Download Presentation
Collisions using separating-axis tests

Loading in 2 Seconds...

1 / 27

# Collisions using separating-axis tests - PowerPoint PPT Presentation

Collisions using separating-axis tests. Christer Ericson Sony Computer Entertainment Slides @ http://realtimecollisiondetection.net/pubs/. Problem statement. Determine if two (convex) objects are intersecting. Possibly also obtain contact information. ?. !. Underlying theory.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
Download Presentation

## PowerPoint Slideshow about 'Collisions using separating-axis tests' - sirius

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
Collisions using separating-axis tests

Christer EricsonSony Computer EntertainmentSlides @ http://realtimecollisiondetection.net/pubs/

Problem statement
• Determine if two (convex) objects are intersecting.
• Possibly also obtain contact information.

?

!

Underlying theory
• Set C is convex if and only if the line segment between any two points in C lies in C.
Underlying theory
• Separating Hyperplane Theorem
• States: two disjoint convex sets are separable by a hyperplane.
Underlying theory
• Nonintersecting concave sets not generally separable by hyperplane (only by hypersurfaces).
• Concave objects not covered here.
Underlying theory
• Separation w.r.t a plane P  separation of the orthogonal projections onto any line L parallel to plane normal.
Underlying theory
• A line for which the projection intervals do not overlap we call a separating axis.
Testing separation
• Compare absolute intervals

Separated if

or

Testing separation
• For centrally symmetric objects: compare using projected radii

Separated if

Code fragment

void GetInterval(Object o, Vector axis,

float &min, float &max) {

min = max = Dot(axis, o.getVertex(0));

for (int i = 1, n = o.NumVertices(); i < n; i++) {

float value = Dot(axis, o.getVertex(i));

min = Min(min, value);

max = Max(max, value);

}

}

Axes to test
• But which axes to test?
• Potentially infinitely many!
• Simplification:
• Deal only with polytopes
• Convex hulls of finite point sets
• Planar faces
Axes to test
• Handwavingly:
• Look at the ways features of A and B can come into contact.
• Features are vertices, edges, faces.
• In 3D, reduces to vertex-face and edge-edge contacts.
• Vertex-face:
• a face normal from either polytope will serve as a separating axis.
• Edge-edge:
• the cross product of an edge from each will suffice.
Axes to test
• Theoretically:
• Consider the Minkowski difference C of A and B.
• When A and B disjoint, origin outside C, specifically outside some face F.
• Faces of C come from A, from B, or from sweeping the faces of either along the edges of the other.
• Therefore the face normal of F is either from A, from B, or the cross product of an edge from either.
Axes to test
• Four axes for two 2D OBBs:
Code fragment

bool TestIntersection(Object o1, Object o2) {

float min1, max1, min2, max2;

for (int i = 0, n = o1.NumFaceDirs(), i < n; i++) {

GetInterval(o1, o1.GetFaceDir(i), min1, max1);

GetInterval(o2, o1.GetFaceDir(i), min2, max2);

if (max1 < min2 || max2 < min1) return false;

}

for (int i = 0, n = o2.NumFaceDirs(), i < n; i++) {

GetInterval(o1, o2.GetFaceDir(i), min1, max1);

GetInterval(o2, o2.GetFaceDir(i), min2, max2);

if (max1 < min2 || max2 < min1) return false;

}

for (int i = 0, m = o1.NumEdgeDirs(), i < m; i++)

for (int j = 0, n = o2.NumEdgeDirs(), j < n; j++) {

Vector axis = Cross(o1.GetEdgeDir(i), o2.GetEdgeDir(j));

GetInterval(o1, axis, min1, max1);

GetInterval(o2, axis, min2, max2);

if (max1 < min2 || max2 < min1) return false;

}

return true;

}

Note: here objects assumed to be in the same space.

Moving objects
• When objects move, projected intervals move:
Moving objects
• Objects intersect when projections overlap on all axes.
• If tifirst and tilast are time of first and last contact on axis i, then objects are in contact over the interval [maxi { tifirst}, mini { tilast}].
• No contact if maxi { tifirst} > mini { tilast}
Moving objects
• Optimization 1:
• Consider relative movement only.
• Shrink interval A to point, growing interval B by original width of A.
• Becomes moving point vs. stationary interval.
• Optimization 2:
• Exit as soon as maxi { tifirst} > mini { tilast}
Nonpolyhedral objects
• What about:
• Spheres, capsules, cylinders, cones, etc?
• Same idea:
• Identify all ‘features’
• Test all axes that can possibly separate feature pairs!
Nonpolyhedral objects
• Sphere tests:
• Has single feature: its center
• Test axes going through center
• (Radius is accounted for during overlap test on axis.)
Nonpolyhedral objects
• Capsule tests:
• Split into three features
• Test axes that can separate features of capsule and features of second object.
Nonpolyhedral objects
• Sphere vs. OBB test:
• sphere center vs. box vertex
• pick candidate axis parallel to line thorugh both points.
• sphere center vs. box edge
• pick candidate axis parallel to line perpendicular to edge and goes through sphere center
• sphere center vs. box face
• pick candidate axis parallel to line through sphere center and perpendicular to face
• Use logic to reduce tests where possible.
Nonpolyhedral objects
• For sphere-OBB, all tests can be subsumed by a single axis test:

Closest point on OBB to sphere center

Robustness warning
• Cross product of edges can result in zero-vector.

Typical test:

if (Dot(foo, axis) > Dot(bar, axis)) return false;

Becomes, due to floating-point errors:

if (epsilon1 > epsilon2) return false;

Results in: Chaos!

(Address using means discussed earlier.)

Contact determination
• Covered by Erin Catto (later)
References
• Ericson, Christer. Real-Time Collision Detection. Morgan Kaufmann 2005. http://realtimecollisiondetection.net/
• Levine, Ron. “Collisions of moving objects.” gdalgorithms-list mailing list article, November 14, 2000. http://realtimecollisiondetection.net/files/levine_swept_sat.txt
• Boyd, Stephen. Lieven Vandenberghe. Convex Optimization. Cambridge University Press, 2004. http://www.stanford.edu/~boyd/cvxbook/
• Rockafellar, R. Tyrrell. Convex Analysis. Princeton University Press, 1996.