Machine Kinematics
solved, then drawn
Concept

Convex decomposition — where it appears

The cutting of a simple polygon into convex pieces, by ear clipping and then a merge. It is what makes a dented part answerable, since the distance between two polygons is the best answer over the pairs of pieces — and a decomposition is checked by its consumer, which re-measures every piece for convexity.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Also named here as separating-axis — the same set of essays touches all of them, so they are one junction rather than several.

Peaucellier's cell: the closest pair at one position. Eight links, ten pins and an exact straight line — the site's densest planar loop. Every joint is where the solver put it, exactly as in the linkage field; the material is the only thing added. The heavy segment joins the two closest points over every pair of parts that is tested — which excludes pairs sharing a pin, since their material surrounds that pin by construction — and its length is the gap: 0.1983 here, between long arm A · crank. A negative value is a penetration depth, the distance the pair would have to be moved apart, and it is drawn in the warning colour.

A gap is a number

A collision test that answers yes or no cannot say by how much, and therefore cannot say what would fix it. The quantity this field is built on is one signed number: positive is a gap, negative is how far the parts would have to be moved to stop overlapping.

bodies · Body A dented shape, cut into 2 convex pieces. Ear clipping cuts the outline into triangles and Hertel–Mehlhorn then deletes every diagonal whose removal leaves both sides convex, which takes this crank to 2 pieces rather than the 4 the triangulation produced. The pieces tile the polygon to 1.8e-16 of its area, and — the part the areas cannot check — a point is inside the pieces exactly when it is inside the polygon, tested at 4,000 random points per shape with 0 disagreements. The gap between two parts is then the best answer over the pairs of pieces, which is why the decomposition has to be right rather than merely plausible.

A shape with a dent in it

The separating-axis theorem is not approximately right about a non-convex shape; it is wrong, and it returns a confident number while being wrong. The repair is to cut the shape into convex pieces — and the shortcut everybody takes instead adds a hundred per cent more material.

bodies · Body

Named alongside it

The objects these essays reach for when they reach for this one.

Convex hullInterferenceLink bodyPenetration depthSeparating-axisSigned clearanceClearanceWitness pair

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