Collapsed Coordinate Transformation

• Collapsed coordinate transformation is a set of techniques that re-map complex spatial or dynamical dependencies into lower-dimensional, canonical representations by absorbing nontrivial structure into a Jacobian or rotor.
• In unbalanced power systems, the method extracts the signal locus from two measurements to collapse an n-dimensional signal into a two-dimensional plane, recovering known frameworks like Clarke and Park transforms.
• The approach also simplifies nonlinear diffusion and relativistic formulations, though care must be taken as not all collapsed mappings (e.g., in transformation optics) yield the intended physical isolation.

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 (Montoya et al., 12 Jun 2025). 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 (Ansini et al., 2019). Related collapse mechanisms also appear in the longitudinal part of Lorentz boosts (Wagner, 2016), in the $0\to R_1$ radial transformation of cylindrical transformation optics (Li et al., 2017), 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 (Smit, 2023).

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 (Montoya et al., 12 Jun 2025)
Nonlinear diffusion	$\sigma_{12}$0 with $\sigma_{12}$1 and $\sigma_{12}$2	Linear mobility in the transformed PDE (Ansini et al., 2019)
Special relativity	Coordinate-free boost projected onto axes aligned with $\sigma_{12}$3	Longitudinal coordinate collapse by $\sigma_{12}$4 (Wagner, 2016)
Cylindrical transformation optics	$\sigma_{12}$5	Exact pull-back of fields, but no cloak (Li et al., 2017)
Collapsed-phase EDT	Singular or regular map from $\sigma_{12}$6 to $\sigma_{12}$7	Schwarzschild-like $\sigma_{12}$8 metric with interior (Smit, 2023)

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 $\sigma_{12}$9 into $\mathbb{R}$0 leaves a nonzero inner-boundary field and therefore cannot produce cylindrical cloaking (Li et al., 2017).

2. Geometric Algebra collapsed frame in unbalanced power systems

The most explicit use of the expression appears in the analysis of unbalanced $\mathbb{R}$1-phase, $\mathbb{R}$2-wire sinusoidal systems. An orthonormal basis $\mathbb{R}$3 is chosen, and an instantaneous electrical quantity is represented by

$\mathbb{R}$4

A bivector $\mathbb{R}$5 represents an oriented plane, and a rotor $\mathbb{R}$6 with $\mathbb{R}$7 rotates multivectors by the sandwich product $\mathbb{R}$8 (Montoya et al., 12 Jun 2025).

For an unbalanced sinusoidal $\mathbb{R}$9-phase system, the trajectory $0\to R_1$0 lies on a $0\to R_1$1D plane in $0\to R_1$2. The method identifies that plane from two samples,

$0\to R_1$3

with $0\to R_1$4 and $0\to R_1$5. The plane bivector is

$0\to R_1$6

If $0\to R_1$7, the signal is collinear, interpreted as zero-sequence. Otherwise one normalizes $0\to R_1$8.

Because in $0\to R_1$9 dimensions two planes need not intersect, the construction uses a two-step rotor. First,

$SO(4)$0

rotates $SO(4)$1 onto $SO(4)$2. Then

$SO(4)$3

contains $SO(4)$4 and therefore intersects the $SO(4)$5-plane. The second rotor is

$SO(4)$6

and the total rotor is

$SO(4)$7

The collapsed transformation of any subsequent sample is

$SO(4)$8

The paper presents this as a direct transformation valid for any degree of unbalance in $SO(4)$9-phase, $SO(4)$0-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

$SO(4)$1

while the Park frame is obtained by the extra planar rotor

$SO(4)$2

The algorithmic procedure is fully specified: measure $SO(4)$3; compute $SO(4)$4; normalize $SO(4)$5 and $SO(4)$6; construct $SO(4)$7, then $SO(4)$8, then $SO(4)$9; and finally apply the sandwich product to each new sample. The stated computational cost is $SO(3)$0 one-off plus $SO(3)$1 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 $SO(3)$2 had all 15 bivector components nonzero, the two-step rotor yielded $SO(3)$3 with $SO(3)$4, and the transformed vectors lay exactly in the $SO(3)$5-plane. In a 3-phase real-time laboratory test on an OPAL-RT platform with phase-phase-ground fault injection, the sample interval was $SO(3)$6, the plane was computed in one step with $SO(3)$7 separation, and the rotor

$SO(3)$8

produced collapsed coordinates that remained stable, zero-sequence removed, and fast-responding under pre-fault and fault conditions (Montoya et al., 12 Jun 2025).

3. Coordinate transformation for nonlinear diffusion with degenerate fast-decay mobility

In nonlinear diffusion on a bounded interval $SO(3)$9, the transformation is designed for a mobility $\mathbf{R}=\mathbf{R}_2\mathbf{R}_1$0 that is $\mathbf{R}=\mathbf{R}_2\mathbf{R}_1$1, vanishes at the endpoints, and satisfies the Osgood condition

$\mathbf{R}=\mathbf{R}_2\mathbf{R}_1$2

The strictly increasing map

$\mathbf{R}=\mathbf{R}_2\mathbf{R}_1$3

then maps $\mathbf{R}=\mathbf{R}_2\mathbf{R}_1$4 diffeomorphically onto $\mathbf{R}=\mathbf{R}_2\mathbf{R}_1$5, with

$\mathbf{R}=\mathbf{R}_2\mathbf{R}_1$6

This is the core coordinate change for fast-decay mobilities (Ansini et al., 2019).

Writing $\mathbf{R}=\mathbf{R}_2\mathbf{R}_1$7, the rescaled density is

$\mathbf{R}=\mathbf{R}_2\mathbf{R}_1$8

and preserves mass: $\mathbf{R}=\mathbf{R}_2\mathbf{R}_1$9 The original equation is

$\mathbf{v}'=\mathbf{R}\mathbf{v}\mathbf{R}^\dagger$0

After setting $\mathbf{v}'=\mathbf{R}\mathbf{v}\mathbf{R}^\dagger$1 and $\mathbf{v}'=\mathbf{R}\mathbf{v}\mathbf{R}^\dagger$2, the transformed equation becomes

$\mathbf{v}'=\mathbf{R}\mathbf{v}\mathbf{R}^\dagger$3

where the new nonlinearity is defined by

$\mathbf{v}'=\mathbf{R}\mathbf{v}\mathbf{R}^\dagger$4

Equivalently,

$\mathbf{v}'=\mathbf{R}\mathbf{v}\mathbf{R}^\dagger$5

The paper’s stated point is that the spatial dependence $\mathbf{v}'=\mathbf{R}\mathbf{v}\mathbf{R}^\dagger$6 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

$\mathbf{v}'=\mathbf{R}\mathbf{v}\mathbf{R}^\dagger$7

Under standard hypotheses—$\mathbf{v}'=\mathbf{R}\mathbf{v}\mathbf{R}^\dagger$8 convex and growing superlinearly, $\mathbf{v}'=\mathbf{R}\mathbf{v}\mathbf{R}^\dagger$9 a confining potential, and appropriate bounds on $n$0 ensuring displacement $n$1-convexity of $n$2 in the $n$3-Wasserstein metric—the JKO scheme is applied: $n$4 Standard compactness and lower-semicontinuity arguments yield a weak limit curve

$n$5

and uniqueness follows from the EVI characterization of $n$6-flows in Wasserstein space: $n$7

Back-translation is given by

$n$8

The resulting $n$9 for all $\sigma_{12}$00 satisfies the original PDE in weak form against test functions vanishing near $\sigma_{12}$01, and no additional boundary conditions at the endpoints are needed because the fast-decay of $\sigma_{12}$02 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 $\sigma_{12}$03, and that higher-dimensional extensions would require replacing the $\sigma_{12}$04D map $\sigma_{12}$05 by a solution of a Liouville-type or elliptic change-of-variable problem (Ansini et al., 2019).

4. Collapse of the longitudinal coordinate in coordinate-free relativity

In the coordinate-free treatment of boosts, the relation between the position-time $\sigma_{12}$06 in one frame and $\sigma_{12}$07 in another moving at velocity $\sigma_{12}$08 is

$\sigma_{12}$09

with $\sigma_{12}$10, $\sigma_{12}$11, and $\sigma_{12}$12 (Wagner, 2016).

The key decomposition is

$\sigma_{12}$13

Then

$\sigma_{12}$14

while the component parallel to the boost obeys

$\sigma_{12}$15

After introducing Cartesian axes with $\sigma_{12}$16 and $\sigma_{12}$17 spanning the orthogonal plane, the familiar form follows: $\sigma_{12}$18

The paper interprets the collapse as a length-contraction effect in the boost direction. For two events simultaneous in frame $\sigma_{12}$19 and separated by $\sigma_{12}$20,

$\sigma_{12}$21

A rod at rest in frame $\sigma_{12}$22 therefore appears in frame $\sigma_{12}$23 to have length

$\sigma_{12}$24

so the parallel coordinate has “collapsed” by $\sigma_{12}$25, 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 (Wagner, 2016).

5. Radial collapse in cylindrical transformation optics and the no-cloak result

In cylindrical transformation optics, the transformation

$\sigma_{12}$26

maps the virtual region $\sigma_{12}$27 into the physical shell $\sigma_{12}$28, with

$\sigma_{12}$29

for the stated endpoint and derivative conditions. The common linear choice is

$\sigma_{12}$30

but the argument is presented for any smooth, monotonic increasing $\sigma_{12}$31 satisfying the above conditions (Li et al., 2017).

The push-forward of Maxwell’s equations yields anisotropic relative parameters in the shell. With $\sigma_{12}$32 and $\sigma_{12}$33, the tensor in the $\sigma_{12}$34 basis is diagonal: $\sigma_{12}$35 For the $\sigma_{12}$36-directed field, the transformed anisotropic equation is

$\sigma_{12}$37

The crucial ansatz is

$\sigma_{12}$38

which, by direct substitution and the chain rule, collapses the anisotropic equation back to the original free-space equation in $\sigma_{12}$39.

Because $\sigma_{12}$40 and $\sigma_{12}$41, 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 $\sigma_{12}$42, one has $\sigma_{12}$43, so

$\sigma_{12}$44

These are exactly the boundary data for a free-space interior solution in $\sigma_{12}$45, and the paper states that the unique solution is nonzero throughout the interior. Its conclusion is therefore negative: energy leaks into $\sigma_{12}$46, the region is not field-free, and the incident wave passes straight through the shell without distortion. Li et al. further state that in many published papers on EM cylinder cloak, the $\sigma_{12}$47 to $\sigma_{12}$48 cylinder radial transformation is wrongfully used for their EM cylinder layer cloak (Li et al., 2017).

6. Singular and regular transformations in the collapsed phase of EDT

In the collapsed phase of Euclidean dynamical triangulations, the starting point is the $\sigma_{12}$49-invariant metric

$\sigma_{12}$50

with

$\sigma_{12}$51

Because $\sigma_{12}$52, the minimal three-sphere radius $\sigma_{12}$53 is finite (Smit, 2023).

The singular transformation introduces $\sigma_{12}$54 through

$\sigma_{12}$55

Two conditions are imposed: $\sigma_{12}$56 Formally solving the resulting system leads, in the limit $\sigma_{12}$57, to the diagonal metric

$\sigma_{12}$58

with

$\sigma_{12}$59

and

$\sigma_{12}$60

The $\sigma_{12}$61 slice is therefore interpreted as a Euclidean black-hole exterior with a zero at $\sigma_{12}$62 and a regular interior of radius $\sigma_{12}$63.

A family of regular transformations is obtained by imposing only $\sigma_{12}$64, integrating $\sigma_{12}$65 numerically at fixed $\sigma_{12}$66, and deferring the diagonal limit to $\sigma_{12}$67. The spatial inverse component has the envelope

$\sigma_{12}$68

while $\sigma_{12}$69 depends strongly on the choice of pole, interpreted as a time re-parametrization. After rescaling $\sigma_{12}$70 so that $\sigma_{12}$71 or $\sigma_{12}$72 in the interior, the limiting $\sigma_{12}$73 coincides with the singular result.

The Einstein tensor also stabilizes in the $\sigma_{12}$74 limit. In the interior,

$\sigma_{12}$75

while the exterior components are nontrivial functions of $\sigma_{12}$76 and satisfy $\sigma_{12}$77. The off-diagonal component $\sigma_{12}$78 vanishes as $\sigma_{12}$79, including at $\sigma_{12}$80 in the distributional sense. The paper interprets $\sigma_{12}$81 as an effective condensate energy-momentum tensor with constant negative energy density in the interior, positive tensions, and nontrivial exterior $\sigma_{12}$82-dependence.

An inverse construction starts from the Hayward regular black-hole metric

$\sigma_{12}$83

and solves

$\sigma_{12}$84

with $\sigma_{12}$85 and $\sigma_{12}$86. The resulting $\sigma_{12}$87 is compared with the EDT scale factor. For $\sigma_{12}$88 and $\sigma_{12}$89 chosen so that $\sigma_{12}$90, the behavior is reported as very similar at small and large $\sigma_{12}$91, and the fit requires $\sigma_{12}$92, matching at the few-percent level over $\sigma_{12}$93 apart from lattice-artifact deviations at very small or very large $\sigma_{12}$94 (Smit, 2023).

7. Structural themes, distinctions, and recurrent misconceptions

Taken together, these papers suggest that a collapsed coordinate transformation is best understood as a structural operation rather than a single named formula. The collapsed object may be the inhomogeneous mobility in a diffusion equation, the signal-locus plane of an unbalanced electrical system, the longitudinal coordinate in a boost, the virtual radial interval of a cylindrical shell, or the foliation of an $\sigma_{12}$95-invariant metric. What is shared is the relocation of nontrivial dependence into a Jacobian, a rotor, a longitudinal projection, or an induced anisotropic tensor so that the transformed description becomes simpler or more revealing.

A central distinction is between analytical simplification and physical equivalence. In diffusion, the transformation removes the explicit space dependence of the mobility and enables existence and uniqueness via Wasserstein gradient-flow techniques (Ansini et al., 2019). In power systems, it identifies the exact signal-locus plane from two samples and yields a two-coordinate reference frame compatible with Clarke and Park limits (Montoya et al., 12 Jun 2025). In relativity, the collapse concerns only the parallel component and is inseparable from the relativity of simultaneity (Wagner, 2016). By contrast, the cylindrical radial collapse in transformation optics preserves the pulled-back field values and therefore, according to Li et al., fails to isolate the interior region from the incident field (Li et al., 2017).

Several recurrent misconceptions are explicitly addressed in the source material. One is that collapsing coordinates automatically implies physical concealment; the cylindrical cloak example is presented as a counterexample. Another is that a vectorial boost law is identical to a coordinate transformation of column vectors; the coordinate-free relativity paper argues that the latter requires an explicit physical specification of coordinate systems. A further misconception would be to treat all collapse procedures as dimensional reduction in the same sense: in the diffusion paper the transformed space is $\sigma_{12}$96 rather than a lower-dimensional manifold, whereas in the power-systems paper the transformed signal is literally confined to two coordinates.

The broader implication is methodological. A collapsed coordinate transformation can expose latent canonical structure—flat Wasserstein mobility, planar sinusoidal dynamics, Schwarzschild-like symmetry—or it can reveal that a formally exact mapping does not produce the intended physical effect. In that sense, the term designates a class of coordinate strategies whose value depends on what is preserved by the transformation: mass, signal plane, Lorentz interval, Maxwell operator, or limiting Einstein tensor.