Concept

Penetration depth — where it appears

The least translation that would separate two overlapping bodies, taken as the smallest overlap over every edge normal of both. It is the number a designer acts on, because it says how much material has to go — and it saturates once two long bars are fully crossed, at the width of one of them.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

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 $The gap is a straight line in the width. Three machines, four widths each, every width a fraction of that machine's own limit. Each set of points is collinear to the last bit of a double — the slopes wander by less than 10⁻¹³ across the range — because a feature-to-feature distance is linear in the corners of the two bodies and a bar's corners are linear in its width. The slope reads the contact: Chebyshev's linkage at -2.692, a crank rocker with a post at -1.350, a crank passing a stud at -1.350. A bar's boss grows 1.35 times as fast as its side, so −1.35 is a boss against something that is not growing and −2.70 is boss against boss. Extending each line to zero gives the widest link the machine will take, and the bisection that finds it the hard way agrees — 0.30113 against 0.30112, inside the bisection's own residual.$

The gap is a straight line in the metal

Thickening every link by the same amount subtracts the same amount from every clearance, exactly, and moves the angle at which the worst one occurs by nothing at all. So a whole swept check can be done once on bars of any width and every other width read off by subtraction — until the closest pair changes hands, and never past zero.

bodies · Body

Named alongside it

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

Interference Link body Signed clearance Convex decomposition Convex hull Separating-axis Witness pair Approximation Clearance Design rule Root-finding Scale invariance

All concepts