Smooth Cameras and Euler's Method

A game camera glides toward a moving target. The game can't solve the motion exactly, so every frame it takes one small straight step. Those steps are Euler's method, and if a frame is too long they overshoot and blow up.

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How it works

Euler's method

In one sentence

Euler's method approximates $y' = f(t, y)$ by $y_{n+1} = y_n + hf(t_n, y_n)$ ; its error at a fixed time is proportional to $h$ , and for $y' = -ky$ it is stable only when $hk < 2$ .

General formula

\frac{d{\htmlData{sym=0}{y}}}{dt} = {\htmlData{sym=1}{k}}\left({\htmlData{sym=3}{\sin t}} - {\htmlData{sym=0}{y}}\right), \qquad {\htmlData{sym=0}{y}}_{n+1} = {\htmlData{sym=4}{y_n}} + {\htmlData{sym=2}{h}}\,{\htmlData{sym=1}{k}}\left({\htmlData{sym=3}{\sin t}}_n - {\htmlData{sym=4}{y_n}}\right)

Each step follows the tangent line for one frame. The true path curves during the frame, so the tangent drifts off it, a little more each step.

What the symbols mean

$y$
Camera position
Where the camera is along one axis.
$k$
Follow strength
How snappily the camera chases its target.
$h$
Frame time
Seconds per frame: the Euler step size.
$\sin t$
Target
The moving point the camera follows.
$y_n$
Position at frame n
The camera's position after n Euler steps.

A smooth camera moves toward its target at a speed proportional to the distance: $y' = k(\text{target} - y)$ .
Each frame the game takes one Euler step with $h$ equal to the frame time.
Euler uses the slope at the start of the frame, so the camera lags the true path; smaller frames mean smaller error.
If $hk > 2$ , each step overshoots by more than it corrects: the camera zigzags and flies off.

Learn the math

Euler's MethodDiff Eq · Lesson 12, First Order Differential Equations