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Quadrature Domain Solver

Domain type

Weight

Domain

singular (0 ∈ Ω)

Riemann map φ(z)

Quadrature function h(w)

Preset:

Thesis example:

h(w) =

Degree m_∞: (−1 = no polynomial part)

Map parameters

PQD power α

α =

Riemann map center φ(0)

Centroid of poles

Manual:

Residue q at origin

q =

|q| 0.000

arg 0.000π

Conformal radius c = φ′(∞)

c

searching…

Solver settings

Boundary samples:

Aggressiveness:

Auto-fit view on solve

Advanced solve behaviour

Auto-switch singular ⇄ non-singular PQD when the boundary crosses the origin

Overlays

→ Vector field:

Critical points

Curvature heat-strip

Annotated phenomena

Faber roots

Search options (advanced)

Phases

Direct algebraic

Continuation in t

Multistart, perturbed

Diverse seeds, log-uniform

Deflation (Brown–Gearhart)

Multistart budget

Newton restarts

Diverse seeds

Deflation seeds

Background search chunks

Chunk size

Keep searching (no chunk cap)

Newton & continuation

Newton max iterations

Newton tolerance

Continuation t-start

Continuation grow factor

Deflation

α (strength)

p (exponent)

Also deflate from valid solutions

Acceptance

Univalence samples

Identity tol

Show non-univalent candidates

Show identity-failing candidates

Auto-escalate to exhaustive on failure

RNG

Seed

Status

Idle

solving…

running…

Analytic oracle

Faber polynomials

Order N

All roots up to N Single F_n, n =

Plot roots on domain Show polynomial

Quadrature ↔ map equations

Formulation: Classical (forward) Schwarz function

Representation: Conjugate over ℚ(i) Real / imaginary split

Fix φ(0) = w₀ in the equations

Max pole order

Alternate solutions searching…

Figure & export

Strip the plot to just the elements a figure needs, recolour the boundary, and export a high-resolution image — for research-paper figures and mathematical art.

Preset

Style

Elements

Axes Grid Tick labels Interior fill Quadrature nodes φ(0) marker Cusp markers Hide diagnostic overlays

Boundary

Custom colour

Width px

Colours

Background Grid Axes

Markers & labels

Node shape size

Labels font size

Export

Resolution or width

Background

Family sweep

Sweep one parameter and overlay the resulting QD boundaries.

Parameter

from to steps

Show non-univalent members (dashed)

Direct problem

Schwarz reflection dynamics

Geometric properties

Boundary singularities

Geometry & accuracy

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