Computing Distance Erin Catto Blizzard Entertainment Convex - PowerPoint PPT Presentation

Computing Distance Erin Catto Blizzard Entertainment

Convex polygons

Closest points

Overlap

Goal  Compute the distance between convex polygons

Keep in mind  2D  Code not optimized

Approach simple complex

Geometry

If all else fails …

DEMO!

Outline 1. Point to line segment 2. Point to triangle 3. Point to convex polygon 4. Convex polygon to convex polygon

Concepts 1. Voronoi regions 2. Barycentric coordinates 3. GJK distance algorithm 4. Minkowski difference

Section 1 Point to Line Segment

A line segment A B

Query point Q A B

Closest point Q P A B

Projection: region A Q A B

Projection: region AB Q A B

Projection: region B Q A B

Voronoi regions region A region AB region B A B

Barycentric coordinates A G B G(u,v)=uA+vB u+v=1

Fractional lengths v=0.5 u=0.5 A G B G(u,v)=uA+vB u+v=1

Fractional lengths v=0.25 u=0.75 A G B G(u,v)=uA+vB u+v=1

Fractional lengths u=1.25 v=-0.25 G A B G(u,v)=uA+vB u+v=1

Unit vector A B n B-A n = B-A

(u,v) from G A G B v u       G-A n B-G n v= u= B-A B-A

(u,v) from Q Q A G B v u       Q-A n B-Q n v= u= B-A B-A

Voronoi region from (u,v) v > 0 u > 0 A G B u > 0 and v > 0 region AB

Voronoi region from (u,v) u > 0 v < 0 G A B v <= 0 region A

Voronoi region from (u,v) u < 0 v > 0 A B G u <= 0 region B

Closet point algorithm input: A, B, Q compute u and v if (u <= 0) P = B else if (v <= 0) P = A else P = u*A + v*B

Aside: center of mass A COM B mass A = mass B

Aside: center of mass A COM B mass A > mass B

Aside: center of mass A COM B mass A < mass B

Aside: center of mass massA massB   COM A B   massA massB massA massB massA>0 massB>0

Section 2 Point to Triangle

Triangle B A C

Closest feature: vertex Q P=B A C

Closest feature: edge B A P Q C

Closest feature: interior B A P=Q C

Voronoi regions B Region B Region AB A Region A Region ABC Region BC Region CA C Region C

3 line segments Q   u ,v AB AB B A   C u ,v BC BC   u ,v CA CA

Vertex regions  u 0 AB B  v 0 BC  A u 0 CA  v 0 AB C  u 0 Using line segment uv’s BC  v 0 CA

Edge regions  u 0 AB  v 0 B AB ? A C Line segment uv’s are not sufficient

Interior region B A ? C Line segment uv’s don’t help at all

Triangle barycentric coordinates B A Q    Q uA vB wC C    u v w 1

Linear algebra solution       A B C u Q x x x x       A B C v = Q       y y y y        1 1 1    w  1 

Fractional areas B A Q C

The barycenctric coordinates are the fractional areas B  w Area(ABQ) A w Q u v  u Area(BCQ)  v Area(CAQ) C C

Barycentric coordinates B  w 0 A Q  u 1  v 0 C

Barycentric coordinates area(QBC)  u area(ABC) area(QCA)  v area(ABC) area(QAB)  w area(ABC)

Barycentric coordinates are fractional line segment : fractional length triangles : fractional area tetrahedrons : fractional volume

Computing Area C B A 1   signed area= cross B-A,C-A 2

Q outside the triangle B A Q C C

P outside the triangle B A w  0 Q v  0 v+w>1 C C

P outside the triangle B A Q u  0 C C

Voronoi versus Barycentric  Voronoi regions != barycentric coordinate regions  The barycentric regions are still useful

Barycentric regions of a triangle B A C

Interior B A Q    u 0, v 0, w 0 C

Negative u B A Q u  0 C

Negative v B A Q v  0 C

Negative w Q B w  0 A C

The uv regions are not exclusive B A P C Q

Finding the Voronoi region  Use the barycentric coordinates to identify the Voronoi region  Coordinates for the 3 line segments and the triangle  Regions must be considered in the correct order

First: vertex regions  u 0 AB B  v 0 BC  A v 0 AB  u 0 CA C  u 0 BC  v 0 CA

Second: edge regions  u 0 AB  v 0 B AB ? A C

Second: edge regions solved  u 0 AB  v 0 B AB  w 0 ABC A C

Third: interior region B A u > 0 ABC v > 0 ABC w > 0 ABC C

Closest point  Find the Voronoi region for point Q  Use the barycentric coordinates to compute the closest point Q

Example 1 Q B A C

Example 1 Q B A C uAB <= 0

Example 1 Q B A C uAB <= 0 and vBC <= 0

Example 1 Q P=B A Conclusion: C P = B

Example 2 Q B A C

Example 2 Q B A C Q is not in any vertex region

Example 2 Q B A C uAB > 0

Example 2 Q B A C uAB > 0 and vAB > 0

Example 2 Q B A C uAB > 0 and vAB > 0 and wABC <= 0

Example 2 Q B A P C Conclusion: P = uAB*A + vAB*B

Implementation input: A, B, C, Q compute uAB, vAB, uBC, vBC, uCA, vCA compute uABC, vABC, wABC // Test vertex regions … // Test edge regions … // Else interior region …

Testing the vertex regions // Region A if (vAB <= 0 && uCA <= 0) P = A return // Similar tests for Region B and C

Testing the edge regions // Region AB if (uAB > 0 && vAB > 0 && wABC <= 0) P = uAB * A + vAB * B return // Similar for Regions BC and CA

Testing the interior region // Region ABC assert(uABC > 0 && vABC > 0 && wABC > 0) P = Q return

Section 3 Point to Convex Polygon

Convex polygon B C A D E

Polygon structure struct Polygon { Vec2* points; int count; };

Convex polygon: closest point B Query point Q C A Q D E

Convex polygon: closest point B Closest point Q C A Q P D E

How do we compute P?

What do we know? Closest point to point Closest point to line segment Closest point to triangle

Simplex 0-simplex 1-simplex 2-simplex

Idea: inscribe a simplex B C A Q D E

Idea: closest point on simplex B P = C A Q D E

Idea: evolve the simplex B C A Q D E

Simplex vertex struct SimplexVertex { Vec2 point; int index; float u; };

Simplex struct Simplex { SimplexVertex vertexA; SimplexVertex vertexB; SimplexVertex vertexC; int count; };

We are onto a winner!

The GJK distance algorithm  Computes the closest point on a convex polygon  Invented by Gilbert, Johnson, and Keerthi

The GJK distance algorithm  Inscribed simplexes  Simplex evolution

Starting simplex B Start with arbitrary vertex. Pick E. C This is our starting A simplex. Q D E