The Sigma Tool

A pair of compatible masters establishes the drawings at the ends of an interpolation interval. It does not guarantee that every drawing between them will have the intended proportions. A counter may close too quickly, a join may gain too much weight, or a curved stroke may become flat before reaching the heavier master. Sigma lets you adjust how corresponding points travel between those endpoint drawings.

Use it when the masters themselves are satisfactory and the correction belongs to the transition. You shape the progression directly, without adding another independently drawn master.

What Sigma Changes

In ordinary interpolation between two masters on one axis, each corresponding point follows a straight line from its position in the first master to its position in the second. At the midpoint of the interval, it is halfway between those positions. The glyph’s contour may be curved, but the movement of each of its nodes and handles is still linear.

A straight route is not always the best way to preserve a letter’s character. As a round letter grows heavier, for example, you may want its counter to retain more space through the middle weights before reaching the heavy drawing. Underware’s illustrated study, Higher Order Interpolation for Variable Fonts (2018), explores the idea behind this freedom: Bézier curves can shape the movement between designs as well as the outlines themselves. You can give the transition the same deliberate attention as the drawings at its ends.

Sigma replaces selected straight movements with controlled curves. A Sigma link identifies a corresponding point, an axis, and a pair of masters. Its trajectory is the path that point follows as the axis value changes. A link can be Linear, Quadratic, with one trajectory control, or Cubic, with two.

The distinction between the two kinds of curve matters:

A contour curve describes the shape of the glyph at one designspace location. Its nodes and handles define the outline you draw.
An interpolation trajectory describes how one of those nodes or handles moves through successive designs. Sigma edits this trajectory.

A cubic outline can therefore have a quadratic Sigma trajectory, and a quadratic outline can have a cubic Sigma trajectory. Choosing Cubic in the Sigma controls does not convert the glyph’s outlines to cubic curves.

A Point between Two Masters

Suppose a node is at (100, 0) in a master at Weight 400 and at (200, 0) in a master at Weight 900. Ordinary interpolation puts it at (150, 0) at Weight 650. A quadratic trajectory with its control at (150, 40) instead places that halfway node at (150, 20):

Weight	Straight trajectory	Example quadratic trajectory
400	`(100, 0)`	`(100, 0)`
650	`(150, 0)`	`(150, 20)`
900	`(200, 0)`	`(200, 0)`

The numbers are font-unit coordinates. Both master positions remain unchanged; the correction grows within the interval and returns to zero at its ends. The trajectory control is not an intermediate master, and the moving point does not generally pass through it. Its purpose is to influence the curve between the endpoints.

Moving a trajectory control away from the straight route bends the movement. Moving it along that route can change how quickly the point advances toward the other master. Two cubic controls provide separate influence near the two ends. This gives you local control over a shape’s development rather than another global axis setting.

A contour depends on several points together. Moving one node differently can alter the stroke’s thickness or curvature relative to its handles and neighboring nodes. Sigma does not automatically know which counter, join, or optical relationship you intend to preserve. Choose the relevant links together and inspect the resulting outline.

Choose the Appropriate Correction

Axis mapping changes how a user’s axis value reaches a designspace location. It changes the progression of the axis as a whole. Sigma changes the trajectories of selected points within a particular master span; it does not remap that axis for every glyph.

A local corrective master supplies another source drawing that the interpolation must reach. Choose it when you want to draw and maintain a particular intermediate design. Sigma is useful when you want to shape the continuous movement between existing drawings instead. A correction to one link need not impose the same trajectory on other points or other master pairs.

Neither approach replaces correspondence. Sigma still requires compatible masters and does not add, remove, or pair contour points for you. When the design should change structure rather than morph continuously, use Bracket Layers. For sparse and font-less corrective sources, see Beyond the Uniform Blend.

Edit a Point’s Path

Prepare compatible masters and preview the interval to identify the points that need a different progression. Resolve correspondence problems before editing Sigma.
Activate the Sigma tool. Click a link on the canvas to choose its axis and master pair. Different master pairs can have different paths.
Click a node or segment to select its links in the active span. Use Shift-click to add or remove links, or drag around nodes to select them.
Choose Quadratic or Cubic in the Sigma controls. The first non-linear edit activates Sigma for the glyph. You can also double-click or Alt/Option-click a link to cycle its path type.
Drag a trajectory control to shape the progression. Drag a selected link, node, or segment to move the selected controls together. Press Esc to cancel a drag.
Preview several positions, including positions near both masters. Check the whole contour, spacing, and any composites that use the glyph. When an intermediate master joins two spans, inspect the approach from both sides: a good correction in one span does not establish the progression in the other.

The lower row of Sigma controls determines how neighboring handle trajectories follow a node-trajectory edit. None leaves them unchanged; Move moves adjacent handle controls with the node control; Nudge adjusts neighboring handle controls proportionally. These choices act during editing. FontLab stores the resulting paths; the selected following mode is not an additional effect applied during interpolation or export.

To remove a correction, select its links and choose Linear. Returning the last non-linear link to Linear restores ordinary interpolation for the glyph. Save the editable project in a native FontLab format. Output and import considerations are covered in OpenType, Variable, and Web Fonts.