Cassini Oval

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

• Plot Cassini ovals — points where the product of distances to two foci equals b * b
• Use the polar form with a square-root guard to handle values that do not exist
• Watch a single constant b change the curve's whole topology: one oval, a figure-eight, or two separate lobes

An ellipse keeps the sum of distances to two focus points constant. A Cassini oval keeps the product constant instead — and that one change creates a shape-shifting family. Depending on how b compares to the focus distance c, you get one peanut-shaped oval, a perfect figure-eight (the lemniscate), or two separate egg-shaped lobes.

Welcome to the Cassini Oval!

The astronomer Giovanni Cassini thought planets traced these curves. He was wrong about the planets — but the curves are amazing: nudge one number and a single oval tears itself into two!

How It Works

Pick two foci at (-c, 0) and (c, 0). A point belongs to the curve when distance1 * distance2 = b * b. In polar coordinates that becomes r*r = c*c*cos(2*theta) + or - sqrt(c**4 * cos(2*theta)**2 + (b**4 - c**4)). For each angle we compute the value under the square root first — if it is negative, no point exists at that angle and we simply skip it. Because r*r can have two valid answers at one angle, we plot dots for both the + and - branches instead of dragging a pen line.

Sample Code

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

c = 100
steps = 400

def draw_cassini(b, color):
    for i in range(steps):
        theta = 2 * math.pi * i / steps
        cos2 = math.cos(2 * theta)
        disc = c ** 4 * cos2 * cos2 + (b ** 4 - c ** 4)
        if disc < 0:
            continue
        root = math.sqrt(disc)
        for sign in [1, -1]:
            r2 = c * c * cos2 + sign * root
            if r2 > 0:
                r = math.sqrt(r2)
                monty.penup()
                monty.goto(r * math.cos(theta), r * math.sin(theta))
                monty.dot(3, color)

draw_cassini(120, 'royalblue')
draw_cassini(100, 'forestgreen')
draw_cassini(90, 'crimson')

What Do You Think Will Happen?

The foci sit at distance c = 100 from the center, and we draw b = 120, b = 100, and b = 90. Which b gives the figure-eight: bigger than c, equal to c, or smaller than c? Guess, then click Run!

Try It Now

▶ Run ↻ Reset

Equal — at exactly b = c the green curve pinches through the center point, forming the famous figure-eight called the lemniscate of Bernoulli. Were you right?

How It Works

Think about a point exactly between the two foci (the origin). Its two distances are both c, so their product is c * c. If the target product b * b is bigger than that, the curve has room to pass over the middle and wraps both foci in one blue peanut. If b * b is smaller, the middle is out of reach — the curve must stay close to one focus or the other, so it splits into two crimson lobes. At exactly b = c the curve just barely touches the origin: the figure-eight. One constant crossing one threshold flips the topology from one piece to two.

Explanation Table

Line	What it does
disc = c ** 4 * cos2 * cos2 + (b ** 4 - c ** 4)	The value under the square root — negative means no point at this angle
if disc < 0: continue	The guard: skip angles where the curve does not exist
for sign in [1, -1]:	Try both the + and - branches — some angles have two radii
monty.dot(3, color)	Plot a dot; dots handle curves that split into pieces better than pen lines

Learning Check

Your Turn — Split the Oval

The program below draws a single purple oval with b = 130. Change one number so the curve splits into two separate lobes. Remember the rule you just learned — should b end up bigger or smaller than c = 100?

▶ Run ↻ Reset

Any b smaller than c works — try b = 80 and the oval tears into two lobes, one hugging each focus.

Experiments

1. Sneak up on the split. Try b = 101, then b = 99. You'll know it worked when 101 gives one very pinched peanut and 99 gives two lobes joined by almost nothing.

2. Shrink b far below c. Set b = 60 in the Learning Check program. You'll know it worked when the two lobes become small tight eggs hugging the black focus dots.

3. Move the foci. Change c = 100 to c = 60 in the main program. You'll know it worked when all three curves become single ovals — with the foci closer together, every b is now bigger than c, so even the crimson curve wraps around both foci.

4. Count the existing angles. Add a counter variable, add 1 each time a dot is drawn for b = 90, and print it at the end. You'll know it worked when the count printed in the output pane is far less than steps * 2 — proof the guard skipped the impossible angles.

Topology Changer!

You watched one constant crossing a threshold rip a single curve into two pieces — mathematicians call that a change in topology, and you coded it with one if guard. Up next: Fermat's Spiral — a square root sends two mirrored arms winding out of the origin.