mathbf{R}_1= \frac{1+\sigma_1\,\hat{\mathbf{v}}_1} {\big\|1+\sigma_1\,\hat{\mathbf{v}}_1\big\|}, \qquad \hat{\mathbf{v}}_1=\frac{\mathbf{v}_1}{\|\mathbf{v}_1\|}, \] rotates $\hat{\mathbf{v}}_1$ onto $\sigma_1$. Then \[ \hat{\mathbf{B}}_\times=\mathbf{R}_1\,\hat{\mathbf{B}}\,\mathbf{R}_1^\dagger \] contains $\sigma_1$ and therefore intersects the $\sigma_{12}$-plane. The second rotor is \[ \mathbf{R}_2= \frac{1+\sigma_{12}\,\hat{\mathbf{B}}_\times^\dagger} {\big\|1+\sigma_{12}\,\hat{\mathbf{B}}_\times^\dagger\big\|}, \] and the total rotor is \[ \mathbf{R}=\mathbf{R}_2\mathbf{R}_1. \] The collapsed transformation of any subsequent sample is \[ \mathbf{v}'(t)=\mathbf{R}\,\mathbf{v}(t)\,\mathbf{R}^\dagger = v'_1(t)\,\sigma_1 + v'_2(t)\,\sigma_2, \qquad (v'_3,\dots,v'_n=0). \] The paper presents this as a direct transformation valid for any degree of unbalance in $n$-phase, $(n+1)$-wire sinusoidal systems, requiring only two measurements taken at different time instants. Clarke and Park appear as special cases. In the balanced three-phase case, the construction recovers the classical Clarke components \[ v'_\alpha=\tfrac23\bigl(v_a-\tfrac12 v_b-\tfrac12 v_c\bigr),\qquad v'_\beta=\tfrac{\sqrt3}{3}(v_b-v_c),\qquad v'_0=0, \] while the Park frame is obtained by the extra planar rotor \[ \exp\!\bigl(-\tfrac{\omega t}{2}\sigma_{12}\bigr). \] The algorithmic procedure is fully specified: measure $\mathbf{v}_1,\mathbf{v}_2$; compute $\mathbf{B}=\mathbf{v}_1\wedge\mathbf{v}_2$; normalize $\mathbf{B}$ and $\mathbf{v}_1$; construct $\mathbf{R}_1$, then $\mathbf{R}_2$, then $\mathbf{R}$; and finally apply the sandwich product to each new sample. The stated computational cost is $\mathcal{O}(n^2)$ one-off plus $\mathcal{O}(n)$ per sample, with only two nearly simultaneous measurements and no quarter-cycle wait or recursive amplitude/phase estimator. The validations emphasize both numerical precision and hardware feasibility. In a 6-phase synthetic test, the computed $\mathbf{B}$ had all 15 bivector components nonzero, the two-step rotor yielded $\mathbf{B}'=\sigma_{12}+\varepsilon$ with $\|\varepsilon\|\approx 10^{-16}$, and the transformed vectors lay exactly in the $\{\sigma_1,\sigma_2\}$-plane. In a 3-phase real-time laboratory test on an OPAL-RT platform with phase-phase-ground fault injection, the sample interval was $100\,\mu\text{s}$, the plane was computed in one step with $1.6\,\text{ms}$ separation, and the rotor \[ \mathbf{R}=0.4597+0.6280\,\sigma_{13}+0.6280\,\sigma_{23} \] produced collapsed coordinates that remained stable, zero-sequence removed, and fast-responding under pre-fault and fault conditions [2506.10835]. ## 3. Coordinate transformation for nonlinear diffusion with degenerate fast-decay mobility In nonlinear diffusion on a bounded interval $(a,b)$, the transformation is designed for a mobility $g(x)\ge 0$ that is $C^1$, vanishes at the endpoints, and satisfies the Osgood condition \[ \int_a^b \frac{dz}{g(z)}=+\infty. \] The strictly increasing map \[ \Phi(x):=\int_{x_0}^x \frac{dz}{g(z)}, \qquad x_0\in (a,b), \] then maps $(a,b)$ diffeomorphically onto $\mathbb{R}$, with \[ \Phi'(x)=\frac{1}{g(x)}>0,\qquad (\Phi^{-1})'(y)=g(\Phi^{-1}(y)). \] This is the core coordinate change for fast-decay mobilities [1902.02764]. Writing $y=\Phi(x)$, the rescaled density is \[ \rho(y,t):=\Phi'(x)\,u(x,t)=\frac{1}{g(x)}u(x,t), \] and preserves mass: \[ \int_{\mathbb{R}}\rho(y,t)\,dy=\int_a^b u(x,t)\,dx. \] The original equation is \[ \partial_t u = \partial_x\Bigl(g(x)^2\,u\,\partial_x[\phi'(u)+W(x)]\Bigr). \] After setting $x=\Phi^{-1}(y)$ and $u(x,t)=\Phi'(x)\rho(y,t)$, the transformed equation becomes \[ \partial_t \rho = \partial_y\Bigl(\rho\,\partial_y[\varphi'(\rho)]\Bigr), \] where the new nonlinearity is defined by \[ \varphi'(\rho):=\phi'(u) \quad\text{with}\quad u=g^{-1}(\Phi^{-1}(y))\,\rho. \] Equivalently, \[ \partial_t \rho=\partial_y\bigl(A(\rho)\,\partial_y\rho\bigr), \qquad A(\rho):=\rho\,\varphi''(\rho). \] The paper’s stated point is that the spatial dependence $g(x)$ has dropped out of the mobility factor, while the nonlinearity described by the original nonlinear mobility is included in the diffusive process. Existence and uniqueness are developed at the level of the rescaled density. The free-energy functional is \[ \mathcal{K}[\rho] = \int_{\mathbb{R}}\varphi(\rho(y))\,dy + \int_{\mathbb{R}}V(y)\,\rho(y)\,dy, \qquad V(y)=W(\Phi^{-1}(y)). \] Under standard hypotheses—$\varphi$ convex and growing superlinearly, $V$ a confining potential, and appropriate bounds on $g,\phi,W$ ensuring displacement $\lambda$-convexity of $\mathcal{K}$ in the $2$-Wasserstein metric—the JKO scheme is applied: \[ \rho^k\in\arg\min_{\rho\in\mathcal{P}_2} \frac{1}{2\tau}W_2^2(\rho,\rho^{k-1})+\mathcal{K}[\rho]. \] Standard compactness and lower-semicontinuity arguments yield a weak limit curve \[ \rho(t)\in L^m([0,T]\times\mathbb{R})\cap L_t^2 H_x^1(\mathbb{R}), \] and uniqueness follows from the EVI characterization of $\lambda$-flows in Wasserstein space: \[ W_2(\rho(t),\bar{\rho}(t)) \le e^{-\lambda t} W_2(\rho^0,\bar{\rho}^0). \] Back-translation is given by \[ u(x,t)=\Phi'(x)\,\rho(\Phi(x),t). \] The resulting $u\in L^m((a,b))$ for all $t>0$ satisfies the original PDE in weak form against test functions vanishing near $x=a,b$, and no additional boundary conditions at the endpoints are needed because the fast-decay of $g$ makes the flux vanish automatically. The paper further states that other degenerate-mobility diffusion equations, including multi-species systems or higher-order nonlinearities, can be tackled by the same collapsing of the inhomogeneous mobility into the Jacobian $\Phi'$, and that higher-dimensional extensions would require replacing the $1$D map $\Phi$ by a solution of a Liouville-type or elliptic change-of-variable problem [1902.02764]. ## 4. Collapse of the longitudinal coordinate in coordinate-free relativity In the coordinate-free treatment of boosts, the relation between the position-time $(\mathbf{r},t)$ in one frame and $(\mathbf{r}',t')$ in another moving at velocity $\mathbf{v}$ is \[ \mathbf r' = \mathbf r + \Bigl[(\gamma-1)(\hat{\mathbf v}\!\cdot\!\mathbf r)-\gamma v t\Bigr]\hat{\mathbf v}, \qquad t'=\gamma\!\Bigl(t-\tfrac{v}{c^2}(\hat{\mathbf v}\!\cdot\!\mathbf r)\Bigr), \] with $v=|\mathbf{v}|$, $\hat{\mathbf v}=\mathbf v/v$, and $\gamma=(1-v^2/c^2)^{-1/2}$ [1611.08018]. The key decomposition is \[ \mathbf r=\mathbf r_\parallel+\mathbf r_\perp, \qquad \mathbf r_\parallel=(\hat{\mathbf v}\!\cdot\!\mathbf r)\,\hat{\mathbf v}, \qquad \mathbf r_\perp=\mathbf r-\mathbf r_\parallel. \] Then \[ \mathbf r'_\perp=\mathbf r_\perp, \] while the component parallel to the boost obeys \[ \mathbf r'_\parallel = \gamma\bigl((\hat{\mathbf v}\!\cdot\!\mathbf r)-vt\bigr)\hat{\mathbf v}. \] After introducing Cartesian axes with $\mathbf e_x\equiv \hat{\mathbf v}$ and $\mathbf e_y,\mathbf e_z$ spanning the orthogonal plane, the familiar form follows: \[ x'=\gamma(x-vt),\qquad y'=y,\qquad z'=z,\qquad t'=\gamma\Bigl(t-\frac{v}{c^2}x\Bigr). \] The paper interprets the collapse as a length-contraction effect in the boost direction. For two events simultaneous in frame $A$ and separated by $\Delta x$, \[ \Delta t'=-\gamma\,v\,\Delta x/c^2, \qquad \Delta x'=\gamma\,\Delta x. \] A rod at rest in frame $B$ therefore appears in frame $A$ to have length \[ \Delta x=\frac{\Delta x'}{\gamma}, \] so the parallel coordinate has “collapsed” by $1/\gamma$, while perpendicular lengths are untouched. The derivation is explicitly tied to a physics-based formulation in which a boost direction is identified through free motions of point particles and coordinate systems are specified in each frame in terms of physical objects and directions. The paper also emphasizes that the vectorial relation differs from the vector-like relationship often treated as a coordinate transformation in the literature, because the latter deals with column vectors of Cartesian coordinates of true vectors [1611.08018]. ## 5. Radial collapse in cylindrical transformation optics and the no-cloak result In cylindrical transformation optics, the transformation \[ \rho_q(\rho)=R_1+Q(\rho),\qquad 0\le \rho\le R_2, \] maps the virtual region $0\le \rho\le R_2$ into the physical shell $R_1\le \rho_q\le R_2$, with \[ Q(0)=0,\qquad Q(R_2)=R_2-R_1,\qquad \frac{dQ}{d\rho}\Big|_{\rho=R_2}=1 \] for the stated endpoint and derivative conditions. The common linear choice is \[ Q(\rho)=\frac{R_2-R_1}{R_2}\rho, \] but the argument is presented for any smooth, monotonic increasing $Q$ satisfying the above conditions [1707.02500]. The push-forward of Maxwell’s equations yields anisotropic relative parameters in the shell. With $\rho=Q^{-1}(\rho_q-R_1)$ and $f(\rho)=\rho_q$, the tensor in the $(\rho_q,\phi,z)$ basis is diagonal: \[ \epsilon=\mu=\operatorname{diag}\!\left[\frac{\rho f'}{f},\, \frac{f}{\rho f'},\, \frac{f'f}{\rho}\right]. \] For the $z$-directed field, the transformed anisotropic equation is \[ \frac{1}{\epsilon_z \rho_q}\partial_{\rho_q} \Bigl(\frac{\rho_q}{\mu_\rho}\partial_{\rho_q}H_z\Bigr) +\frac{1}{\epsilon_z\rho_q^2}\partial_\phi^2 H_z +k^2\mu_z H_z=0. \] The crucial ansatz is \[ H_z(\rho_q,\phi,z) = H_z^{\mathrm{inc}}\!\bigl(\rho=Q^{-1}(\rho_q-R_1),\phi,z\bigr), \] which, by direct substitution and the chain rule, collapses the anisotropic equation back to the original free-space equation in $\rho$. Because $Q(R_2)=R_2$ and $Q'(R_2)=1$, the field and its normal derivative match the free-space field at the outer boundary, yielding the no-scattering condition. The paper then evaluates the same analytic solution at the inner boundary. Since $Q(0)=0$, one has $\rho=0$, so \[ H_z(R_1,\phi,z)=H_z^{\mathrm{inc}}(\rho=0,\phi,z)\ne 0, \qquad \partial_{\rho_q}H_z|_{\rho_q=R_1}=0. \] These are exactly the boundary data for a free-space interior solution in $\rhoh. \] The $t=0$ slice is therefore interpreted as a Euclidean black-hole exterior with a zero at $r=h$ and a regular interior of radius $h$. A family of regular transformations is obtained by imposing only $g_{rt}(r,t)=0$, integrating $y'=P(r,y)$ numerically at fixed $t$, and deferring the diagonal limit to $t\to 0$. The spatial inverse component has the envelope \[ \frac{1}{g_{rr}(r,t)}\to \begin{cases} F(r), & r>h,\\ 1-r^2/h^2, & r