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During the past month, I have found myself in the position of having to explain the contents of this article to six different persons, either at work or over the Internet. Though there are a lot of articles on the subject, it’s still as if almost everyone gets it wrong.

I was still polishing this article when I had the opportunity to explain it. And two days ago a coworker told me the source code of a certain framework disagreed with me The kind of framework that probably has three NDAs preventing me from even thinking about it. Well that framework got it wrong, too. So now I’m mad at the entire world for no rational reason other than the ever occurring realisation of the amount of wrong out there, and this article is but a catharsis to deal with my uncontrollable rage.

A simple example Imagine a particle with position Pos and velocity Vel affected by acceleration Accel. Let’s say for the moment that the acceleration is constant. This is the case when only gravity is present. A typical game engine loop will update position with regards to a timestep (often the duration of a frame) using the following method, known as Euler integration.

Why this is wrong When doing, computing position from acceleration is an integration process. First you integrate acceleration with respect to time to get velocity, then you integrate velocity to get position. The integral of a function can be seen as the area below its curve.

So, how do you properly get the integral of our velocity between t and t + dt, ie. The green area below? It’s not by doing new_velocity * dt (left image).

It’s not by doing old_velocity * dt either (middle image). It’s obviously by doing (old_velocity + new_velocity) * 0.5 * dt (right image). And now for the correct code This is what the update method should look like.

It’s called Velocity Verlet integration (not strictly the same as Verlet integration, but with a similar error order) and it always gives the perfect, exact position of the particle in the case of constant acceleration, even with the nastiest framerate you can think of. Even at two frames per second. Well I am glad you asked. And are part of a family of iterative methods known as the, respectively of first order and second order.

There are many more for you to discover and study. • Richard Lord did this nice and instructive about several integration methods. • Glenn Fiedler also explains in why idiots use Euler, and provides a nice introduction to RK4 together with source code. • Florian Boesch did a of various integration methods for the specific application of gravitation (it is one of the rare cases where Euler seems to actually perform better). In practice, Verlet will still only give you an approximation of your particle’s position. But it will almost always be a much better approximation than Euler. If you need even more accuracy, look at the method.

Your physics will suck a lot less, I guarantee it. Acknowledgements I would like to thank everyone cited in this article, explicitly or implicitly, as well as the commenters below who spotted mistakes and provided corrections or improvements. • Posted: 2011-12-14 00:39 (Updated: 2011-12-15 14:34) • Author: • Categories.

Hi Sam, I did, not a long time ago, a small survey on popular explicit integration methods. What I found in terms of terminology (unintended wordplay) was that your 'Velocity Verlet' is actually the same as the 'midpoint method'. You can find some of these popular integrators mentioned in Appendix A of this paper ->.


As I understood it, Velocity Verlet requires using a velocity prediction to find the future position, then correct/update the velocity to obtain a somewhat more synchronized pair (pos, vel). Thus, Velocity Verlet would evaluate the acceleration twice per step: the first time to get the position, then to get the corrected velocity using the updated position and the velocity prediction. Is this view correct, or have I misunderstood it badly?

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