Collision Detection and Response

1. Collision Detection and Response

2. How to do this? Here is where I want my object to be Here is where my object is Here is where my object is going to be

3. Axis Aligned Bounding Boxes(AABB) • 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? Initial Airplane Orientation Airplane Orientation 2

4. AABB • Do a bunch of ifs • if( px<= maxx &&py<= maxy &&pz<= maxz &&px>= minx &&py>= miny &&pz>= minz ) then collide = true; else collide = false minx maxx maxy P miny

5. Comparing AABBs • if mins/maxes overlapthen collide = trueelse collide = false; Bminx Bmaxx Bmaxy Amaxx Aminx Bminy

6. AABB Approach Initialization: Iterate through vertices and find mins and maxes After Transformations: Iterate through AABB vertices and find mins and maxes

7. AABB Approach • 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

8. Shoot a projectile • 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

9. Collision Detection: Planes and Points • 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 theplane? • 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] [A,B,C] • P D + -

10. Manually Transforming Objects for Collisions • 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 BB glPopMatrix() The p here is already a position in world coordinates! YAY! RT matrix 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

11. Another way to do the transform • 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 BB glPopMatrix() Multiply p by (RT)-1 or (-T)(-R)p And do the same its direction vector v 2) do collision detection and response calculations with the untransformed BB 3) Put p back into world coordinates with RTpand its direction vector v Watchout for non-uniform scaling – for this you would need do do multiplications of the form M-1Tv RT matrix

12. Simple Collision Response • 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) Refldirection N Current Position Next Position Raydirection Rayorigin

13. Ray-Plane Intersections • 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 Refldirection N θ i Raydirection Rayorigin θ adjacent

14. Calculate a Reflection • Really we should use physics here but… • Think back to lighting • Refldirection=-2dot(N, Raydirection) *N + Raydirection Refldirection N θ i Raydirection Rayorigin θ adjacent

15. Tips for making Assignment 3 easier • 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

16. Oriented Bounding Boxes (OBB) • 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 Airplane Orientation 1 Airplane Orientation 2

17. Creating the Plane Equations • 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 v2 v1 v3

18. Collision Detection: Planes and Points • 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 theplane? • Axp+Byp+Czp+ D = signed distance(assuming [A,B,C] is length 1) [A,B,C] • P D + -

19. OBB – Point Collision • If( a point evaluates to be <=0 distance from all 6 planes that make up the box • Then collide = true • Else collide = false

20. Collision Detection: OBB and OBB • 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…

21. Separating Axes • 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…

22. Transforming OBB Planes • 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?