ComplexPlot3D

<ComplexPlot3D value="1/(z^2 + 1)"> -- Wolfram's ComplexPlot3D: |f(z)| as a surface over the complex plane, each face colored by arg f(z) on the same gradient <ComplexPlot> paints (gradient, default phase, the hue wheel; reverse turns it round). A pole is a spike that rises to max-height with every hue winding round it; a zero is a dimple the hues wind round the other way.

value is Epsil; LaTeX is accepted inside a $…$ island. Sampled in a WebGPU compute shader through the same complex lowering the portrait uses, falling back to the CPU -- the base package's complex evaluator, or the engine's own numeric evaluation for a head it has no lowering for -- where WebGPU is missing or the expression doesn't lower. gpu="false" forces the CPU; a number sets the GPU's samples. Drag rotates, ctrl/⌘ + wheel zooms, double-click resets the view.

A grid past CANVAS_THRESHOLD samples a side is painted on a canvas rather than serialised as SVG: the same projection and painter's order, but no DOM node per face, so a GPU-resolution surface still turns under the pointer.

The surface is cut at max-height rather than left to grow without bound near a pole, as Wolfram's is, so a single pole does not flatten everything else to the floor. A face's hue is the circular mean of its corners, so a cell straddling arg = ±π takes the hue between its sides rather than the opposite one.

packages/components/src/notatio-complex-plot-3d.ts· draws ComplexPlot3D

Stories

Poles and zeros

#A rational function

Two poles at ±i, clipped at the ceiling, and the hue running once round each.

source

ComplexPlot3D(1 / (z ^ 2 + 1))

notebook cells, the CLI

#Zeros too

The cube roots of unity are the three dimples; the poles are the two spikes. Reading the colour wheel round a point gives its order, and the direction of the winding says which.

source

ComplexPlot3D((z ^ 3 - 1) / (z ^ 2 + 1), (z, -2 - 2i, 2 + 2i), (Samples, 60))

notebook cells, the CLI

Special functions

#The gamma function

Poles at 0, −1, −2, …, each a spike; to the right the surface climbs as the factorials do.

source

ComplexPlot3D(Gamma(z), (z, -4 - 3i, 4 + 3i), (Samples, 60), (MaxHeight, 6))

notebook cells, the CLI

#The Riemann zeta function

The pole at z = 1, and up the critical line the first non-trivial zeros at ½ + 14.13i, ½ + 21.02i, ½ + 25.01i -- the dimples every hue winds round.

source

ComplexPlot3D(Zeta(z), (z, -2, 3 + 30i), (Samples, 70), (MaxHeight, 3))

notebook cells, the CLI

GPU evaluation

#A dense rational surface

The rational function again at 160 samples a side, so the spikes come to a point and the hue is continuous across each face.

source

ComplexPlot3D((z ^ 3 - 1) / (z ^ 2 + 1), (Gpu, 160), (MaxHeight, 3))

notebook cells, the CLI

#An essential singularity

Exp(1/z) near the origin: the surface climbs to the ceiling on the right and falls to the floor on the left, and between them every hue passes infinitely often.

source

ComplexPlot3D(
               ExponentialE ^ (1 / z),
               (z, -1 - i, 1 + i),
               (Gpu, 200),
               (MaxHeight, 3),
               (Elevation, 30),
             )

notebook cells, the CLI

#The trigamma function

PolyGamma(1, z) through the analytic kernels on the GPU: double poles at 0, −1, −2, …, so the hue winds twice round each spike.

source

ComplexPlot3D(PolyGamma(1, z), (z, -4 - 2i, 2 + 2i), (Gpu, 160), (MaxHeight, 5))

notebook cells, the CLI

As an expression

#Through an expression, corner-to-corner

The head is the component: an evaluated ComplexPlot3D(f, (z, a + b i, c + d i)) draws itself, the iterator's complex corners becoming the domain -- the same picture as the first story, written the other way.

source

ComplexPlot3D(1 / (z ^ 2 + 1), (z, -2 - 2i, 2 + 2i))

notebook cells, the CLI

Color

#The argument on the sinebow gradient

Faces take arg f(z) on the gradient; phase, the hue wheel, by default.

source

ComplexPlot3D(1 / (z ^ 2 + 1), (z, -2 - 2i, 2 + 2i), (ColorFunction, "Sinebow"))

notebook cells, the CLI

Props

Prop	Type	Default	Notes
value	string	""	The complex-valued expression to draw, in Epsil.
var	string	"z"	The complex variable; defaults to z.
domain	string	"-2,2,-2,2"	The rectangle to sample, as re0,re1,im0,im1.
samples	number	40	Samples per side, clamped to 2..400.
gpu	string	""	"false" samples on the CPU even where WebGPU is available. A number sets samples for the GPU.
maxHeight attribute max-height	number	4	Height ceiling for |f|, since a pole goes to infinity.
axes	string	"true"	"false" hides the axes frame and its range labels.
azimuth	number	45	Rotation about the vertical, in degrees. Dragging changes this.
elevation	number	25	Viewing angle above the plane, in degrees. Dragging changes this.
zoom	number	1	View scale; ctrl/cmd + wheel or a pinch changes this. Double-click resets.
label	string	""	Caption drawn above the surface.
gradient	string	"phase"	The gradient the argument is colored with: phase (the hue wheel, the default), sinebow, turbo, ….
reverse	boolean	false	Run the gradient from its last color to its first.

As a web component

Plain custom elements, no framework required: import @enumeratio/frontend for the side effect and write <notatio-complex-plot-3d> in any HTML. Attributes are kebab-case and always strings; the props table above names each one's attribute where it differs from its prop.

Settable only as a JS property, not an attribute -- bindings: bind with :bindings.prop="…" directly on the tag.