1. Week 13 - Wednesday
CS361

2. Last time What did we talk about last time? Before that… Exam 2!
Polygonal techniques
Tessellation and triangulation
Triangle strips, fans, and meshes
Simplification
Static
Dynamic
View-dependent

3. Questions?

4. Project 4

5. Student Lecture: Intersection Tests and Bounding Volumes

6. Intersection Test Methods

7. Intersection testing
We need to test for various intersections for lots of reasons
What object is the mouse hovering over (or your gun pointing at)? (Called picking)
Do these two objects collide?
Does a ray hit a bounding box?
As with everything in real time graphics, it needs to be efficient

8. Definitions

9. Bounding volumes
It's a challenging (and computationally expensive) task to see if a ray intersects with, say, a demon model
Instead, models are often surrounded by bounding volumes that are easy to test intersection against
If a ray or another model (or another model's bounding volume) intersects with a bounding volume, we can compute a more exact intersection
If not, it doesn't matter

10. Bounding volume examples
Bounding Sphere
Bounding Box
Bounding Cow
Easy to do intersection tests with
Poor fit for many rectangular objects
More costly intersection tests
Better fit for rectangular objects
No computational savings at all
Perfect fit for cow

11. Ray
A ray r(t) is defined by an origin point o and a direction vector d
d is usually normalized
Negative t values are behind the starting point of the ray and usually don't count
If d is normalized, positive t values give the distance of the point from o
It is also common to store l, the maximum distance along the ray we want to look

12. Surface
An implicit surface is one described by a vector equation where any point on the surface has a value of 0
f(p) = f(px, py, pz) = 0
Implicit sphere: f(p) = px2 + py2 + pz2 - r2 = 0
An explicit surface is one parameterized by two parameters
Explicit sphere:

13. Axis-aligned bounding box
An axis-aligned bounding box (AABB) (also known as a rectangular box) is a box whose faces have normals pointing the same way as the x, y, and z axes
It's a non-rotated box in 3 space
It can be defined with two points (lower corner and upper corner)

14. Oriented bounding box
An oriented bounding box (OBB) is an AABB that has been arbitrarily rotated
It can be described by
A center point bc
Three normalized vectors bu, bv, and bw giving the side directions of the box
And half-lengths (from center to wall) huB, hvB, and hwB

15. k-DOP
A k discrete oriented polytope (k-DOP) has k/2 normalized normals ni
For each ni, there are two values dimin and dimax which defines a slab Si that is the volume between the two planes
The k-DOP is the intersections of all the slabs

16. Separating axis test
For two arbitary, convex, disjoint polyhedra A and B, there exists a separating axis where the projections of the polyhedra are also disjoint
Furthermore, there is an axis that is orthogonal to (making the separating plane parallel to)
A face of A or
A face of B or
An edge from each polyhedron (take the cross product)
This definition of polyhedra is general enough to include triangles and line segments

17. Creating Bounding Volumes

18. AABB and k-DOPs An AABB is the easiest A k-DOP is not much harder
Take the minimum and maximum points in each axis and, BOOM, you've got an AABB
A k-DOP is not much harder
Take the minimum and maximum values in each of the k/2 axes and use those to define the slabs
You have to have axes in mind ahead of time

19. Bounding spheres Not as simple as you might think One approach: Or:
Make an AABB and use the center and diagonal of the corners to make your sphere
Or:
Make the AABB and do another pass through the vertices, taking the one furthest from the center as the radius
There are other more complicated ideas

20. Geometric probability
The relative probability that a random point is inside an object is proportional to the object's volume
However, the relative probability that a random ray intersects an object is proportional to its surface area

21. OBB Messiest yet! Find the convex hull of the object
Find the centroid of the entire convex hull
Compute a matrix using math from the book
The eigenvectors of the matrix are the direction vectors of the box
Use them to find the extreme points for each axis
There are other approaches

22. Rules of thumb
Perform computations early that could easily get a reject or accept
If possible, reuse results from previous tests
If you are using multiple tests, experiment with the order in which you apply them
Postpone expensive calculations (trig, square roots, and divisions) until you absolutely need them
Reduce the problem to a lower dimension whenever possible
If one ray or object is being tested against many others, try to compute whatever you can for that one item a single time
Expensive intersection tests should always come after some simpler BV test
Use empirical timing data to see what really works best
Write robust code, particularly with respect to floating point issues

23. Intersection Methods

24. Ray/sphere intersection
We can write the implicit sphere equation as f(p) = ||p – c|| – r = 0
p is any point on the surface
c is the center
r is the radius
By substituting in r(t) for p, we can eventually get the equation t2 + 2tb + c = 0, where b = d • (o – c) and c = (o – c) •(o – c) – r2
If the discriminant is negative, the ray does not hit the sphere, otherwise, we can compute the location(s) where it does

25. Optimized ray/sphere
Looking at it geometrically, we can optimize the test
Find the vector from the ray origin to the center of the sphere l = c – 0
Find the squared length l2 = l • l
If l2 < r2, then o is in the sphere, intersect!
If not, project l onto d: s = l • d
If s < 0, then the ray points away from the sphere, reject
Otherwise, use the Pythagorean theorem to find the squared distance from the sphere center to the projection: m2 = l2 – s2
If m2 > r2, the ray will miss, otherwise it hits

26. Ray/box intersection
Ray box intersection is a key element to have in your arsenal
Bounding boxes are a very common form of bounding volume
A ray/box intersection is often the first test you will use before going down deeper
There are a couple of methods
Slabs method
Line segment/box overlap test

27. Slabs method
First find the t value where the ray intersects each plane
The box is made up of 3 slabs
Find the min t and max t for each slab
The final tmin is the max of all the tmin values
The final tmax is the min of all the tmax values
If tmin ≤ tmax, the ray intersects the box, otherwise it does not
The idea can be extended to frustums and k-DOPs

28. Separating axis test
For two arbitary, convex, disjoint polyhedra A and B, there exists a separating axis where the projections of the polyhedra are also disjoint
Furthermore, there is an axis that is orthogonal to (making the separating plane parallel to)
A face of A or
A face of B or
An edge from each polyhedron (take the cross product)
This definition of polyhedra is general enough to include triangles and line segments

29. Line segment/box overlap test
This method uses the separating axis test and only works for AABBs and line segments
The AABB has its center at (0,0,0) and size half vector h
The line segment is defined by center c and half vector w
If |ci| > wi + hi for any i x,y,z then disjoint
There is another test for each axis that is the cross product of the x, y, and z axis and w
If any test passes, then disjoint
Only if all tests fail, then overlap

30. Triangle representation
One way to represent a triangle is with barycentric coordinates
For triangles, barycentric coordinates are weights that describe where in the triangle you are, relative to the three vertices
These weights are commonly labeled u, v, and w and have the following properties
u ≥ 0, v ≥ 0, w ≥ 0 and u + v + w ≤ 1

31. Ray triangle intersection
We represent a point f(u,v) on a triangle with the following explicit formula
f(u,v) = (1 – u – v)p0 + up1 + vp2
Then, setting the ray equal to this equation gives
o + td = (1 – u – v)p0 + up1 + vp2
This is simply a vector representation of three equations with three unknowns
If the solution has a positive t, and u and v between 0 and 1, it's an intersection

32. Ray/polygon intersection
First we compute the intersection of the ray and the plane of the polygon
Then we determine if that point is inside the polygon (in 2D)
The plane of the polygon is np • x + dp = 0
np is the plane normal
dp is the distance along the normal from the origin to the plane
Intersection is found by:
np • (o + td) + dp = 0
Then project onto xy, xz, or yz plane (whichever maximizes polygon area) and see if the point is in the polygon

33. Crossings test
If you want to see if a point is inside a polygon, you shoot a ray from that point along the positive x axis
If it intersects edges of the p0lygon an even number of times, it is outside, otherwise it is inside
Problems can happen if a ray intersects a vertex, so we treat all vertices with y ≥ 0 as being strictly above the x axis

34. Plane/box intersection
Simplest idea: Plug all the vertices of the box into the plane equation n • x + d = 0
If you get positive and negative values, then the box is above and below the plane, intersection!
There are more efficient ways that can be done by projecting the box onto the plane

35. MonoGame Intersection Tools

36. MonoGame primitives
There are a few basic primitives that MonoGame uses for intersection testing
Plane
Defined by a normal and a distance from the origin along that normal
Or you can make one with three points
Ray
Defined by a starting point and a direction vector
Vector3
Used (unsurprisingly) for points

37. MonoGame bounding volumes
BoundingBox
Axis aligned bounding box
Defined by two points
BoundingSphere
Defined by a center point and a radius
BoundingFrustum
Defined by a matrix (view x projection)

38. Upcoming

39. Next time… Finish intersection test methods
Bounding volume intersections

40. Reminders
Keep working on Project 4
Keep reading Chapter 16