Elm Bezier

This package is for working with cubic Bezier curves and is specifically made as a dependency for Elm Animator.

A good amount of this code was borrowed from Elm Geometry, though the Bezier.Spring module is largely unique (though it was written with a considerable amount of help from Ian Mackenzie, for which I am very grateful!)

Tests

Run the automated spring regressions from the repository root:

elm-test tests/SpringTest.elm

Use an explicit test path: tests/visual-tests is a separate Elm application, and default recursive test discovery also picks up its playground modules.

Visual playgrounds

From tests/visual-tests, run elm reactor, then open:

• http://localhost:8000/src/BezierSegments.elm — interactive spring controls, sampled positions, fitted segments, and segment time boundaries.
• http://localhost:8000/src/Playground.elm — preset spring comparisons and Bezier splitting examples.

Both use the local library source. The graphs use milliseconds horizontally and position vertically (positive downward). Each page includes a color legend. The numerical reference uses 0.16ms integration steps and is still approximate.

In the interactive playground, vary the end position through positive, zero, and negative values: the trace should still end at the estimated settling-time marker. Try both signs of initial velocity and compare the sampled positions with the numerical reference and fitted curves. These are static plots, not animated previews; samples cover 0–1000ms and large excursions can leave the fixed viewport. The automated tests check velocity and restart continuity.

Spring integration notes

• Time arguments and segment x coordinates are in milliseconds; initial and returned velocities are in position units per second. Converting a segment's slope (position per millisecond) to a spring velocity requires multiplying by 1000.
• segments traces from zero to settlesAt spring, not to a caller-supplied duration. Its endpoint is a sample of the spring, not an exact snap to the target. settlesAt is an estimate based on spring parameters, independent of the initial position and velocity. In particular, its overdamped estimate is known to be inaccurate. new's settleMax should not be treated as a hard timing guarantee.
• Segment fitting is approximate. Peak detection uses 5ms sampling and an existing 15ms offset; fitted curves do not guarantee exact velocity continuity. The number of segments depends on the spring and is not capped at ten.
• at uses an analytical solution for underdamped springs, but numerical stepping for critical and overdamped springs (10ms and 20ms steps respectively). Those branches have step-size-dependent accuracy and stability. stepOver needs a positive step size when advancing time.
• Use finite, positive mass, stiffness, and damping for settling traces. Parameters are not validated; undamped springs do not settle and are unsuitable for segments.

Regression history

In tag 1.0.0, segment generation confuses target position with elapsed time, and at ignores initial velocity. The 2.0.0 rewrite uses initial velocity, but retained the time/position mix-up for critical, overdamped, and near-critical springs. It also introduced an underdamped velocity formula inconsistent with its position formula. The automated suite covers the corrected time bounds, initial state, position derivative, numerical reference motion, and restart continuity.