Epicycloid Star

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

• Plot an epicycloid using x = (R+r)cos(t) - r*cos((R+r)/r * t)
• Understand how R/r (an integer) gives the number of cusps
• Compare epicycloids (outer rolling) to hypotrochoids (inner rolling)

An epicycloid is traced by a point on a small circle rolling around the outside of a larger circle. When R/r is a whole number, the result is a star-like shape with exactly R/r pointed cusps.

Welcome to the Epicycloid Star!

While the Spirograph rolls a circle inside, the epicycloid rolls a circle outside! When the ratio is a whole number, the result is a perfect star shape. Let's code it together!

How It Works

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

R is the base circle radius, r is the rolling circle radius. When R/r = k (an integer), the curve has k cusps.

Sample Code

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

R = 100
r = 20
steps = 2000
colors = ['crimson', 'darkorange', 'gold', 'forestgreen', 'royalblue']

monty.pencolor('crimson')
monty.pensize(2)
monty.penup()

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

What Do You Think Will Happen?

R/r = 100/20 = 5 (an integer). The rule says we get R/r cusps — how many? Make your guess — then click Run to find out!

Try It Now

▶ Run ↻ Reset

5 pointed cusps — a 5-pointed epicycloid star. Were you right?

How It Works

Each cusp is formed when the rolling circle's contact point momentarily stops (the speed of the point equals zero). This happens R/r times per revolution. At each stop, the curve makes a sharp outward point — a cusp.

Explanation Table

Line	What it does
R = 100, r = 20	R/r = 5 → 5 cusps
(R + r) * cos(t)	Position of rolling circle center
r * cos((R+r)/r * t)	Point on the rolling circle
(R+r)/r * t	Rolling circle's own rotation angle

Learning Check

Your Turn — Change to 7 Cusps

Change r = 20 to make R/r = 7. What value of r gives 7 cusps from R=100? Hint: 100 / r = 7, so r = ?

▶ Run ↻ Reset

R=105, r=15 → R/r = 7 → 7 cusps. A 7-pointed epicycloid star!

Experiments

1. Try a non-integer ratio. Use R=100, r=30. The curve won't close in one revolution. You'll know it worked when an irregular looping shape appears.

2. Draw a cardioid. Use R=80, r=80 (ratio 1:1). A cardioid has just one cusp and looks like a heart. You'll know it worked when one cusp appears.

3. Overlay multiple epicycloids. Use R=100 and loop over r values 10, 20, 25, 50 with different colors. You'll know it worked when 4 different star shapes overlap.

4. Compare hypo vs epi. Draw both a hypotrochoid (Spirograph) and an epicycloid on the same canvas. You'll know it worked when both types of curve appear together.

Stars from Rolling Circles!

You built a family of star curves from rolling circle geometry! These epicycloids appear in planetary gear systems and the design of rotary engines. Up next: Category 8 — Optical Illusions!