2. Here is where my object is Here is where my object is going to be Here is where I want my object to be

3.  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.  Do a bunch of ifs ◦ if( p x = min x && p y >= min y && p z >= min z ) then collide = true; else collide = false P min x max x max y min y

5.  if mins/maxes overlap then collide = true else collide = false; Bmin x Bmax x Bmax y Bmin y Amin x Amax x

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

7.  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: (min x, min y,min z ), (max x,max y,max z ), but not required

8.  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.  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 = (x p,y p,z p )  What is the shortest distance from p to the plane?  Ax p +By p +Cz p + D = signed distance (assuming [A,B,C] is length 1)  For AABBs, normals are always going to be parallel to a principle axis e.g., x-axis: [A,B,C] = [1,0,0] [A,B,C] D P + -

10.  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() Get the vertices of this BB and multiply them by the RT matrix: e.g., RTv i for each of the 8 vertices, v i. This will put the BB into world coordinates RT matrix The p here is already a position in world coordinates! YAY!

11.  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() RT matrix 1)Multiply p by (RT) -1 or (-T)(-R)p 2)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 RTp and its direction vector v Watchout for non-uniform scaling – for this you would need do do multiplications of the form M -1T v

12.  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) Ray direction Ray origin N Refl direction Current Position Next Position

13.  Make sure N and Ray direction are normalized!  Adjacent = A*Ray originX + B*Ray originY + C*Ray originZ +D  adjacent / cos(θ) = hypotenuse ◦ That is, dot (Ray direction, N) = cos(θ)  Ray origin +Ray direction *hypotenuse = i Ray direction Ray origin N Refl direction θ θ adjacent i

14.  Really we should use physics here but…  Think back to lighting ◦ Refl direction =-2dot(N, Ray direction ) *N + Ray direction Ray direction Ray origin N Refl direction θ θ adjacent i

15.  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.  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.  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 ◦ Av1 x + Bv1 y + Cv1 z = - D v1 v2 v3

18.  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 = (x p,y p,z p )  What is the shortest distance from p to the plane?  Ax p +By p +Cz p + D = signed distance (assuming [A,B,C] is length 1) [A,B,C] D P + -

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

20.  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.  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 BB A and all the planes of the BB B ◦ 2) Test the collisions between all the vertices of BB B and all the planes of the BB A ◦ 3) Then test collisions between all the vertices of BB A and BB B and all cross products of each pair of edge normals of BB A and BB B This actually works for any convex polyhedron. There are optimizations for OBBs…optimizations

22.  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 -1T n, n= [A,B,C,0] T // this is a vector!  Transform the plane equation ◦ p = [A,B,C,D] ◦ Matrix M = arbitrary transforms  M -1T p ◦ OR, if you don’t have any non-uniform scaling  Mp  What would happen if you had non uniform scaling?