Rose Curve

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

• Plot a rose curve using r = cos(k * theta) in polar coordinates
• Understand how odd k gives k petals and even k gives 2k petals
• Convert polar (r, theta) to Cartesian (x, y) for plotting

A flower-like curve made from the simple equation r = cos(k*θ). Changing k changes the petal count — odd k gives k petals, even k gives 2k petals.

Welcome to the Rose Curve!

One short equation produces these petal patterns: r = cos(k * theta). Math at its most beautiful — a whole flower from three characters! Let's code it together!

How It Works

For theta from 0 to 2π (or 4π for even k), compute r = scale * cos(k * theta). Convert to Cartesian: x = r * cos(theta), y = r * sin(theta). Connect the points to trace the petals.

Sample Code

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

k = 5
scale = 140
steps = 1000
colors = ['crimson', 'darkorange', 'gold', 'forestgreen', 'royalblue']

for petal in range(k):
    monty.pencolor(colors[petal % len(colors)])
    monty.penup()
    monty.pensize(2)

monty.pencolor('crimson')
monty.penup()
for i in range(steps + 1):
    theta = 2 * math.pi * i / steps
    r = scale * math.cos(k * theta)
    x = r * math.cos(theta)
    y = r * math.sin(theta)
    if i == 0:
        monty.goto(x, y)
        monty.pendown()
    else:
        monty.goto(x, y)

What Do You Think Will Happen?

With k=5 (odd), the rule says we get k petals. How many petals is that? Make your guess — then click Run to find out!

Try It Now

▶ Run ↻ Reset

5 petals — odd k gives exactly k petals. Were you right?

How It Works

When cos(k*theta) < 0, r is negative, which in polar coordinates places the point 180° opposite. This doubles back through the center creating the petal shapes. For even k, the 180° reflection creates an additional set of petals.

Explanation Table

Line	What it does
r = scale * math.cos(k * theta)	Polar equation of the rose curve
x = r * math.cos(theta)	Convert polar r, theta to x
y = r * math.sin(theta)	Convert polar r, theta to y
k = 5	Odd: 5 petals; try k=4 for 8 petals

Learning Check

Your Turn — Try k=4 (even)

Change k = 5 to k = 4. Even k gives 2k petals — so how many petals will appear? Also change range to range(steps * 2) and divide by steps * 2 for even k. Predict, then run it!

▶ Run ↻ Reset

k=4 → 8 petals (2×4). Even-k rose curves need theta to go from 0 to 4π to complete all petals.

Experiments

1. Try k=3. You'll know it worked when 3 petals appear in a triangle arrangement.

2. Use sin instead of cos. Replace math.cos(k * theta) with math.sin(k * theta). You'll know it worked when the petals rotate 90°/k.

3. Draw all odd k from 1 to 9. Use a loop and different colors. You'll know it worked when 5 overlapping rose curves appear.

4. Fill the petals. Call begin_fill() and end_fill() around the drawing loop. You'll know it worked when the petals are filled in.

Blooming with Math!

One equation — r = cos(k*theta) — draws all these flower patterns! This is the beauty of polar coordinates: complex shapes from simple equations. Up next: Spirograph — rolling circles inside circles.