A Beautiful Trick to Avoid Trig 😄

A pulley, a slider, and a small constraint that quietly removes trigonometry from the problem.

Three years ago, Kenneth and I built one of my favorite iOS mini-projects: a pulley system with two wheels spinning in sync as a slider moves.

The challenge was simple but subtle:

How do we make circles rotate naturally without using sin or cos at all?

We didn’t want trigonometry in the implementation — not even once.

So we asked a different question:

Can rotation come purely from rope length?

The answer turned out to be yes.

The key idea

The slider controls a single quantity:

Then everything follows from one geometric identity:

So we rearrange it:

No sine. No cosine. Just length and ratio.

The mechanism

As the rope moves:

• we measure how much rope is displaced
• divide by each wheel’s radius
• use that angle to rotate the colored sectors

What looks like trigonometry is actually just bookkeeping of arc length.

Result

Kenneth recorded a 19-second clip: two wheels spinning in perfect sync as a floating slider moves up and down. The sectors glide smoothly like a tiny physics system on iOS.

🌱 Dream Team
Small circle. Big dreams.

11/17/2025