For some reason, information on collision detection & response is quite sparse compared to all the other subjects. Worse yet, there are some bad information out there that are praised to be good resources. Pieces of information are also often discoherent, such as resources on collision detection says nothing about collision response and vice versa. It took me a while to find good information on this subject, and I have finally implemented a working collision system that works reasonably well for games.
A new collision object representation
This time, I chose convex polyhedrons to be the collision representation for the small objects in my game, such as trees, rocks, and so on. Concave objects, such as houses, are broken down into convex polyhedrons. I am aware of some of the convex decomposition techniques, but I didn’t want to do them, as I think that would take too much time. Instead, I manually decompose concave mesh into convex polyhedrons in Blender.
As for terrain, I use a completely different representation. This time, I render the terrain from top down in orthographic view, then store away the depthmap into some texture I can easily use later. Therefore, querying terrain height at any arbitrary point becomes very fast.
Unlike last time, we don’t have the fortune to have a uniform representation for all collidables in our game world, so we have to write two separate solutions. However, it is widely more efficient compared to last time. Instead of feeding huge numbers of polygons into the collision system, I can approximate them with cheap convex shapes now. As for terrain, it just becomes four texture reads.
First, let’s talk about how per-object collision works.
GJK is a powerful technique that queries the minimal distance between two convex objects, and it is also what I mainly use for my per-object collision system. I will only briefly summarize it here since there are already really good information on this topic, which I will link later.
First, imagine convex objects to be made of infinite small points. If a convex object A, and a convex object B are considered colliding, that must mean some of the points in this space belong to both A and B. If we subtract all points belong to B from all points belong to A, we must be subtracting a small subset of points from themselves. Therefore, some of these subtractions result in zero.
Imagine the subtractions we did form a new shape, C. For A and B to intersect, C must contain the origin. That is due to some of the subtractions resulting in zero. If A and B do not intersect, then C does not contain origin.
Now, to check if A and B intersect, we can check if C contains the origin. This is the essence of GJK; we have reduced the problem to checking whether or not the origin is contained by some convex shape.
How do we obtain C? Do we have to subtract all the points from B by all the points from A? No. If we can find support functions for A and B, then we can combine them to be the support function for C.
Given the support function of C, we can use it to map directions to extreme points in C. It can also be shown that for any convex shape, if it contains an origin, a tetrahedron made up of four points from the convex shape can capture the origin (meaning contain the origin). That means, we can use the support functions to find extreme points, and try to let these extreme points form a tetrahedron that contains the origin. That is what GJK does.
GJK does this by evolving a group of points into that tetrahedron that would capture the origin, we will call this group of points a simplex. For the 3D case, a simplex can be a point, a line segment, a triangle, or a tetrahedron.
In each iteration of GJK, it finds which voronoi region the origin is in relative to the simplex, then keep only the subset of that simplex which is contained by that voronoi region’s feature, and then change the search direction towards that voronoi region.
We can check the progress by finding the closest point on the simplex to the origin. Functions that do this, such as ClosestPointOnLineSegmentToPoint(), ClosestPointOnTriangleToPoint(), and ClosestPointonTetrahedronToPoint(), are well covered in Ericson’s book “Real Time Collision Detection”. The distance from this closest point to the origin tells us our progress. If this closest point stops getting closer, that means we will never be able to capture the origin. In that case, we return the distance as our minimal distance.
I know in Casey’s lecture on GJK, the termination condition is quite different. He simply checks if the extreme point in the search direction can get to the other side of the origin. That is enough for determining whether or not two objects collide, but it is not enough to find the minimal distance and the closest point on the simplex, which is vital to us, since we will need these information to resolve collision. Furthermore, some of the interesting optimizations I did will require GJK to reach the simplex that has the minimal distance in order to work.
Lastly, a word of warning. What I said above is purely in theory. There is more to termination condition in 3D GJK, such as clever ways of checking progression. I always find it reassuring to actually keep a minimal distance every iteration. Furthermore, you will also need to watch out for degenerate simplices, since most algorithms that determines a point’s voronoi regions will have unexpected behaviors if a simplex is degenerate, such as when a triangle becomes collinear and a tetrahedron becomes collinear in certain faces or coplanar. These are indications that GJK cannot make further progresses, because the new points is still within the current simplex.
In addition, a naive GJK implementation doesn’t work well with quadric shapes, such as spheres or ellipsoids, which are often shapes used to represent a player’s collision volume. In order for GJK to work for them, a tolerance value must be added to the progression check.
More details on GJK, read Erin’s talk.
Accounting for movement in GJK
As I described. GJK is an algorithm for computing the minimal distance between two convex shapes. It cannot be directly used as a collision detection algorithm in games, since it doesn’t account for movement.
One simple trick in Ericson’s book can change that, though. Keep in mind that GJK only requires the geometry’s support function to work, it does not care about the explicit definition of that geometry. We can turn one of the object to be swept across space by its motion by changing the support function for that object.
support function only returns points from B if search direction D is along motion vector
Consider the above swept sphere. The support function of the convex hull of the moving sphere will only return points from sphere B if the search direction is along the motion vector. In other words, we can simply do a dot product between the search direction and motion vector and use it to determine whether to use A’s support mapping or B’s support mapping. This effectively gives us a support function for a swept geometry.
Incorporating GJK into game’s collision detection
In my game, I am using a capsule to approximate player’s collision volume and convex polyhedrons for other geometries in the world. There are two pieces of information that we need to handle collision correctly each frame. One is time-of-impact (TOI) and contact point. We use TOI to stop the player right before collision and use contact point to compute the normal plane, which is then used to reflect player’s motion in order to achieve “gliding”.
My use of capsule is intentional, as it simplifies things a lot.
To obtain time of impact, one GJK isn’t enough. It only tells me whether my moving capsule will collide with any object, but doesn’t say when it will collide with them. As recommended by Ericson in his book, bisection method can be used here to obtain a relatively accurate TOI. The concept of bisection method is simply halving the interval to ensure that the margin within the interval is where the possible TOI is. It is analogous to binary search.
finding TOI with bisection method
However, as I have stated before, quadric shapes don’t play well with GJK. We can simplify the problem a lot here by reducing the capsule into its inner line segment. This is inspired by Randy Gaul’s answer to my gamedev.net post. His suggestion is reduce the capsule to a line segment in order to compute contact point, but I realized it can also work in collision detection phase.
I am not sure if this is the canonical definition of a capsule, but the “capsule” that I am using is defined by two parameters: the inner line segment and a radius. The inner line segment defines its height, and the radius expands that line segment out radially so it actually has a volume.
Given the above definition, it is clear that an object can only intersect with the capsule if the minimal distance between the inner line segment and the object is less or equal to the capsule’s radius. Therefore, instead of feeding a capsule into GJK, I can instead feed its inner line segment to GJK, and then compare the capsule’s radius with the returned minimal distance to detect collision.
Obtaining contact point
After determining TOI, we move the capsule forward by that amount, and we now determine the contact point between capsule and the closest object. As Randy Gaul suggested, the capsule is now reduced to line segment to find the contact point. That gives the advantage of only having to deal with non-quadric shapes and expect a non-tetrahedron simplex in the last iteration of GJK, since it’s impossible for the two to intersect when the radius of the capsule is non-zero, and our capsule’s radius is obviously non-zero.
But how do we compute closest point on either the capsule or the colliding object? Think back to the GJK algorithm. If the two objects are non-intersecting (which is obviously the case here), GJK is terminated by detecting a lack of progression (namely, comparing the minimal distance with the last iteration). Recall that we keep a closest point on the simplex to origin. This is not the closest point we need, though, as it is a point belong to the minkowski difference instead of the two original convex shapes. At the last terminating iteration, we convert the point into its barycentric form relative to the simplex. Because the current closest point on the simplex is a direct subtraction between A and B, we must be able to find two simplices on A and B that correspond to the end minkowski simplex as “source”, with the same dimension.
there must be a 1 to 2 mapping for simplex kept during GJK
In order to find the corresponding simplex in A or B, we have to keep track of the source points when we construct the simplex in GJK. By retrieving these points and applying the barycentric coordinates to them, we can obtain the closest point on either A or B.
After obtaining the contact point, the rest is old news; reflecting the motion back towards the collision normal so that the resultant motion vector is along the collision normal plane, and all that goodies.
That is all for player vs object collision detection & response routine. I wanted to include terrain collision in this post as well, but it’s getting quite long, so I will save it for next time. And as always, a video of the working collision system. The pink spheres are debug drawings of the closest point computed by this algorithm: