Separating Axis Test (SAT) and Support Points in 2D

Randy Gaul

Special thanks to Dirk Gregorius for his GDC 2013 lecture and slides

Overview - Separating Axis Test (SAT)

• Why use SAT

• Theory

• Implementation

• Support Points

• Implementation

Why use the SAT?

• Robust

• Versatile• Axis of minimum penetration

• Penetration distance

• Contact points

• Efficient

• Easy to conceptualize

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”

Develop Intuition with Circles

𝒅 = 𝒕 − (𝒓𝟏 + 𝒓𝟐)

𝒕

r1 r2

(r1 and r2 as scalar radii)

Develop Intuition with Circles

𝒕

r1

r2

• Collision when d is positive

• No collision when d is negative

Develop Intuition with Circles - Axis Projection

• Separating Axis• An axis with separation of projections

• For circles project onto axis from centers

No penetration Penetration

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

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

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

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

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;

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

Support Points

• Vertex of polygon farthest along a given direction• Use dot project to compute projected distance

dir

Support( dir )

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;

}

Optimized Penetration Calculation

Distance point to plane:d = n ∙ (p1 – p2)

p2 : any point along the plane

p1 p2n

Support( -n )

p

n

d = n ∙ (Support( -n ) – p)

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;

}

Finding Contact Points

• Point of contact required for resolution• At most, two points can be used

A B

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

Finding Contact Points : Incident Face

• To find contact points:• Find incident face

• Incident face most faces axis of minimum penetration

Incident face

A B

Finding Contact Points : Clipping

• Clip incident face against reference face side planes• Use Sutherland-Hodgman clipping (see references)

A B

Finding Contact Points : Culling

• Only keep points behind reference face• Compute signed distance, test sign (positive or negative)

A B

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

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

References

• Dirk Gregorius’s 2013 GDC Slides – Box2D.org downloads

• Box2D source code

• Patrick Moghames – Graphics professor (Sutherland-Hodgmanclipping)

• Impulse Engine – randygaul.net

• All of Erin Catto’s GDC Slides – Box2D.org