Collision Detection and Response

How to do this?

Here is where my object is

Here is where my object is going to be

Here is where I want my object to be

A box that is ◦ Defined by the min and max

coordinates of an object◦ Always aligned with the

coordinate axes How can we tell if a point

p is inside the box?

Axis Aligned Bounding Boxes(AABB)

Initial Airplane Orientation

Airplane Orientation 2

Do a bunch of ifs◦ if(

px<= maxx &&py<= maxy &&pz<= maxz &&px>= minx &&py>= miny &&pz>= minz )then collide = true;else collide = false

AABB

P

minx maxx

maxy

miny

if mins/maxes overlapthen collide = trueelse collide = false;

Comparing AABBsBminx Bmaxx

Bmaxy

Bminy

Aminx Amaxx

AABB ApproachInitialization: Iterate through vertices and find mins and maxes

After Transformations: Iterate through AABB vertices and find mins and maxes

Initialization◦ iterate through all vertices of your model to find the mins

and maxes for x, y, and z During runtime

◦ Test if any of the AABB mins/maxes of one object overlap with another object’s AABB mins/maxes MAKE SURE THAT THE AABB VALUES ARE IN THE SAME

COORDINATE FRAME (e.g., world coordinates)! If they aren’t, then manually transform them so they are. This is equivalent to multiplying the 8 points by a matrix for

each object Then make sure to recalculate your mins/maxes from the 8

transformed points! Note: it is possible to do this with only 2 points from the box:

(minx,miny,minz), (maxx,maxy,maxz), but not required

AABB Approach

Keep a position p and a unit vector v. Each frame add the vector to the position p+v*speed, This is essentially how the camera works in my

latest code sample on my webpage How about gravity?

◦ Add a gravity vector (e.g., g = [0,-1,0]◦ v+=v+g*gravity◦ p +=v*speed

◦ glTranslatefv(p)◦ where gravity and speed are float

Shoot a projectile

Equation: Ax+By+Cz+D = 0 ◦ [A,B,C] is the normal of the plane◦ D is how far from the origin it is

p = (xp,yp,zp) What is the shortest distance from p to the

plane? Axp+Byp+Czp+ D = signed distance

(assuming [A,B,C] is length 1)

For AABBs, normals are alwaysgoing to be parallel to a principle axise.g., x-axis: [A,B,C] = [1,0,0]

Collision Detection: Planes and Points

[A,B,C]

DP

+ -

Manually (i.e., make your own matrix multiplication functions) transform all things collidable into the same coordinate frame (e.g. world coordinates)

E.g., if you have :

gluLookat(…) glPushMatrix()

◦ glTranslatefv(p);◦ Draw sphere projectile

glPopMatrix()glPushMatrix();◦ glRotate(a,x,y,z)◦ glTranslate(tx,ty,tz)◦ Draw a BBglPopMatrix()

Manually Transforming Objects for Collisions

Get the vertices of this BB and multiply them by the RT matrix: e.g., RTvi for each of the 8 vertices, vi. This will put the BB into world coordinates

RT matrix

The p here is already a position in world coordinates! YAY!

Manually transform the projectile into the BB’s object coordinate frame (less work for cpu)

E.g., if you have :

gluLookat(…) glPushMatrix()

◦ glTranslatefv(p);◦ Draw sphere projectile

glPopMatrix()glPushMatrix();◦ glRotate(a,x,y,z)◦ glTranslate(tx,ty,tz)◦ Draw a BBglPopMatrix()

Another way to do the transform

RT matrix

1) Multiply p by (RT)-1 or (-T)(-R)p2) And do the same its direction

vector v2) do collision detection and response calculations with the untransformed BB3) Put p back into world coordinates with RTp and its direction vector v

Watchout for non-uniform scaling – for this you would need do do multiplications of the form M-1Tv

collide = true; // then what?◦ Calculate ray intersection with the plane◦ Calculate reflection vector (sound familiar?)◦ Calculate new position

Rays are made of an origin (a point) and a direction (a vector)

Simple Collision Response

Raydirectio

nRayorigin

NRefldirection

Current Position Next Position

Make sure N and Raydirection are normalized! Adjacent = A*RayoriginX + B*RayoriginY + C*RayoriginZ +D adjacent / cos(θ) = hypotenuse

◦ That is, dot (Raydirection , N) = cos(θ) Rayorigin+Raydirection*hypotenuse = i

Ray-Plane Intersections

Raydirectio

nRayorigin

NRefldirection

θ

θ

adjacent

i

Really we should use physics here but… Think back to lighting

◦ Refldirection =-2dot(N, Raydirection) *N + Raydirection

Calculate a Reflection

Raydirectio

nRayorigin

NRefldirection

θ

θ

adjacent

i

1) test collisions and response on untransformed objects before trying it with applied transforms◦ Actually, the only requirement is projectile

transformations. 2) use spheres for the projectiles ( you can

basically treat these as points) and then you do not need to implement separating axes with OBB

3) draw your bounding boxes so you can see them (this is actually required is the project)

4) graduates : Don’t worry, I don’t expect terrain collisions

Tips for making Assignment 3 easier

A box that◦ Stays oriented to the model

regardless of transformations◦ These are often defined by artists in

the 3D modeling program◦ There are algorithms to compute the

minimum OBB, but this is out of scope for this class

◦ How to create the initial box? 1) Either:

Iterate through vertices (same as AABB Make a nice box with a modeling

program 2) Convert to plane equations

Oriented Bounding Boxes (OBB)

Airplane Orientation 1

Airplane Orientation 2

Take 3 vertices from one side of your box Compute the normal

◦ [v3-v1] X [v2-v1] = [a,b,c]◦ Normalize the normal◦ [A,B,C] =[a,b,c] / ||[a,b,c]||

Solve the following:◦ Ax +By+ Cz + D = 0

Plug in a point we know is on the plane◦ Av1x + Bv1y + Cv1z = - D

Creating the Plane Equations

v1v2

v3

Equation: Ax+By+Cz+D = 0 ◦ [A,B,C] is the normal of the plane◦ D is how far from the origin it is

p = (xp,yp,zp) What is the shortest distance from p to the

plane? Axp+Byp+Czp+ D = signed distance

(assuming [A,B,C] is length 1)

Collision Detection: Planes and Points

[A,B,C]

D

P

+ -

If( a point evaluates to be <=0 distance from all 6 planes that make up the box

Then collide = true Else collide = false

OBB – Point Collision

Test whether any of the 8 points that make up one box collide with the other◦ Do this for both boxes.◦ This won’t always work in 3D…

Collision Detection: OBB and OBB

In ANY of the following cases, if all of the collision tests evaluate to positive, then assume no intersection◦ 1) Test collisions between all the vertices of BBA and all the

planes of the BBB

◦ 2) Test the collisions between all the vertices of BBB and all the planes of the BBA

◦ 3) Then test collisions between all the vertices of BBA and BBB and all cross products of each pair of edge normals of BBA and BBB

This actually works for any convex polyhedron. There are optimizations for OBBs…

Separating Axes

Again, you will have to make sure that all planes, points, etc are in the same coordinate frame when computing collisions.

Think about normals◦ Vertex: Mv as usual◦ Normal: M-1Tn, n= [A,B,C,0]T // this is a vector!

Transform the plane equation◦ p = [A,B,C,D]◦ Matrix M = arbitrary transforms

M-1Tp◦ OR, if you don’t have any non-uniform scaling

Mp What would happen if you had non uniform scaling?

Transforming OBB Planes