QMLMathPlot

An embeddable function-plotting component for Qt Quick apps. The curve is drawn per pixel by a fragment shader (implicit signed-distance model): the per-frame cost is ∝ the pixel count and independent of the function's frequency, so oscillating functions (e.g. sin(1/x)) cannot drag the frame rate down through polyline aliasing, and no CPU-side sampling is needed.

The state is a camera on an infinite canvas: a Plot holds a Camera (where you look) and a list of Curves (what is drawn), and the QML component PlotView binds to it. Several panels are several Plots in a Qt layout — there is no figure, no axes box and no page.

Quick start

One line in a QtWidgets layout (least effort):

from PySide6.QtWidgets import QApplication, QVBoxLayout, QWidget
from qmlmathplot import MathPlotWidget

app = QApplication([])
window = QWidget()
box = QVBoxLayout(window)
plot = MathPlotWidget("sin(1/x)")        # drag to pan / wheel zooms anchored at the cursor
box.addWidget(plot)
window.show()
app.exec()

An un-themed plot looks like matplotlib (white background, tab10 colours, grey grid, black axes through the origin with the tick labels riding them) — the default style sheet is applied at construction, so plot.theme = "default" is the reset.

add_curve() (and Curve.expression) accept the expression either as a string or as a sympy expression: plot.add_curve(sp.sin(1 / sp.Symbol("x"))) is the same curve as plot.add_curve("sin(1/x)"). The stored and reported form is always the canonical string — QML, a project file and the text field want one — while the parsed expression is cached and exposed as Curve.expr, which is what re-baking and the matplotlib hand-off use. The object form exists so a curve can be composed without string interpolation:

curve = plot.add_curve("sin(x)")
plot.add_curve(sp.diff(sp.sympify(curve.expression, locals={"x": sp.Symbol("x")})), label="f'")

MathPlotWidget keeps one curve's worth of convenience (expression, error, line_width, curve_color, background_color, aspect, view_bounds(), reset_view(), zoom(), pan_pixels()); everything else lives on the model behind it:

plot.plot.curves.add_curve("tan(x)", color="#ff8866", label="tan")   # a second curve
plot.plot.camera.aspect = 1.0          # square units ("auto" by default)
plot.plot.grid = False
plot.plot.theme = "ggplot"

Standalone window / command line:

uv run qmlmathplot "sin(1/x)"                    # equivalent to python examples/minimal.py …
uv run qmlmathplot --backend d3d11 "sin(1/x)"    # force a specific backend
python examples/explorer_qtwidgets.py            # function input box + sibling widgets

The examples live in examples/: minimal.py is the smallest window; explorer_qtwidgets.py deliberately puts the plot widget into a "grabby" environment (input box, QScrollArea, QSplitter, overlay) to verify the widgets do not fight each other; explorer_qtquick.py (plus explorer_qtquick.qml) is the pure Qt Quick demo.

Embedding into an app

QtWidgets: MathPlotWidget is an ordinary QWidget (internally it bridges the QML component into the widgets world with a QQuickWidget and renders into its own FBO, so it can be stacked on top of / coexist with sibling widgets). Properties and methods: expression / error / line_width / curve_color / background_color / aspect, view_bounds() / reset_view() / zoom(delta, u, v) / pan_pixels(dx, dy), the signals expressionChanged / errorChanged, and the model itself as .plot.

Pure QML (Qt Quick): create the model in Python and inject it (MVVM, see below).

from PySide6.QtCore import QUrl
from PySide6.QtGui import QGuiApplication
from PySide6.QtQuick import QQuickView
from qmlmathplot import Plot, qml_component_path, register_qml_types

app = QGuiApplication([])
register_qml_types()                       # registers Plot, Camera, Curve and PlotView

plot = Plot()                              # the model (owned by the app)
plot.add_curve("sin(1/x)")
view = QQuickView()
view.setSource(QUrl.fromLocalFile(qml_component_path()))
view.rootObject().setProperty("plot", plot)
view.show()
app.exec()

import QmlMathPlot 1.0

PlotView {                       // inject a Plot, or let the component create its own
    anchors.fill: parent
    plot: myPlot

    // Arbitrary QML in world coordinates: the transform is the view's own
    Rectangle {
        x: mapToScreen(0, 0).x - 4
        y: mapToScreen(0, 0).y - 4
        width: 8; height: 8; radius: 4; color: "#ffcc00"
    }
}

For a standalone view with no injected model, the curve is explicit:

PlotView { defaultExpression: "sin(x)"; anchors.fill: parent }

There is no hidden default curve: the component creates its own empty Plot when none is injected, and defaultExpression is the only way it adds one.

PlotView exposes mapToScreen(x, y) / mapFromScreen(point) plus the forwarders camera and curves; input handling (pan by left-drag, wheel zoom anchored at the cursor) is built in and switchable through camera.panEnabled / camera.zoomEnabled.

The model:

Object	Members
Plot	camera, curves, xlim/ylim/aspect (forwarded), add_curve() / add_implicit() / remove_curve(), available_themes, and the style properties: background, grid, grid_color, grid_width, grid_alpha, grid_style, axis_color, axis_width, axes_position, color_cycle, line_width, text_color, tick_color, tick_length, font_size, tick_font_size, title, xlabel, ylabel, title_color, title_font_size, title_bold, ticks_visible, theme
Camera	centre, zoom (world units per logical pixel), aspect, xlim/ylim (derived), ticks_x/ticks_y, zoomStep / panEnabled / zoomEnabled, animationDuration, input (a ViewInput), slots set_viewport(w, h) / zoom_by(delta, u, v, w, h) / zoom_at(factor, u, v, w, h) / pan_pixels(dx, dy) / animate_zoom(...) / animate_pan(dx, dy, ms) / stop_animation() / reset()
ViewInput	keyboardEnabled, keyPanStep, panLeftShortcut / panRightShortcut / panUpShortcut / panDownShortcut / zoomInShortcut / zoomOutShortcut, matches(key, modifiers, shortcut)
Curve	expression (string or sympy expression), expr (the parsed expression, read-only), implicit, color, lineWidth, visible, label, error (read-only), the baked vertexShader / fragmentShader
CurveListModel	the curves as a model (curve, expression, color, lineWidth, visible, vertexShader, fragmentShader roles) so QML puts one ShaderEffect behind each

Every setter is a Qt Property with a notify signal, so the same calls work from QML, from a Qt Designer slot or from a QTimer:

QTimer.singleShot(1000, lambda: setattr(curve, "expression", "sin(3/x)"))

Design notes

The design — the camera model, the Qt property/signal contract, the object model and the build order — is specified in docs/api-design.md. In short: the Matplotlib vocabulary is kept where it is vocabulary (limits, grid, title, ticks), while the mechanism is refused — no Figure/Axes/subplots in the core (one view per widget, no bounded page), no redraw loop (the scene is live), no toolbar and no legend (host chrome), and an export is a canvas region at a pixel size rather than a re-rendered page (§9, implemented — see Export below). A thin compatibility facade (Figure, Axes, FigureCanvasQt, qmlmathplot.pyplot) provides the familiar names for porting scripts; it maps onto the same model and refuses the mechanisms that do not fit (see Matplotlib-style facade below).

Matplotlib-style facade

For porting scripts and for code that expects Matplotlib's vocabulary, the package ships a thin compatibility layer. It maps onto the same Plot/Camera model and refuses the mechanisms that do not fit an infinite, live canvas.

from qmlmathplot import Figure, FigureCanvasQt

fig = Figure(figsize=(6.4, 4.8))       # stored for the window size only; no page is implied
ax = fig.gca()                         # exactly one axes per figure
ax.plot("sin(1/x)", label="sin(1/x)")
ax.set_xlim(-1, 1)
ax.set_ylim(-2, 2)
ax.set_title("sin(1/x)")
ax.set_xlabel("x")
ax.set_ylabel("y")
ax.grid(True)
ax.set_aspect("equal")

canvas = FigureCanvasQt(fig)           # a QWidget, like MathPlotWidget
layout.addWidget(canvas)
canvas.draw()                          # no-op: the Qt scene is already live

The same model through a pyplot-shaped module (global current figure, for scripts and notebooks; applications should keep the object API):

import qmlmathplot.pyplot as plt

plt.figure()
plt.plot("sin(1/x)")
plt.xlim(-1, 1)
plt.title("sin(1/x)")
plt.xlabel("x")
plt.grid(True)
plt.savefig("out.png")                 # Qt raster export
plt.show()                             # blocks in a script; use block=False in a host

What maps: one figure/one axes (subplots() and subplot(111)), plot(expression), xlim/ylim, title/xlabel/ylabel, grid, aspect, savefig/to_image, themes. plt.gcf()/gca()/sca()/close()/show() follow Matplotlib's current-figure shape.

What is deliberately different:

• One view per figure. subplots(2, 1)/add_subplot(211) raise NotImplementedError; several panels are several Figure objects in a Qt layout, not a GridSpec.
• Expressions, not x/y arrays. ax.plot("sin(x)") is native; ax.plot(x, y) is refused because the Qt renderer evaluates an expression per pixel. For sampled data use Plot.to_matplotlib() (the hand-off), or add a data-series artist.
• draw() / draw_idle() are no-ops. The scene repaints from model signals; there is no deferred renderer or blitting.
• savefig is a region at a pixel size, not a page at a dpi. width/height are device pixels and dpi is a multiplier; backend="matplotlib" is the vector route.
• No legends, annotations, log axes or manual ticks yet. legend(), annotate(), set_xscale() and set_xticks() raise with the reason; chrome belongs to the host.
• Themes are per figure, not rcParams; plt has no global style state beyond the current figure.
• mpl_connect/blit are absent. Qt signals are the event system.

Implicit curves

An expression whose free symbols are x and y is drawn as the level set F(x, y) = 0. The artist follows the expression, whoever sets it: add_curve routes it by itself, assigning to curve.expression re-routes the same curve (so a text field can turn sin(x) into a circle), and add_implicit asks for the level-set artist explicitly — also for a function of x alone, which is a pair of vertical lines. Only a third free symbol is refused: substitute parameters first.

plot.add_curve("x**2 + y**2 - 1")        # a circle: {x, y} routes to the level-set artist
plot.add_implicit("x*y - 1")             # the same artist, asked for by name
plot.curves.at(0).expression = "x**2 + y**2 - 4"   # re-routes the curve that is already there
plot.add_curve("a*sin(x)")               # ValueError: substitute `a` first

The renderer is the same idea as the 1-D stroke, one dimension up: one evaluation of F and its two partials per pixel give a candidate distance to the level set in pixels, d = |F| / sqrt((Fx·dx)² + (Fy·dy)²), and the coverage is clamp(lineWidth/2 + 0.5 - d, 0, 1) — analytic antialiasing and a stroke width that does not depend on the angle, with no sampling grid (the unit circle measures the same 3 device pixels at 0°, 45°, 90° and 135° for lineWidth=2 at DPR 1.5).

That quotient is a pseudo-zero at a pole and at a diverging domain boundary, so it is certified by projecting onto the level set: two damped Newton steps from the pixel (or, for a pixel outside the domain, from the nearest in-domain sample), accepting only a converged residual (|F| <= 0.25·R·|grad F|) and using the distance to the projected point. The domain therefore clips the curve, not the stroke: the half of the band lying outside is painted, so x-log(y-1) does not thin out as it approaches y = 1 and y-log(x) reaches x = 0 in a deep view. A POLE(x,y) factor (e.g. cos(y) for tan(y)) rejects a normal probe that crosses a singularity, so x-tan(y) does not paint its asymptotes; far-out garbage lines are still refused (the stroke is allowed one radius outside the domain, no more).

The pixel is also a marching-squares cell, borrowed from GeoGebra's ImplicitCurveMarchingRect/AdaptiveQuadTree: the four corner values are evaluated when every corner is in the domain and no pole factor changes sign between them; the edge crossings are interpolated and the cell's contour segment is stroked (the saddle case is paired using the cell centre's sign). The first-order/Newton path still supplies the sub-pixel width; the cell term adds the coverage of a thin or tangent branch whose centre projection fails. A singular cell (|Fx| + |Fy| < 1e-3 in world units, plus |F| < 0.05) gets a small disc, so an isolated point such as x**2 + y**2 = 0 is not left to the projection alone.

Still not ported: GGB's Bernstein subdivision (polynomial-only), its adaptive quad-tree refinement and the ContourAssembler/ContourLinker (they exist to produce a selectable vector path, not to rasterise). A closed loop that fits entirely inside one pixel with no corner sign change is not seen by the four-corner test; the projection is the only fallback there. The matplotlib hand-off skips implicit curves (a level set has no polyline to sample), while the Qt export renders them like anything else.

The camera (centre, zoom, aspect)

Camera stores three numbers that do not depend on the widget: centre (world coordinates in the middle of the view), zoom (world units per logical pixel on x) and aspect (the ratio of the y scale to the x scale). xlim / ylim are derived from them and the last reported size, so they always say what is on screen; assigning them moves the camera.

• aspect = "auto" (the default) keeps the two scales independent: the widget's shape decides how much canvas is visible, and neither axis is tied to the other. A number keeps the ratio fixed (matplotlib's convention: 1.0 = square units, 2.0 = the y-unit twice as tall). Setting it only ever shows more — the camera keeps its centre and expands, never crops.
• Resizing never zooms: the scales are kept and the visible range follows the widget, so dragging a window edge or a splitter shows more or less canvas instead of scaling the curve.
• camera.reset() returns to the home view (the classic 12 x 4 window, resolved for the current size); the aspect stays as configured.
• The home is a view, so the first size report turns it into scales for that widget: opening a narrow window shows the same window of canvas rather than a sliver of it.

plot = MathPlotWidget("sin(1/x)", aspect=1.0)     # square units
plot.aspect = "auto"                              # back to following the widget's shape
plot.plot.camera.zoom_by(120, 0.5, 0.5, w, h)     # one wheel notch, anchored at the centre

Keyboard, touch and animation

Clicking the plot gives it focus; while it has focus the arrow keys pan (10% of the viewport per press by default), +/= zooms in and -/_ zooms out. Every command is a property on the camera's input object, so a host can rebind it — the value is a QKeySequence string, and | separates alternatives (empty disables the command). input.keyPanStep, input.keyboardEnabled, panEnabled and zoomEnabled shape the behaviour the same way.

camera = plot.plot.camera
input = camera.input
input.panLeftShortcut = "A"           # rebind one arrow
input.panRightShortcut = "D"
input.zoomInShortcut = "Ctrl+="
input.zoomOutShortcut = "Ctrl+-"
input.keyPanStep = 0.2                # 20% of the viewport per arrow press
camera.animationDuration = 220        # ms; 0 disables easing

Wheel and keyboard moves are eased (scale interpolates geometrically, so a zoom feels like a constant rate); repeated wheel notches accumulate onto the in-flight target instead of restarting from the last frame. Mouse drags and touch gestures are deliberately direct — the pointer or the fingers are the animation.

Touch is built in: one finger pans, two fingers pinch to magnify and pan together, keeping the world point under the finger centroid. They need no focus and work in both embedding modes; like the mouse, they cancel an in-flight eased move as soon as they start.

Themes

Plot.theme applies a style sheet to the style properties above; the default is matplotlib's own default style, so an untouched plot already looks like matplotlib and plot.theme = "default" is the reset:

plot.theme = "ggplot"                # any of the bundled sheets
plot.theme = "default"               # back to matplotlib's look
plot.theme = "my-style"              # themes/my-style.mplstyle next to the package
plot.theme = "path/to/any.mplstyle"  # any file in the same format
plot.available_themes                 # the names, for a host's picker (both demos use one)

The 26 bundled sheets are Matplotlib's own styles, vendored verbatim into qmlmathplot/themes/*.mplstyle and converted at load time: Matplotlib's rcParams keys are mapped onto this library's property names, and keys describing things this library does not have (spines, figure size, dpi, legend, marker styles, …) are ignored. themes.names() lists them ("default" plus the 26 sheets). An unknown name raises KeyError and leaves the current theme alone.

Axes, grid and ticks

• The grid is drawn in QML (a Canvas), one line per tick position, in grid_color / grid_width / grid_alpha / grid_style (-, --, :, -.).
• The axes are the lines through world (0,0) — the x-axis at y = 0, the y-axis at x = 0 — drawn in axis_color / axis_width, and the tick marks and labels ride those axes instead of hugging the item's edges. When 0 is off screen the axis sticks to the nearest edge, so its labels are never lost (and a label never leaves the item); the grid stays where it is. plot.axes_position = "edge" restores the edge-pinned behaviour.
• camera.viewport reports the last size the view sent, so a host can size an export from it.
• The tick values are nice numbers computed in Python (Camera.tick_values() / nice_ticks(), steps of 1/2/5 x 10^n) and pushed through ticksChanged; QML only positions and formats them, so a frame never recomputes ticks.
• plot.ticks_visible = False hides the whole furniture (axes, marks and labels).

Export

savefig() writes a file, to_image() returns a QImage; both take the same keywords:

plot.savefig("out.png", xlim=(-1, 1), ylim=(-1, 1), width=1200, height=400)
plot.savefig("hi.png", dpi=2.0)                        # the live view, twice the pixels
image = plot.to_image(width=800, height=600)           # -> QImage
plot.to_image()                                        # the live view, its own size

• xlim / ylim default to the camera: the live centre and scale, extended to the export size. The region is a parameter, so an export can cover more canvas than the widget shows.

• width / height are device pixels and default to the live view's device size; dpi multiplies both (dpi=2.0 doubles the pixels).

• adjustable reconciles a requested range with a requested size when their aspects disagree — Matplotlib's parameter, and its own default:

  adjustable	What happens
  "box" (default)	one scale, taken from the live scale ratio, chosen so the requested range fits: the range is centred and the leftover pixels stay background — nothing is distorted
  "datalim"	the camera's own scales, with the limits expanding (never cropped) to fill the size
  "stretch"	the requested range maps onto the requested size exactly — independent scales, so a mismatched aspect distorts (the "report figure" mode)

  With no explicit range all three agree, because the range derived from the camera already has the export size's aspect.

• transparent=True clears to alpha 0.

• Everything else follows the live styling (curves, grid, axes, theme), so an export cannot drift from what the user sees. Rendering goes through an offscreen QQuickWidget, so the export needs a QApplication (a clear error is raised instead of a crash if there is none).

Vector output, matplotlib and sympy

to_matplotlib() hands the plot to matplotlib (one Axes) and savefig(..., backend="matplotlib") renders through it, so .svg/.pdf work; to_sympy() builds the same figure through sympy.plotting, which is what you want when the figure should live in sympy's world:

figure = plot.to_matplotlib(xlim=(-1, 1), ylim=(-1, 1))    # -> matplotlib.figure.Figure
plot.savefig("out.svg", backend="matplotlib")              # vector, matplotlib's own savefig
handle = plot.to_sympy(xlim=(-1, 1))
handle.save("out.png")

matplotlib is an optional extra (pip install "qmlmathplot[matplotlib]"); nothing in the package imports it until one of these is called, and the error names the extra. sympy itself is a hard dependency, so to_sympy() needs matplotlib only because its plotting backend does. Neither hand-off needs a Qt application: a Plot works in a plain script, which is what makes it usable next to a notebook or a matplotlib-only tool.

This is a second renderer, and a sampled one. matplotlib draws polylines, so each curve is evaluated at samples points (default 2000) and an oscillating function such as sin(1/x) aliases — exactly what the Qt view avoids by evaluating the expression per pixel. SymPy does not solve it either, and its adaptive mode does not help: measured on sin(1/x) over [-1, 1], sympy.plotting.plot(adaptive=True) uses 289 points (fewer than a fixed 2000), emits no warning in sympy 1.14, and draws the same block of vertical connectors. The difference is how the two fail: a sampled renderer draws an artefact, the Qt view draws the envelope the data actually supports. (to_sympy() returns the object sympy made, so its own adaptive mode is available there.) Fonts and antialiasing are matplotlib's, not the Qt view's, so the two renderers cannot look identical, and the style mapping is partial: background, grid (on/off and colour/width/alpha/dash), per-curve colour and width, tick and text colours, font sizes, title, axis colour/width, the limits, the aspect with matplotlib's own adjustable, and axes_position="zero" (spines through the origin, no box). What it buys is vector output and matplotlib's ecosystem — nothing else.

RHI backends

--backend {d3d11,d3d12,vulkan,metal,opengl,null}; if unspecified, Qt's default backend is used. The implementation sets QSG_RHI_BACKEND before QGuiApplication (Qt reads it during platform initialisation, and a later setGraphicsApi() has no effect). On this machine (Intel driver 32.0.101.8826) d3d11 and opengl measured pixel-for-pixel identical.

Structure (model / view)

The layers are strict about Qt. model.py (expression → GLSL), themes.py (style-sheet conversion) and mpl.py (the sampled hand-off) import no Qt at all — they are unit-testable without a QApplication. Camera, Curve and Plot are the ViewModel: QObjects whose Propertyes carry notify signals, which is precisely what the QML component binds to — and why a Python-side assignment updates the view with no call in between. view.py, PlotView.qml, export.py, widget.py and canvas.py are the View/embedding layer and do need Qt; the last three import QtWidgets lazily, so a pure-QML host never loads it.

Layer	File	Responsibility
Model	src/qmlmathplot/model.py	expression → GLSL, shader source template (no Qt dependency, unit-testable)
Camera	src/qmlmathplot/camera.py	Camera: centre / zoom / aspect, the derived ranges, pan/zoom/reset, the nice-number tick algorithm
Input	src/qmlmathplot/input.py	ViewInput: keyboard switch, pan step, rebindable shortcuts, key matching
Animation	src/qmlmathplot/animation.py	CameraAnimator: the one eased interpolator behind wheel/keyboard moves
Curves	src/qmlmathplot/curve.py	Curve (expression → baked .qsb, style, error; later edits bake on the thread pool) and CurveListModel
Plot	src/qmlmathplot/plot.py	Plot: the camera, the curves and the styling the view binds to
Facade	src/qmlmathplot/figure.py	Figure / Axes: Matplotlib-shaped names over one Plot (no Qt beyond the model)
Facade host	src/qmlmathplot/canvas.py	FigureCanvasQt: a MathPlotWidget attached to a Figure (QtWidgets, lazy)
Facade scripting	src/qmlmathplot/pyplot.py	qmlmathplot.pyplot: current figure, plot/xlim/show/savefig for scripts
View	src/qmlmathplot/qml/PlotView.qml (+ PlotCurves.qml, PlotFurniture.qml, PlotInput.qml, view.py)	composition: model injection, camera mapping and keyboard focus; curves, furniture and pointer input live in the three child components (one ShaderEffect per curve)
Themes	src/qmlmathplot/themes.py + themes/	the vendored Matplotlib style sheets, resolved onto the Plot's properties
Export	src/qmlmathplot/export.py	Plot.savefig() / to_image(): the canvas region rendered offscreen at a pixel size (QtWidgets is imported lazily)
Hand-off	src/qmlmathplot/mpl.py	Plot.to_matplotlib() / to_sympy() / savefig(backend="matplotlib"): a sampled second renderer for vector output (matplotlib is an optional extra, imported lazily)
Bake	src/qmlmathplot/qsb.py	GLSL → .qsb (PySide6 ships qsb.exe), cached by source hash
Widget	src/qmlmathplot/widget.py	MathPlotWidget: QQuickWidget bridge, drops straight into QtWidgets layouts
Entry	src/qmlmathplot/app.py	command line / standalone window (the qmlmathplot script, examples/minimal.py)
Examples	examples/	minimal.py (minimal), explorer_qtwidgets.py (input box + widget coexistence), explorer_qtquick.py (Qt Quick)

Algorithm (fixed-x sampled polyline + the interval column)

The supported path is GeoGebra's column idea implemented on the GPU: sample the function on a sub-pixel x grid, stroke the polyline, and fill the exact interval enclosure where a column cannot be represented by a line.

1. Sub-pixel polyline stroke. Each pixel evaluates F at 16 samples over ±2 columns (0.25 logical-px spacing). Adjacent valid samples form a polyline, and coverage is R - distance to the nearest segment — Desmos' stroke rule. The sample spacing is sub-pixel, so the vertices themselves are sub-pixel and the segments are diagonal: the edge is smooth at any zoom, not a per-column staircase. ±2 columns (rather than the old ±1) cover the perpendicular foot, which a steep line displaces along x by up to the stroke radius; measured k·x 3.00/3.00/3.00/2.94 device px at screen slopes 2/10/40/200 against a nominal 3.0.
2. Pole/jump segment guard. A segment is skipped when the expression's pole factor changes sign between its endpoints (POLE(prev_x) * POLE(xi) <= 0) — the branches of tan/1/x never join across a pole — and, for a dependent quotient such as x/sqrt(x**2), when the endpoint jump exceeds the join radius. The flat branches are still drawn; measured x/sqrt(x**2) draws its ±1 branches without the old 454-px connector at x = 0.
3. Interval topology — GeoGebra's column rules on the GPU. The expression is evaluated in interval arithmetic over the pixel's column (generated from sympy in interval.py, compiled to the GLSL IVal FI(xlo, xhi)): a conservative y-enclosure with topology connected / two rays around a gap / whole / empty. The shader reproduces the reference plotter's rules:
   • connected, sub-column oscillation (smoothstep(1.5, 3.0, sturns)): GeoGebra's DrawInterval.drawJoined path — the vertical join at x.low and the diagonal across the column, choosing drawUp/drawDown from the high edge; an independent column is a vertical at x.high. These are strokes, not a filled rectangle.
   • divergence (log's boundary): IntervalPath.drawNormalInfinity → leftToBottom/leftToTop, only when the finite bound is on screen: one segment from (x.low, finite bound) to (x.middle, viewport edge). If the interval covers the viewport but the finite bound is above/below it (the deep log view), the column is effectively whole and gets drawWhole (a vertical at x.low) — an extension needed because the Java branch draws nothing there although the visible curve lies inside the column.
   • inverted (tan/1/x poles): DrawInvertedInterval — the top segment at x.high from the viewport top to the gap's high, the bottom at x.low from the gap's low to the viewport bottom; no filled ray.
   • whole: DrawInterval.drawWhole — one vertical segment at x.low. Deliberate deviations, because this is a fragment shader and not the Java tuple list: the 16-sample polyline supersamples GGB's one interval per column for resolved stretches; a dependent quotient's whole column is still suppressed (the fixed-x polyline draws the real branches, avoiding the x/sqrt(x**2) connector); and a joined inverted column uses those sample branches instead of replaying JoinLines.inverted's neighbour decisions.
4. Domain. DOM/DOMS (from sympy) guard every F evaluation, so log(negative) is never evaluated; a pixel just outside the domain still gets the band, which is centred on the boundary.

An ordinary pixel costs 16 F evaluations plus two interval evaluations; there is no march, no tangent-foot projection and no column probe on the supported path. Expressions with an operation the interval backend does not implement (floor, sign, …) compile the legacy sampled band + ray + march path (#define USE_INTERVAL 0); it is the only user of the arc-length march now.

Tests

uv run pytest                      # everything
uv run pytest -m "not gui"         # skip the render smoke tests that open a window
QSG_RHI_BACKEND=opengl uv run pytest -m gui
uv run ruff check src tests examples   # bug-level lint (undefined names, unused imports)

• test_model.py: GLSL generation (small integer powers as repeated multiplication, avoiding pow(x, y)'s undefined behaviour for x<0), shader injection (including the camera uniform and the mapping), and the camera math (the home view at the first size, resize keeping the scale, an anchored zoom that is exactly reversible, the aspect expanding but never cropping, ticks).
• test_curve.py: a failed expression keeps the last working shaders, the style setters are idempotent, and the list model reports the right roles — plus the Expr/string round-trip accounting (a string is parsed once, an expression never).
• test_implicit.py: the routing by free symbols (and the ValueError for a third symbol), the level-set shader text (same uniform block as the 1-D one, no sampling loop), the rendered geometry of a unit circle (crossings a quarter of the height from each edge, a blank middle, radius 1 on both axes, and a stroke of the same width at 0°/45°/90°/135°), and the two domain/pole guards: x-tan(y) has no connector at its poles or far pixels in a wide view, x-log(y-1) keeps its stroke on both sides of y = 1, and y-log(x) reaches the axis in a deep view.
• test_plot.py: the plot's properties convert, notify exactly once and are wired (a PySide property with an unresolved notify name silently loses it), the defaults are matplotlib's default style, the limits are forwarded to the camera, and a theme applies (and re-applies as a reset) while an unknown name is rejected.
• test_themes.py: the vendored style sheets resolve to this library's properties.
• test_qsb_cache.py: the cache location/info/clear API and the age + LRU prune; unrelated files in the temp directory are not touched.
• test_shader_bake.py: the stage of a baked .qsb must match its purpose, and the four backend targets GLSL/HLSL/MSL/SPIR-V must all be covered.
• test_render_smoke.py: open a window, render, read pixels; verify the thin-line shape, sin(1/x) filling its ±1 envelope, no |y|>1 artefacts outside the central columns, poles not connected, nothing drawn outside the domain, and the QML furniture (grid lines exactly at the tick positions, the axes crossing at the origin with their labels on them, the axis pinning to the edge once the origin is panned away, the title).
• test_widget.py: MathPlotWidget coexisting with neighbouring widgets — the plot appearing inside a layout, an expression change / an invalid expression keeping the previous graph, curves added/hidden/removed through the model, the wheel over a scroll area being consumed by the plot (zoom instead of scrolling the parent), the fixed aspect being undistorted by the widget's shape, and the plot staying out of the Tab focus chain. conftest.py provides a session-scoped QApplication (QtWidgets needs one, and a process may only have one).
• test_qtquick_coexistence.py: the same drag conflict in Qt Quick (a Flickable must not steal the pan).
• test_handoff.py: the matplotlib/sympy hand-off — the figure reflects the limits, aspect, adjustable, grid, title, spines at zero and per-curve style; the sampling is uniform and NaN outside the domain (so log(x) breaks); savefig(backend="matplotlib") writes real .svg, .pdf and .png; to_sympy().save() writes a file and keeps per-curve colours; importing qmlmathplot does not import matplotlib, and hiding matplotlib produces an error naming the extra.
• test_async_bake.py: a later expression assignment bakes on the thread pool, a stale bake is discarded, and a failed bake keeps the drawn curve.
• test_compat.py: the Matplotlib-shaped facade — expressions (and the refusal of x/y arrays), limits/title/labels/grid/aspect, the single-axes rule, pyplot current-figure semantics, FigureCanvasQt rendering and export, and that importing qmlmathplot does not load QtWidgets.
• test_export.py: to_image()/savefig() — the image is exactly the requested device size (dpi multiplies it), the no-argument form reproduces the live centre and scale, the three adjustable modes reconcile range and size as documented (a world 45-degree line stays at 45 in "box", with background margins; "stretch" fills the frame and distorts), the live camera and background survive an export, and transparent clears to alpha 0.

Pitfalls near the sin(1/x) singularity (fixed — do not repeat)

All of them (macro arguments, GPU sin/cos, the fixed-x-grid sampling era's stroke saturation, per-column phase noise, the jump-discontinuity connector) are recorded in AGENTS.md as symptom → cause → fix with the measurements that settled them. After the fixes (900×600 logical, default view): the singularity column of sin(1/x) is lit over a ratio of 0.50 (= half a column filling the ±1 envelope) with flat edges, and inside the central 41 columns no solid pixel falls outside |y|>1 (7961 before the envelope band existed).

• Macro arguments must be parenthesized.
• GPU sin/cos are unreliable for large arguments.
• Tangent-based distance saturates on steep columns.
• Per-column sampling is phase noise for oscillating functions.
• The join invariant replaces pole detection: tangent-foot seeding and arc-length marching, so a connector of any origin (poles, jumps) cannot be formed (AGENTS.md item 44).

Domain, poles and stroke (Desmos-style)

Desmos's engineering blog spells it out: sample the function into points, join them into segments, stroke along the segments; "break where neighbouring points suggest a jump". The same approach, with the sampling made uniform in arc length (the march above):

• Stroke = distance to the joined polyline segments (not to the tangent), plus the stroked column segments for what the march cannot reach: a pixel also measures the vertical y-range of the two neighbouring columns and strokes it with the same ramp. That is GeoGebra's per-column segment with a vector line width, on the GPU; a steep wall (1/x/tan at large |y|, log's descent) keeps its nominal width even though its x span is narrower than a pixel and its march seed is a whole view up the arc.
• Break on jumps — by construction, not by detection. The join invariant (a segment only joins samples within 2.5 px) refuses a connector of any origin: the poles of 1/x/tan(x) and plain jump discontinuities alike (x/sqrt(x**2) had no pole denominator for the old guards to find; measured: a 454-px column at x=0). The interval band does not reintroduce one: a dependent quotient (x/sqrt(x**2)) is left to the march, and a sign-changing pole (tan, 1/x) is a gap or a whole column, not a chord.
• Domain (derived by sympy, injected as DOM for the scalar march): denominator ≠ 0, log argument > 0, sqrt/even-root ≥ 0 (a negative power also excludes 0), asin/acos ∈ [−1,1], x**(1/3) all reals. The interval evaluator is authoritative for the band: it returns empty outside the domain, [-∞, log h] at a log boundary, and the half-open values at a root endpoint. A pixel just outside the boundary is painted from the stroke footprint (x ± R) and the neighbouring column segments — not by a symbol-derived ray. The legacy ray remains only in the fallback shader (USE_INTERVAL 0).
• Undersampled columns use the interval enclosure of the pixel's column: no ±8-column window, no violation count, no chord-stroke. The midpoint probe survives only as an interval-dependency check for a wide finite column (x * 1/[a,b] overestimation); the band and the neighbour column segments are always the exact interval ranges.

Measured: sin(1/x)'s transition overfill went from 1.66 to 0.0099 world units (threshold-20 pixels; the remaining edge is the stroke/AA radius); x**(1/3) draws both branches; the log(x) descent measures 3.00 device px at every sampled row without the ray; 1/x/tan(x) in a y∈[50,60] view are 3 full columns (2 before the neighbour segments); x/sqrt(x**2) still draws no connector (worst column 4 px); k·x holds one width at every slope (the slope test at 2/10/40/200); tan(x) at an enormous y span fills the screen (the interval is whole, matching GeoGebra); sin(x)/x^2/exp(-x²)sin(10x) are unaffected.

Performance (measured)

Startup to the first frame is about 230 ms, attributed as follows (Windows / D3D11 / 900×600):

Stage	Time	Notes
Qt graphics device init (RHI/D3D11)	~160 ms	exactly what the first window's show→expose costs; a bare window (no ShaderEffect) is the same, so it is not the shader's fault
Window creation + QML load + first frame	~80 ms	from the second window on (device already created) this is the order of magnitude
qsb bake	0 / ~126 ms	cached by source hash: 0.15 ms on a hit; later expression assignments bake on Qt's thread pool, so only the first curve construction blocks (tests force the synchronous path)
Expression analysis (sympy)	0.6~3 ms	unbounded_edges only analyses polynomial boundaries (tan(x)'s cos(x)=0 takes 94 ms via solve+limit, while tan diverges fast and sampling is naturally deep enough, so that ray is not needed)

Per-frame grab cycles (900×600, grabToImage — frame + readback, so an upper bound on the GPU cost), steady state on the same window: sin(1/x) default 13.6 ms, tan(x) default 12.4 ms, sin(1/x) at zoom 90 (worst case) 12.5 ms. The previous march/interval renderer measured 27/34/33 ms on the same machine: 16 samples are cheaper than the march's seed + tangent projection + walk + probe, and the march is now compiled only for the legacy path. The 1-D fragment source is about 21 KB.

Per-pixel F evaluations. The supported path is fixed: 16 samples for the polyline (plus two interval evaluations for the column and the stroke footprint); there is no march budget and no band-window evaluation. The old fixed-grid/march table is superseded and no longer describes the shipped path.

Shader cache

qsb binaries are cached in the platform temp directory:

import qmlmathplot

qmlmathplot.cache_dir()    # e.g. C:\Users\me\AppData\Local\Temp\qmlmathplot-qsb
qmlmathplot.cache_info()   # {'path': ..., 'files': ..., 'bytes': ...}
qmlmathplot.clear_cache()  # remove this package's entries, returns bytes freed

The location is tempfile.gettempdir()/qmlmathplot-qsb: %TEMP% on Windows, the per-user /var/folders/.../T on macOS, and $TMPDIR or /tmp on Linux. Set QMLMATHPLOT_QSB_CACHE to use another directory.

It is not left to grow without bound. The first bake in a process, and at most once per minute afterwards, runs prune_cache():

• entries older than 30 days are removed;
• if the cache is over 32 MiB (default), the least-recently-used .qsb files are removed until it is at 75% of the cap — so an existing 60 MB cache drops to about 24 MB on the next bake;
• interrupted-bake leftovers (*.qsb.tmp, *.vert, *.frag) older than an hour are removed.

Both knobs are overridable: QMLMATHPLOT_QSB_CACHE_MB (size cap) and qmlmathplot.prune_cache(max_bytes=..., max_age_days=...) for an explicit pass. The cache contains compiled shaders only; the GLSL source and temporary outputs are deleted after each bake. Concurrent processes share the directory safely: writes use unique temp names and an atomic replace, and pruning only removes entries by age/LRU.

On macOS/Linux qsb is looked up in the PySide6 package (Qt/bin, Qt/libexec) and on PATH; if the wheel does not ship it, install Qt Shader Tools (qt6-shadertools on most distributions) or add qsb to PATH. Only the Windows wheel has been confirmed to ship it.

Pitfalls already hit (do not repeat)

AGENTS.md at the repo root carries all the records (symptom → cause → fix, with the measurement that settled each one). The Qt/packaging ones that shaped the code:

• qsb decides the stage from the source file suffix (see qsb.bake() and test_shader_bake.py).
• Read pixels with the async QQuickItem.grabToImage(), never QQuickWindow.grabWindow().
• Expressions are macros rather than GLSL user functions: a pixel evaluates the same x 9 times, and macros are preprocessor expansion with no call semantics.
• PySide ties a property's notify signal to the Signal object; a name string silently leaves the property without one (and QML bindings then never refresh).
• Screen-space furniture is QML: background Rectangle, grid Canvas, axes/ticks Rectangles, labels/title Text; curves are ShaderEffect passes. Snap its lines to device pixels with an integral device width (_deviceWidth/_snapLine), not logical half-pixels: the 1.07-logical axis is 2 device px at DPR 1.5 and otherwise alternated between one and two rows as it drifted with the zoom.
• QQuickWidget.setProperty(name, …) sets it on the widget, not on the QML root (the widget has no such property, so it silently does nothing): inject the model with widget.rootObject().setProperty("plot", plot), as MathPlotWidget does.

Releasing

1. Bump version in pyproject.toml and commit.
2. Tag and push: git tag v0.2.0 && git push origin v0.2.0.
3. .github/workflows/publish.yml then runs the non-GUI test suite, builds the sdist and wheel with uv build, and uploads them to PyPI. The tag must match pyproject.toml or the run fails before anything is published.

Authentication is PyPI's trusted publishing (OIDC) — no API token, no repository secret. Configure it once at https://pypi.org/manage/account/publishing/ (and the same on test.pypi.org) with the owner, the repository name, the workflow file name publish.yml and the environment pypi. A project that does not exist on PyPI yet uses the "pending publisher" form there and is created by the first successful run. Running the workflow manually (Actions → Publish to PyPI → Run workflow) publishes to TestPyPI by default, so a release can be rehearsed:

pip install -i https://test.pypi.org/simple/ qmlmathplot

If the publish step fails with invalid-publisher: valid token, but no corresponding publisher, the trusted-publisher entry is missing or does not match. The action prints the claims it presented (repository, workflow_ref, environment) — the entry must match them exactly, including the environment name, and it must exist on the index the run targets: a manual run goes to test.pypi.org, a tag push to pypi.org, so both need an entry.

The workflow runs on Windows: that is the platform this library is verified on (the shader baker qsb ships inside the PySide6 wheel and is only confirmed to be present there, see qsb.py), and the GUI tests need a GPU scene graph no runner has. The artifact is py3-none-any, so the platform does not change what is published. The GUI tests stay a developer-machine thing: uv run pytest (everything) versus uv run pytest -m "not gui" (what CI runs).