Lesson 28 · Polygon versus polygon - Build a Physics Engine

Polygon versus polygon

Now the pieces combine: test every edge normal of both polygons, and if none separates them, the shallowest overlap is the contact. Today you build full polygon-polygon SAT.

The goal

Detect collision between two convex polygons and return the minimum-penetration axis as the manifold.

Start here - the target

TO DO

Scenario: Two squares overlapping

Giventhe unit square at the origin and the same square translated to center {1.5, 0}

Whenthey are tested for collision with SAT

Thenthey collide with normal {1, 0} and penetration 0.5

Andwhen the second square is at center {3, 0} there is a separating axis and no collision

Background

Full SAT for two convex polygons is the assembly of the last three lessons: the only axes that can separate them are the edge normals of the two shapes, so you project both polygons onto each candidate normal and measure the overlap. The instant any axis shows a gap, you return “no collision” - most non-colliding pairs bail out on the first or second axis, which is what makes SAT fast. If every axis overlaps, the shapes intersect.

Among all those overlaps, the smallest one is the shortest distance you would have to push the shapes to separate them, so its axis is the contact normal and its size is the penetration. One detail: an edge normal points outward from its own polygon, so flip it if needed to point consistently from A toward B, comparing the shape centers. Two overlapping unit squares separate along x with penetration 0.5. This one function handles every convex polygon pair, rotated or not.

Make it work

func PolygonVsPolygon(a, b []Vec2) Manifold {
  best := math.Inf(1)
  var normal Vec2
  for _, axis := range append(Normals(a), Normals(b)...) {
    o := Overlap(Project(a, axis), Project(b, axis))
    if o <= 0 { return Manifold{} } // separating axis found
    if o < best { best, normal = o, axis }
  }
  // flip normal to point from A's center toward B's center; penetration = best
}

CheckpointDONE

Two convex polygons collide via SAT, reporting the shallowest-penetration normal. Commit and stop here.