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How to Plot Cassini Ovals

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

The process of plotting Cassini ovals is the same as any other mathematical curve. This lesson will explain the process of plotting Cassini ovals using equations and specific values within the equation. We will also use an example to practice.

Cassini Ovals

Let's let math infiltrate our breakfast for a moment! A common breakfast staple is eggs. If you place two eggs next to each other, with their narrow ends facing one another, and look down at it as a two-dimensional image, you're actually looking at a mathematical curve!


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In mathematics, this curve is a Cassini oval, or sometimes a Cassini ellipse or an egg curve. Cassini ovals are a set of points that are described by two fixed points. That is, the product of the distances between any point on a Cassini oval and each of the fixed points is constant.


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The general equation of a Cassini oval is as follows:

  • (x2 + y2 + a2)2 - 4a2x2 = b4

Where a is half the distance between the two fixed points that describe the curve, and b is the square root of the product of the distances between each of the points and any point on the curve.

As with any mathematical curve, we can plot or graph Cassini ovals. Let's take a look at how to do this!

How to Plot Cassini Ovals

Plotting, or graphing, Cassini ovals takes on the same steps as graphing any mathematical equation, and those are as follows:

  1. Determine the general shape of the graph by examining its equation and information given.
  2. Plot several points that satisfy the equation, and connect the points in a smooth curve appropriately.

First, let's examine step one. When it comes to Cassini ovals, the general shape of the graph is determined by the values of a and b.

  • If a < b, the graph is a single loop that is the shape of an ellipse or of a shelled peanut.
  • If a = b, the graph is the shape of the infinity symbol.
  • If a > b, the graph is two loops in the the shape of two eggs with their narrow ends facing each other.


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For instance, if we look back at the graph of our opening example, it is in the shape of two eggs facing each other, so we know that for that graph's equation, a > b.

This tells us that step one of plotting a Cassini oval is a matter of identifying a and b from the equation and information given, and then comparing a and b to determine the general shape of the graph.

Now consider step two. Once we've figured out the general shape of the graph, we can find several points that satisfy the equation of the Cassini oval, plot them, and then connect them accordingly.

Hmmm…this doesn't sound too hard. The equation is a little different than what we may be used to, but the general plotting process is the same as any other mathematical curve or equation. Let's look at an example to help solidify our understanding of plotting Cassini ovals.

Example

Consider the following equation of a Cassini oval, in which a = 2 and b = 2.

  • (x2 + y2 + 4)2 - 16x2 = 16

To plot this Cassini oval, we simply take it through our steps. First, we identify a and b, which are given as a = 2 and b = 2. If a and b are not given, then we would just need to examine and/or rewrite the equation to identify a and b.


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Now, we compare them. Since a = 2 and b = 2, we have that a = b, so we know that the general shape of the graph is going to be that of the infinity symbol.

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