Sketch a motion.
Find its machine.

The target and coupler trace are resampled in traversal order at 48 equally spaced arc-length points. We center each path and normalize its RMS radius. The score minimizes ordered RMS distance over 48 cyclic shifts, either traversal direction, and a proper planar rotation. A similarity transform returns the complete linkage to the target’s position and scale. Reflections are not allowed. Cyclic starting-point invariance is approximate at this finite resolution; drawing speed and point density do not set the timing.

The score matches path shape and order, not crank timing, drawing speed, or point-cloud proximity. Reversing the score’s traversal correspondence never changes the assembly branch or the physical crank direction. Animation always increases the input angle. Search curves use 96 input-angle samples; displayed diagnostics and trace use 360. A denser 1,440-sample audit score is reported after each search; differences reveal discretization limits.

Native ground pivots are O=(0,0), D=(1,0). The crank places A=a(cos θ,sin θ). B is one fixed-side intersection of circles centered at A and D, with radii b and c. The tracer is P=A+u(B−A)+v·perp(B−A). Thus u and v are fractions of the coupler length. Branch ± is the sign of cross(D−A,B−A).

For positive a, the distance |D−A| ranges from |1−a| to 1+a. Strict circle intersections through an entire revolution require |1−a|>|b−c| and 1+a<b+c. Search adds a 0.02-ground margin. Tangent or partial cycles are rejected during search. The partial-cycle inspector retains missing poses, breaks the trace, and stops playback at an invalid interval. No assembly-branch jumps reconnect a broken cycle.

Full geometric closure does not guarantee collision-free motion or usable forces. The margin concerns circle-intersection tangency, not every kind of toggle or rocker reversal. Primary references: METU vector-loop equations, CMU planar linkages, and Cedarville coupler-curve atlas.

Dragging P inverts the displayed similarity transform and the coupler frame to set u and v. Numeric edits also retain the transform and cyclic/reversed target correspondence; they recompute fixed-alignment error without re-optimizing it. The camera remains fixed until Fit view or an explicit new result. Invalid manual cycles remain visible and unscored. Manual provenance records the search or inspection it was derived from.

A/B snapshots deep-copy their target, geometry, alignment and provenance. Restore replaces the active target and candidate, restores its recorded search limits/seed/budget (or defaults when no search was recorded), and cancels active work; later edits cannot mutate a saved slot. Scores from different targets are not ranked. The comparison uses the same input angle and common sketch scale, not fitted timing. Snapshots last for this page session; export a motion study to retain a candidate beyond it.

The motion-study export is one self-contained HTML file with embedded parameters and geometry. It opens locally without a server or network, starts paused, and computes the actual linkage pose at each angle. The study retains missing poses and its stated geometric limitations.

Every admissible proposal is considered without changing the random sequence or optimizer. We retain at most eight options in fit-error order, accepting each only when its branch differs or its native A/B/P positions are sufficiently separated from every already-kept option of the same branch. Separation is RMS Euclidean distance over eight uniform crank angles, divided by ground length 1; the threshold is 0.18. The best found candidate is always retained. This bounded online archive is a diversity heuristic, not exhaustive or representative coverage.

Candidate IDs show the evaluation at which that geometry was discovered. The completed or cancelled run count is recorded separately. Choose current restores that candidate’s target and search settings; Send to A/B captures it without replacing the current mechanism or angle. Drawing a new target clears the gallery while saved slots remain. A new search starts an empty archive, so an exhausted run cannot present older alternatives as new results.

Ground length is 1. Default search bounds: a∈[0.12,0.75], b∈[0.45,1.9], c∈[0.35,1.7], u∈[−0.3,1.5], v∈[−1,1], branch ±. The constraint panel can narrow or expand these within the manual-edit envelope, or pin exact values and a branch. Every initial, random and local proposal obeys the recorded configuration. Fixed dimensions with one allowed branch are evaluated once; either branch means two distinct assemblies. A run with no admissible candidate reports bounded-search exhaustion and retains the previous candidate as prior work. It does not prove impossibility. The seed controls a bounded random-and-local search; no global optimum is claimed. Rejected proposals remain counted. Cancelling retains the most recently delivered candidate and the exact delivered proposal count. Drawing or selecting a new target clears the previous result immediately.

The generated target comes from an explicitly feasible four-bar, but its exact parameters are not injected into the search. A fit need not recover the generating parameters. Sharp corners and arbitrary hand drawings may not be closely achievable in this family. Poor fits remain visible. This is a kinematic sketch studio, not a fabrication plan.