Astroid

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

• Plot the astroid using its parametric form x = R*cos³(t), y = R*sin³(t)
• Draw a "string art" family of line segments whose envelope is the astroid
• Explain what a curve envelope is and how a ladder sliding down a wall traces one

The astroid is a four-cusped star — the path traced by a point on a small circle rolling inside a circle four times its size (a hypocycloid, cousin of the Spirograph). It hides a beautiful secret: if a ladder slides down a wall, the ladder's positions outline the astroid. We'll draw both the star and the sliding "strings" that reveal it.

Welcome to the Astroid!

Cube a cosine, cube a sine — get a four-pointed star! Then we'll overlay classic string art: dozens of straight lines that magically outline the same curved star. No curve is drawn by the strings, yet the star appears. Ready?

How It Works

First the string art: draw straight segments from (R*cos(a), 0) on the x-axis to (0, R*sin(a)) on the y-axis. Every segment satisfies a² + b² = R² — like a ladder of fixed length R sliding down a wall. Then the star itself: for t from 0 to 2π, plot x = R * cos(t)**3 and y = R * sin(t)**3. The cubing pulls points toward the axes, pinching the circle into four sharp cusps — and that curve is exactly the envelope the strings outline.

Sample Code

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

R = 170

monty.pencolor('silver')
monty.pensize(1)
n_lines = 40
for i in range(n_lines + 1):
    a = 2 * math.pi * i / n_lines
    monty.penup()
    monty.goto(R * math.cos(a), 0)
    monty.pendown()
    monty.goto(0, R * math.sin(a))

monty.pencolor('crimson')
monty.pensize(3)
steps = 400
monty.penup()
for i in range(steps + 1):
    t = 2 * math.pi * i / steps
    x = R * math.cos(t) ** 3
    y = R * math.sin(t) ** 3
    if i == 0:
        monty.goto(x, y)
        monty.pendown()
    else:
        monty.goto(x, y)

What Do You Think Will Happen?

The star comes from cos(t)**3 and sin(t)**3. How many sharp points (cusps) will it have? Count the places where cosine or sine hits 1 or -1... guess, then click Run!

Try It Now

▶ Run ↻ Reset

4 cusps — one on each axis, at (±R, 0) and (0, ±R), exactly where cosine or sine reaches ±1. Were you right?

How It Works

None of the silver segments is curved, yet together they outline the star. Each segment is tangent to the astroid — it kisses the curve at exactly one point without crossing it. A curve that every member of a family of lines touches like this is called the family's envelope. The sliding-ladder picture explains the physics: as a ladder of length R slides from upright to flat, its midsection sweeps through the region under the curve, but its outer boundary — the farthest the ladder ever reaches — is the astroid. The cubes in cos³ and sin³ come from the double-angle algebra of a circle of radius R/4 rolling inside a circle of radius R.

Explanation Table

Line	What it does
x = R * math.cos(t) ** 3	The astroid's parametric form — cubing pinches the circle into cusps
monty.goto(R * math.cos(a), 0)	One end of the "ladder" on the x-axis
monty.goto(0, R * math.sin(a))	The other end on the y-axis — the two ends always satisfy a² + b² = R²
n_lines = 40	How many ladder positions to draw; more lines make a smoother envelope

Learning Check

Find the Bug

This program is supposed to draw the four-pointed star — but it draws a plain straight line instead! Look carefully at the exponents. Find the bug and fix it so the star appears.

▶ Run ↻ Reset

The exponents say ** 2 instead of ** 3. Squares are never negative, and since cos² + sin² = 1 every point lands on the straight line x + y = R — change both exponents to 3 and the star returns.

Experiments

1. Use more strings. Change n_lines = 40 to n_lines = 120. You'll know it worked when the silver web becomes so dense the envelope looks like a smooth gray curve.

2. Try a different hypocycloid. Replace the star loop's x and y with x = R * (0.667 * math.cos(t) + 0.333 * math.cos(2 * t)) and y = R * (0.667 * math.sin(t) - 0.333 * math.sin(2 * t)). You'll know it worked when a three-cusped star (the deltoid) appears instead.

3. Color the quadrants. Before drawing each string, set the pencolor based on i (for example silver for i < 10, lightblue for i < 20, and so on). You'll know it worked when each quadrant's string fan has its own color.

4. Cube higher. Change both ** 3 to ** 5. You'll know it worked when the star's arms become even skinnier and the cusps sharper — higher odd powers pinch harder.

Star of the Gallery!

You drew a curve two ways at once: directly from cos³ and sin³, and as the envelope of forty straight lines — and they matched perfectly. That's real mathematics made visible. Browse the Turtle Graphics Gallery for your next challenge!