Epitrochoid

By the end of this lab you'll be able to:

• Plot an epitrochoid — the curve traced by a pen on a circle rolling outside a bigger circle
• Explain how the sign difference between (R + r) and (R - r) separates outer rolling from inner rolling
• Predict the number of loops from the ratio R / r

A small circle of radius r rolls around the outside of a big circle of radius R, with a pen attached at distance d from the small circle's center. The pen traces flower-like loops called an epitrochoid — the outside-rolling cousin of the Spirograph's hypotrochoid.

Welcome to the Epitrochoid!

In the Spirograph lab a circle rolled around the inside of a ring. This time the little circle rolls around the outside — and just two sign changes in the math flip the whole pattern inside out!

How It Works

At time t, the small circle's center sits at distance R + r from the origin (the two circles touch, so their centers are one big radius plus one small radius apart). The pen swings around that moving center, giving:

x = (R + r) * cos(t) - d * cos((R + r) / r * t) y = (R + r) * sin(t) - d * sin((R + r) / r * t)

Compare with the Spirograph (hypotrochoid): inner rolling uses (R - r) and + d in the x line. That sign difference is the entire distinction between rolling inside and rolling outside. The curve makes R / r loops before it closes.

Sample Code

import turtle
import math
monty = turtle.Turtle()
monty.speed(0)
monty.hideturtle()

R = 100
r = 20
d = 40
steps = 1200
colors = ['crimson', 'darkorange', 'gold', 'forestgreen', 'royalblue']

monty.pensize(2)
monty.penup()
for i in range(steps + 1):
    t = 2 * math.pi * i / steps
    x = (R + r) * math.cos(t) - d * math.cos((R + r) / r * t)
    y = (R + r) * math.sin(t) - d * math.sin((R + r) / r * t)
    monty.pencolor(colors[(i * 5) // (steps + 1)])
    if i == 0:
        monty.goto(x, y)
        monty.pendown()
    else:
        monty.goto(x, y)

What Do You Think Will Happen?

The curve makes R / r loops before it closes. With R = 100 and r = 20, how many loops will you see? Make your guess, then click Run!

Try It Now

▶ Run ↻ Reset

5 loops — R / r = 100 / 20 = 5, and each loop gets its own color. Were you right?

How It Works

Every time the small circle rolls once around its own center, the pen swings out and back — that swing is one loop. Rolling around the outside of the big circle takes exactly R / r = 5 of those swings, so the curve closes after 5 loops. The pen distance d controls how dramatic each loop is: d bigger than r gives big open loops, d smaller than r gives gentle bumps, and d equal to r gives sharp points (that special case is called an epicycloid).

Explanation Table

Line	What it does
x = (R + r) * math.cos(t) - d * math.cos((R + r) / r * t)	Center of the rolling circle plus the pen's swing — (R + r) means rolling outside
d = 40	Pen offset from the small circle's center; bigger d means bigger loops
(R + r) / r * t	How fast the pen spins — the rolling circle turns 6 times per trip around
colors[(i * 5) // (steps + 1)]	Splits the 1200 steps into 5 color bands, one per loop

Learning Check

Your Turn — Roll the Circle Inside

The program below uses the same R, r, and d, but this is the Spirograph (inner-rolling) formula. Run it and compare with the curve above. Then convert it to the outer-rolling epitrochoid: change every (R - r) to (R + r) and the + d in the x line to - d. Which way will the loops point after your edit?

▶ Run ↻ Reset

The inner-rolling curve tucks its 5 loops inward and stays small; after your sign changes the loops flip outward and the whole curve grows — same three numbers, opposite rolling.

Experiments

1. Change r = 20 to r = 25. Now R / r = 4. You'll know it worked when the curve closes with exactly 4 loops.

2. Set d = 20 (equal to r). You'll know it worked when the round loops sharpen into 5 pointed cusps — an epicycloid.

3. Try d = 10 (smaller than r). You'll know it worked when the loops flatten into 5 gentle bumps with no crossing.

4. Set r = 30 so R / r is not a whole number. Change 2 * math.pi to 6 * math.pi so the curve has time to close. You'll know it worked when 10 loops weave around before the curve meets itself.

Rolling On the Outside!

You just discovered that one plus-or-minus sign decides whether a circle rolls inside or outside — and completely changes the picture. Up next: Superellipse — one exponent that morphs a star into a circle into a rounded square.