Calculate gears rotation for a realtime simulation

Question

Hi I'm trying to do a game with real time simulations of gears. There is a big Gear with inside a smaller gear.

I managed to draw gears with different diameters but equal size teeth, but if i try to move the smaller one inside the bigger one the movement is odd.

see the animated gif.

the biggest gear is in center C1 and the small in the center C2. I calculate C2 position in this way:

C2.x = C1.x + C1_RADIUS-C2_RADIUS) * cos(t);
C2.y = C1.y - C1_RADIUS-C2_RADIUS) * sin(t);

for t that goes from 0 to TWO_PI in n steps.

I apply as rotation the angle t, but maybe it is wrong and i have to calculate another rotation for get a perfect joint.

EDIT:

I am using radians for rotations and i think this is OK.

My main problem is that i want to achive a perfect joint between teeth like this example. My teeth are overlapping in some odd way.

EDIT

a working example can be found here: http://www.openprocessing.org/sketch/83665

Answer 1

The large gear has circumference 2πR1 and the small one has circumference 2πR2, so when the small gear has made a full circle, it has turned R1/R2 times around itself, minus one time because it turned inside the large gear (it would be plus one if the gear was outside).

So when the centre of the inner gear is rotated by angle t, the gear itself needs to be rotated by angle (R1/R2-1)t.

If the values you handle are tooth counts N1 and N2 you can replace R1/R2 with N1/N2.

There is also the issue of the initial rotation. Let a be the angle value atan2(C2.y - C1.y, C2.x - C1.x). Apply initial rotation a to gear 1. If gear 2 is inside, apply rotation a to it, too. If gear 2 is outside, apply a π+a+h rotation where h = π/(2*N2) is the half angle of a tooth.