While building my physics engine from scratch, understanding continous collision detection wasn't an easy task. Especially finding the resources since there are few resources about the practical side of the topic. Most of the resources are highly academical and aims audiance from the researchers. After spending several days on it, here are the insights actually helped me.

**CCD is a detection problem, not a resolution problem.**

This sounds obvious but end of the first day i realized that i lost my sight of it instead i found myself trying to understand both aspects in the same time. So don't forget that you're trying to detect collisions "between" discrete timesteps not how to respond to it. Resolution comes after.

**The core idea is casting something and finding intersection time with another something.**

For sphere-sphere CCD, you need to find a time within your timestep where the collision is expected to occur. Maybe the first thing that comes to mind that casting a dynamic sphere to a dynamic sphere and finding the intersection but it's hard to solve.

So you simplify: calculate the relative velocity between the two spheres, treat sphere B as static, and reduce sphere A to a point (a ray). Then using the Minkowski Sum, you expand sphere B's radius by sphere A's radius, giving you a sphere C. Now the problem is just ray-sphere intersection, which is straightforward.

But the math gives us parametric time results like t = 0 for ray origin and t=1 for the furthest ray distance point. Like if you shoot the ray from A to B then it’s at A when t = 0 and at B at t = 1. If it’s collide a sphere at some point in between it returns to us a t value in \[0,1\].

This is completely irrelevant for our physics engine because we need exact time not some ratio to calculate exact collision points and also sorting the collisions by their time otherwise we could resolve collisions in wrong order and we can observe ghost collisions or unexpected collisions.

So we scale the t1 and t2 result of the intersection test by dt.

Okay we found the collision times in two seperate frames now what? How do we manage this fragmented situation?

**We do time slicing!**

We’re in a frame.

* As usual apply forces or impulses.
* Test contacts. (we know until this point)
* Now, we need an contact array to sort the contacts. So add found contacts to this array from the previous step.
* Sort contacts.
* Here’s the hard part: We have some contacts and their times that would happen before the next frame but not in this frame that we need to handle.
* start a stopwatch.
* So for each contact integrate all of the bodies to that “contact time” and resolve only the current contact. Add the passed time to the stopwatch.
* if the remaining time(dt - stopwatch time) is bigger than 0 then update all of the bodies by the remaining time.

Well, that's it, i'm not an expert at all but I feel happy if it helped even a little bit. :)

if you're interested further I'm documenting my 3d physics engine development journey on my devlog. [https://derkkek.github.io/posts/cacti3d-Physics-Engine/](https://derkkek.github.io/posts/cacti3d-Physics-Engine/)

CCD is hard, here's my understanding and hope it helps :)