Game Engine Architecture

638 12. Collision and Rigid Body Dynamics

12.4.4.1. Explicit Euler

One of the simplest numerical solutions to an ODE is known as the explicit

Euler method. This is the intuitive approach oft en taken by new game program-

mers. Let’s assume for the moment that we already know the current velocity

and that we wish to solve the following ODE to fi nd the body’s position on

the next frame:

vr()t=().t (12.4)

Using the explicit Euler method, we simply convert the velocity from meters

per second into meters per frame by multiplying by the time delta, and then

we add “one frame’s worth” of velocity onto the current position in order to

fi nd the new position on the next frame. This yields the following approxi-

mate solution to the ODE given by Equation (12.4):

r rv() () () .tt^211 =+t^ Δt (12.5)

We can take an analogous approach to fi nd the body’s velocity next frame

given the net force acting this frame. Hence, the approximate explicit Euler

solution to the ODE

is as follows:

(12.6)

Interpretations of Explicit Euler

What we’re really doing in Equation (12.5) is assuming that the velocity of the

body is constant during the time step. Therefore, we can use the current velocity

to predict the body’s position on the next frame. The change in position Δr be-

tween times t 1 and t 2 is hence Δr = v(t 1 ) Δt. Graphically, if we imagine a plot of the

position of the body versus time, we are taking the slope of the function at time

Δr

Δt

t

r(t 1 )

rapprox(t 2 )

r(t 2 )

r(t)

t 1 t 2

slope

ΔΔrt= v(t 1 )

Figure 12.23. In the explicit Euler method, the slope of r(t) at time t 1 is used to linearly

extrapolate from r(t 1 ) to an estimate of the true value of r(t 2 ).

a()t F ()t v()

m

==net t

vv() ()t 21 =+t Fnetm()t^1 Δt.