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Bézier outline geometry

Exact command model

GlyphOutline preserves moves, lines, quadratic curves, cubic curves, and closes. It does not silently turn a curve into sampled points. The selected control polygons in The exact Bézier outline and selected control polygons of a shaped glyph come from those exact commands.

Quadratic and cubic segments

For \(0 \le t \le 1\), a quadratic segment is

\[ Q(t) = (1-t)^2 P_0 + 2(1-t)t P_1 + t^2 P_2, \]

and a cubic segment is

\[ C(t) = (1-t)^3 P_0 + 3(1-t)^2t P_1 + 3(1-t) t^2 P_2 + t^3 P_3. \]

Both reproduce their endpoints exactly. Their endpoint tangents are parallel to \(P_1-P_0\) and \(P_n-P_{n-1}\).

GlyphOutline(glyph_id, commands, bounds, units_per_em, variation_coordinates=())

Stores exact, renderer-neutral contours in font coordinates; compound glyphs are decomposed during extraction.

MoveTo(point)

Moves to the first point of a new contour without drawing a segment.

LineTo(point)

Draws a straight segment to the specified endpoint.

QuadraticTo(control, point)

Draws a quadratic Bézier segment using one control point.

CubicTo(control1, control2, point)

Draws a cubic Bézier segment using two control points.

Closes the active contour and makes its final edge explicit.

Adaptive flattening

flatten(outline, *, tolerance) → tuple[Polyline, ...]
outlineGlyphOutline

The exact outline to approximate.

tolerancefloat

Largest distance between a control point and its chord, in font units.

Recursively subdivides a curve until every control point lies within the requested tolerance of its chord. The result is a tuple of open or closed Polyline values.

Tolerance and subdivision

Let \(d(P, A B)\) be the distance from control point \(P\) to chord \(A B\). A quadratic is accepted when \(d(P_1,P_0 P_2) \le \epsilon\); a cubic requires both interior controls to satisfy the bound. Otherwise de Casteljau subdivision at \(t=1/2\) creates two smaller curves. The tolerance is in font units, so consumers should convert a layout-space or pixel tolerance before flattening.

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