Topographic Map

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

• Plot closed contour rings in polar coordinates with math.sin() and math.cos()
• Perturb a circle's radius with sine waves to make natural, wobbly coastlines
• Paint elevation bands by filling rings from the outside in

Real hikers' maps show mountains as nested contour rings: each ring is a line of equal elevation, and the colors march from lowland green through brown rock up to a white summit. This lab draws a whole island that way — eight wobbly rings, filled from the outside in so each higher elevation paints on top of the last.

Welcome to the Topographic Map!

Map makers squash 3D mountains onto flat paper using contour lines — and you're about to be the cartographer. Grab your compass: we're drawing an island that never existed, one elevation at a time!

How It Works

Each ring is a circle in polar coordinates whose radius wobbles as the angle sweeps around: r = base_r * (1 + 0.10*sin(3*theta) + 0.06*sin(7*theta + 1)). Adding two sine waves of different frequencies (3 and 7) makes the coastline look natural instead of perfectly round, and because the wobble is a percentage of base_r, the bumps are big on the outer rings and gentle near the peak — and the rings can never cross. The loop draws the biggest ring first and works inward, so each smaller filled ring paints over the one below it, exactly like stacking elevation layers.

Sample Code

import turtle
import math
monty = turtle.Turtle()
monty.speed(0)
monty.hideturtle()
screen = turtle.Screen()
screen.bgcolor('lightsteelblue')

colors = ['darkgreen', 'forestgreen', 'yellowgreen', 'darkkhaki', 'peru', 'sienna', 'rosybrown', 'white']

def contour_r(base_r, theta):
    wobble = 0.10 * math.sin(3 * theta) + 0.06 * math.sin(7 * theta + 1)
    return base_r * (1 + wobble)

steps = 90
for ring in range(8):
    base_r = 160 - ring * 18
    monty.penup()
    monty.pencolor('saddlebrown')
    monty.fillcolor(colors[ring])
    monty.pensize(1)
    for i in range(steps + 1):
        theta = 2 * math.pi * i / steps
        r = contour_r(base_r, theta)
        x = r * math.cos(theta)
        y = r * math.sin(theta)
        if i == 0:
            monty.goto(x, y)
            monty.pendown()
            monty.begin_fill()
        else:
            monty.goto(x, y)
    monty.end_fill()

What Do You Think Will Happen?

The wobble formula includes math.sin(3 * theta), the strongest wiggle. As you travel once around the island's coastline, how many big bulges (peninsulas) will you pass — 3, 7, or 10? Make your guess, then click Run!

Try It Now

▶ Run ↻ Reset

3 big peninsulas — sin(3 * theta) repeats 3 times per lap, and the faster sin(7 * theta) just adds smaller ripples on top. Were you right?

How It Works

Reading the color list tells the whole elevation story: ring 0 (the largest, drawn first) is 'darkgreen' lowland, and each ring inward steps through lighter greens, khaki, and browns until ring 7 is the 'white' summit. This works only because of paint order — turtle fill has no idea about "height", but filling outside-in means every smaller ring covers the middle of the bigger one, leaving just a colored band showing. Real topographic maps use the same trick: the band between two contour lines gets one elevation color. The blue bgcolor plays the ocean around the outermost coastline.

Explanation Table

Line	What it does
wobble = 0.10 * math.sin(3 * theta) + 0.06 * math.sin(7 * theta + 1)	Two sine waves make a bumpy but smooth coastline
return base_r * (1 + wobble)	Wobble scales with ring size, so rings never cross
base_r = 160 - ring * 18	Each ring is 18 pixels smaller — outside in, low to high
monty.fillcolor(colors[ring])	Elevation bands: green shores, brown slopes, white peak

Learning Check

Your Turn — Reshape the Island

In the program below, math.sin(3 * theta) has been changed to math.sin(6 * theta). Predict: how many big peninsulas will the new coastline have? Run it and count them!

▶ Run ↻ Reset

Six peninsulas — the first number inside sin(n * theta) is exactly how many times that bump pattern repeats around the island.

Experiments

1. Make it craggier. Change 0.10 to 0.20. You'll know it worked when the coastline grows dramatic fjords and the rings pinch close together.

2. Shift the terrain. Change math.sin(7 * theta + 1) to math.sin(7 * theta + 4). You'll know it worked when the small ripples slide around to different spots on the coast.

3. Deepen the ocean. Change screen.bgcolor('lightsteelblue') to screen.bgcolor('midnightblue'). You'll know it worked when the island glows against a dark night-chart sea.

4. Flip the elevation. Reverse the color list (put 'white' first and 'darkgreen' last). You'll know it worked when the map becomes a snowy ring-shaped crater with a green valley floor at the center.

Master Cartographer!

Polar coordinates, layered sine wobbles, and outside-in painting — you turned math into a map worthy of an explorer's backpack. Browse the Turtle Graphics Gallery for your next challenge!