Q: How can you spot a bald-faced liar in computer graphics? · Q: How can you spot a bald-faced...

Q: How can you spot a bald-faced liar in computer graphics?

A: Ask if they have fully working code for…

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Collision Handling

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Collision Handling

Consists of two steps:

• collision detection:

• what has collided?

• what is colliding?

• what will collide?

Collision Handling






• collision response: what to do about it?

Collision Handling






• collision response: what to do about it?

Completely different problems and methods

Collision Detection Taxonomy

Discrete: are objects colliding right now?

• check every frame

Issues:

• penetration artifacts



Issues:

• penetration artifacts

• tunneling





Continuous-time collision detection (CTCD): when will the next collision occur?




Continuous-time collision detection (CTCD): when will the next collision occur?

• correct but slow


Discrete vs continuous-time

Rigid vs deforming objects

• rigid objects allow some precomputation


Discrete vs continuous-time

Rigid vs deforming objects

• rigid objects allow some precomputation

Collision vs promixity

Broad Phase vs Narrow Phase

Collision detection is O(n^2) problem



Broad phase: detect collision candidates (hopefully n log n) Narrow phase: check if the candidates actually collide



Broad phase: detect collision candidates (hopefully n log n) Narrow phase: check if the candidates actually collide In practice: BVH with kDOPs very popular/fast broad phase

Discrete CD for Convex Objects

Do these objects collide?

How can you tell?


Algorithm 1: check that all points of one object are outside the other


Algorithm 1: check that all points of one object are outside the other

• shoot a random ray, count intersections

• (better method exist based on winding #)


Algorithm 2: find a separating axis


Algorithm 2: find a separating axis

• for each side of object 1, extent to line

• check all verts on object 2 on correct side

Discrete Collision Detection

What if objects aren’t convex?



• compute convex decomposition

• then check collisions between pieces



• compute convex decomposition

• far from trivial in 3D…


What if you want continuous-time detection?

1. Find separating line/plane




2. Calculate earliest time an object hits axis





• is certificate no earlier collision possible





3. Try to find new line





3. Try to find new line

4. Repeat until objects “close enough”

Discrete Collision Detection II

Alternate approach for rigid objects:

• signed distance field

precompute on grid distance to object boundary positive outside, negative inside




precompute on grid distance to object boundary positive outside, negative inside grid moves as object moves query grid to tell if point is inside




precompute on grid distance to object boundary positive outside, negative inside grid moves as object moves query grid to tell if point is inside also useful for computing response

Discrete Collision Detection III

What about deformable objects?

Discrete Collision Detection III

What about deformable objects?

Objects intersect if and only if some triangle pair intersects… so check all pairs

(Hopeless without a broad phase)

Continuous Collision Detection

Let’s start with the easiest case…



find when point hits infinite line




wait, two roots?




for each positive root, check if point inside segment



This algorithm has at least 4 bugs


for each positive root, check if point inside segment


Bug #1:


Bug #1:

Bug #2:


Bug #1:

Bug #2:

becomes zero when segment collapses


Bug #3:

problem:


Bug #3:

problem:

just check A = 0 and handle specially, right?


Bug #4:

has a collision occurred?


for each vertex v of O1:

for each edge e of O2:

CTCD(e, v)




CTCD(e, v)



CTCD(e, v)

CTCD is still Buggy

thermal exhaust port

CTCD is still Buggy

thermal exhaust port

(literal) corner case causes collisions to (literally) slip through the cracks




CTCD(e, v)



CTCD(e, v)

for each vertex v1 of O1:

for each vertex v2 of O2:

CTCD(v1, v2)

Do Any of These Cases Matter?

Bug #1:

Bug #2:

“zero-probability events”

Do Any of These Cases Matter?

Bug #1:

Bug #2:

“zero-probability events”

Harmon’s Law

When dealing with collisions, “zero-probability events” will occur

100% of the time

Harmon’s Law

CTCD in 3D

Triangle-triangle collision can be decomposed into two types of checks:

Vertex-Face

1. Find when vertex and triangle coplanar

2. Check barycentric coords of vertex

Vertex-Face

1. Find when vertex and triangle coplanar

2. Check barycentric coords of vertex

cubic equation

1. Find when edges coplanar

2. Check barycentric coords of intersection

Edge-Edge

1. Find when edges coplanar

2. Check barycentric coords of intersection

Classic problem: parallel edges

Edge-Edge

CTCD in 3D

For each pair of triangles:

• 6 vertex-face tests

• 9 edge-edge tests

CTCD in 3D

For each pair of triangles:

• 6 vertex-face tests

• 9 edge-edge tests

Also needed for robustness:

• 18 vertex-edge tests

• 9 vertex-vertex tests

Solving These Polynomials

Must find all the roots

• missed roots -> missed collisions

Solving These Polynomials

Must find all the roots

• missed roots -> missed collisions

“Cubic formula” exists but not robust

Best practical code: Jenkins-Traub method

• can find FORTRAN code online

Collision Response

Once collision detected, how to prevent it?

Guiding principles:

• conservation of momentum

• conservation of energy

Collision Response


Guiding principles:



Other concerns: friction

Importance of Friction

Key ingredient of spin / “English”

Obviously essential to stable stacking

Collision Response


Guiding principles:



Other concerns: friction coefficient of restitution

Coefficient of Restitution

Ratio

Measures dissipation due to heat, sound, vibrations etc

Coefficient of Restitution

Ratio

Measures dissipation due to heat, sound, vibrations etc

Perfectly elastic collision: CoR = 1

Perfectly inelastic collision: CoR = 0

Collision Response


Two main approaches:

• penalty method: push apart over time

• impulse method: stop animation and correct velocities instantly

Main idea: insert springs that push objects apart over time

Spring endpoints connect to closest pair of points on the two objects

Penalty Method

Penalty Method

Pros:

• easy to code

• great for gentle, persistent contact

Cons:

• must pick stiffness of springs:

• too soft objects visibly penetrate

• too hard very expensive (blows up)

Impulse Method

Freeze at time of collision

Change the velocities so collision prevented

• change in velocity called impulse

Impulse Method

Freeze at time of collision

Change the velocities so collision prevented

• change in velocity called impulse

Rigid billiard model: apply equal and opposite impulses along contact normal

Impulse Method

Pros:

• stops collisions before they occur

• friction and CoR easy(ier) to include

Cons:

• Zeno issues

Zeno Paradoxes

Infinite steps in finite time

Zeno Paradoxes


• might need infinite impulses per frame

Zeno Paradoxes


• might need infinite impulses per frame

Also arises when CoR < 1

• inelastic collapse phenomena

Other Issues

Rigid object assumption is suspect

Other Issues


Collisions involving multiple objects

• pool breaks, stacks of blocks, etc

[IncredibleScience]

[Stéphane Dorbolo]

Other Issues


Collisions involving multiple objects

• pool breaks, stacks of blocks, etc

Friction handling is extremely complex

• paradoxes abound

Even physicists don’t fully understand some phenomena…

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