Separating Axis Test (SAT) and Support Points in 2D. Randy Gaul Special thanks to Dirk Gregorius for his GDC 2013 lecture and slides

Transcription

1 Separating Axis Test (SAT) and Support Points in 2D Randy Gaul Special thanks to Dirk Gregorius for his GDC 2013 lecture and slides

2 Overview - Separating Axis Test (SAT) Why use SAT Theory Implementation Support Points Implementation

3 Robust Versatile Axis of minimum penetration Penetration distance Contact points Efficient Easy to conceptualize Why use the SAT?

4 SAT Versatility Collision detection is easy Resolving collision is hard Realistic bouncing, stacking Resolution requires: Axis of minimum penetration Penetration distance Contact points SAT can give us all info needed for resolution Gathering this information is called manifold generation

5 Develop Intuition with Circles t r1 r2 d = t (r1 + r2) (r1 and r2 as scalar radii)

6 Develop Intuition with Circles Collision when d is positive No collision when d is negative t r2 r1

7 Develop Intuition with Circles - Axis Projection Separating Axis An axis with separation of projections For circles project onto axis from centers No penetration Penetration

8 SAT - Definition Collision Projections onto all axes are not separating (overlapping) No collision At least one axis contains separating projections Note about No Collision case: Any positive number of axes can be separating

9 Which Axis to Test? A separating axis will always be a face of a polygon Circles have many face normals; just test vector from center to center

10 SAT Algorithm Given two shapes A and B Project A onto face axes of B Record each signed projection overlap Project B onto face axes of A Record each signed projection overlap Return greatest signed distance found Collision if distance is less than zero

11 SAT Implementation (Pseudo Code) float FindLeastPenetration( Polygon A, Polygon B ) { float bestdistance = -FLT_MAX; for(int i = 0; i < A->faceCount; ++i) Vec2 p = A->faces[i]; float d = Project( A, B, p ); } // Store greatest distance if(d > bestdistance) bestdistance = d; return bestdistance; Call twice, flip objects A and B

12 SAT Implementation : Note The largest signed distance will be penetration depth If less than zero then colliding Keep track of face index that supplied penetration depth This face s normal will be resolution vector This face is the axis of minimum penetration // Store greatest distance if(d > bestdistance) bestdistance = d; bestindex = i; return bestdistance AND bestindex;

13 SAT : Optimization Projection is expensive Must transform vertices of both shapes and compute overlap Computing support points is faster Instead of overlap, compute point to line distance Support points act as alternative to actual projections

14 Support Points Vertex of polygon farthest along a given direction Use dot project to compute projected distance Support( dir ) dir

15 Support Points Implementation (Pseudo Code) Vec2 Polygon::GetSupport( const Vec2& dir ) { float bestprojection = -FLT_MAX; Vec2 bestvertex; for(int i = 0; i < vertexcount; ++i) Vec2 v = vertices[i]; real projection = Dot( v, dir ); if(projection > bestprojection) bestvertex = v; bestprojection = projection; } return bestvertex;

16 Optimized Penetration Calculation Distance point to plane: d = n (p1 p2) Support( -n ) p2 : any point along the plane n p1 n p2 p d = n (Support( -n ) p)

17 SAT Collision Test bool SAT( Polygon A, Polygon B ) { if(findleastpenetration( A, B ) > 0.0f) return false; if(findleastpenetration( B, A ) > 0.0f) return false; } return true;

18 Finding Contact Points Point of contact required for resolution At most, two points can be used A B

19 Finding Contact Points : Reference Face Identify reference face Reference face is face with axis of minimum penetration Reference face is always on object A A B Reference face

20 Finding Contact Points : Incident Face To find contact points: Find incident face Incident face most faces axis of minimum penetration A B Incident face

21 Finding Contact Points : Clipping Clip incident face against reference face side planes Use Sutherland-Hodgman clipping (see references) A B

22 Finding Contact Points : Culling Only keep points behind reference face Compute signed distance, test sign (positive or negative) A B

23 Checkpoint Can detect collisions with SAT Understand support points Can optimize SAT with support points Record axis of minimum penetration Normal to this axis is resolution vector Compute contact points with clipping

24 Optimizations If separating axis is found Store this axis, check axis next frame between same two objects Fast early out Can detect if face normals are nearly parallel Testing parallel axes is redundant, don t do it! Can cull redundant axes from both shapes, not just one Hill climbing for Support search Test first vertex and both adjecent Loop in direction with greatest delta Stop searching once delta is positive

25 References Dirk Gregorius s 2013 GDC Slides Box2D.org downloads Box2D source code Patrick Moghames Graphics professor (Sutherland-Hodgman clipping) Impulse Engine randygaul.net All of Erin Catto s GDC Slides Box2D.org