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Parabolic Cylindrical Coordinates

An orthogonal 3D coordinate system formed by extruding a confocal parabolic coordinate web along a Cartesian axis, with a quadratic planar map and explicit branch domain.

Version
v1 · 2026-09-28 · History
Domain-specific #
11192
Domain group
Formal Sciences
Origin domain
Mathematics
Subdomains
Orthogonal Curvilinear Coordinates, Potential Theory → Mathematics
Aliases
Parabolic cylinder coordinates

Core Idea

Parabolic cylindrical coordinates use two variables whose constant curves are confocal parabolas and a third ordinary axial variable. The resulting constant-coordinate surfaces are orthogonal parabolic cylinders and planes.

Quadratic mapping creates useful symmetry for boundary-value problems but also sign redundancy and singular points. Scale factors, Jacobian, vector basis, and parameter ranges must accompany the coordinate formulas.

Structural Signature

Sig role-phrases:

  • Coordinate parameters — Provide two parabolic variables and one axial variable. It is parameters. Counterfactual: Unrestricted signs can duplicate points.
  • Quadratic transformation — Maps parameters into Cartesian x,y. It is coordinate map. Counterfactual: Changing signs or factors changes convention.
  • Axial coordinate z — Extrudes the planar web into cylinders. It is third dimension. Counterfactual: Without it the system is parabolic coordinates in a plane.
  • Domain/branch convention — Makes representation one-to-one except singular sets. It is validity frame. Counterfactual: Ignoring branch choice causes double coverage.
  • Scale factors — Encode metric, Jacobian, and differential operators. It is geometric invariant. Counterfactual: Cartesian derivative formulas do not transfer unchanged.
  • Parabolic coordinate surfaces — Give geometric meaning and symmetry alignment. It is recognition. Counterfactual: One isolated parabola does not define the system.

What It Is Not

  • It is not one parabolic cylinder.
  • It is not merely 2D parabolic coordinates.
  • It is not elliptic cylindrical coordinates.
  • Coordinate domains cannot be omitted.
  • Closest near-miss. Parabolic coordinates are the planar web; parabolic cylindrical coordinates add an unchanged axial coordinate.

Scope of Application

  • Mathematical physics. Separates PDEs with parabolic boundaries.
  • Vector calculus. Expresses operators in an orthogonal chart.
  • Electromagnetism. Models parabolic-cylinder structures.
  • Wave mechanics. Uses separated parabolic-cylinder functions.

Clarity

State Cartesian map, inverse/branch, parameter ranges, axis/orientation, coordinate surfaces, scale factors, Jacobian, basis vectors, singularities, handedness, and PDE boundary alignment.

Manages Complexity

The system trades Cartesian simplicity for geometric alignment, moving difficulty into a nonuniform metric and branch management.

Abstract Reasoning

  1. Choose axis and map convention.
  2. Set a one-to-one parameter domain.
  3. Derive coordinate surfaces and basis.
  4. Compute scale factors and Jacobian.
  5. Transform operators and verify singular boundaries.

Knowledge Transfer

Coordinate solutions transfer only after map scaling, rotation, domain, metric factors, and boundary surfaces are matched.

Examples

Canonical

Using x=(σ²−τ²)/2, y=στ and z=z, constant σ and τ form orthogonal confocal parabolic cylinders with planar scale factor sqrt(σ²+τ²).

Mapped back: parameters → σ,τ,z; map → quadratic; axis → z; domain → declared; metric → orthogonal.

Applied / In Practice

Extruding one physical parabolic curve creates a surface but not a coordinate system because no two-family orthogonal web and invertible chart are provided.

Mapped back: surface → one cylinder; coordinate web → absent.

Structural Tensions

T1 — Symmetry Alignment versus Coordinate Singularity. The web simplifies parabolic boundaries while scale factors vanish or mapping duplicates at special sets.

Diagnostic: Is the computational domain covered one-to-one?

T2 — Formula Convention versus Geometric Invariance. Rotations and swapped variables preserve geometry but change equations and domains.

Diagnostic: Which transformation and handedness are used?

Structural–Framed Character

Parabolic Cylindrical Coordinates are structural as an orthogonal quadratic chart and physically framed by symmetry choice.

Structural Core vs. Domain Accent

The core is chart, domain, basis, metric, and surfaces; applications supply boundaries, fields, and separated equations.

This entry is a kind of Orthogonal coordinates.

  • Approved root. No reviewed parent entails this coordinate chart.

  • Related — parabolic coordinates, orthogonal coordinates, cylindrical coordinates, Jacobian, and parabolic-cylinder function. They provide base web, family, contrast, metric, and solution.

Relationships to Other Abstractions

Local relationship map for Parabolic Cylindrical CoordinatesParents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.Parabolic CylindricalCoordinatesDOMAINDomain-specific abstraction: Orthogonal coordinates — is a kind ofOrthogonalcoordinatesDOMAIN

Current abstraction Parabolic Cylindrical Coordinates Domain-specific

Parents (1) — more general patterns this builds on

  • Parabolic Cylindrical Coordinates is a kind of Orthogonal coordinates Domain-specific

    Parabolic Cylindrical Coordinates is a strict kind of Orthogonal coordinates: its extruded confocal parabolic coordinate surfaces meet orthogonally.

Hierarchy path (1) — routes to 1 parentless root

Neighborhood in Abstraction Space

Parabolic Cylindrical Coordinates sits in a crowded region of the domain-specific corpus (29th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Physical & Geometric Dynamical Quantities (29 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With

  • Parabolic coordinates. Tell: Are two-dimensional.
  • Elliptic cylindrical coordinates. Tell: Use confocal ellipses/hyperbolas.
  • Paraboloidal coordinates. Tell: Have different 3D surfaces.
  • Parabolic cylinder. Tell: Is one surface, not the coordinate system.

References

  • Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Parabolic_cylindrical_coordinates (revision 1316751836).
  • Preserved source candidate: https://archive.org/details/mathematicsofphy0002marg
  • Preserved source candidate: https://archive.org/details/mathematicsofphy0002marg/page/186
  • Preserved source candidate: http://mathworld.wolfram.com/ParabolicCylindricalCoordinates.html

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.