Plotting with Mathematica

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Points Equidistant from Two Others

Finish the question from last time about proving that the set of points equidistant from (a,0,0) and (b,0,0) is the plane parallel to the yz-plane and having x = (a+b)/2.

Last time, we started by using the distance formula to find the distances from general point (x,y,z) to (a,0,0) and (b,0,0), and set those distances equal. Then we did some algebra, crucially including a step where we took square roots of both sides to remove some squaring, and eventually found that a = b:

The a = b result basically says that if you make both points the same, then all points anywhere in space are equidistant from them/it. This is true, but not the result we wanted to check.

Remember that step where we took square roots to eliminate squaring? We didn’t actually do that completely: if (x-a)² = (x-b)², it could be because x-a = x-b, which is what we pursued, but it could also be because x-a = -(x-b). Let’s see where that takes us…

Now we ended up with the equation for the plane parallel to the yz-plane and having x = (a+b)/2. So another way to get points equidistant from (a,0,0) and (b,0,0), if a and b aren’t necessarily equal, is to have the points lie in that plane, like we conjectured.

This example turns out to be an excellent one for how mathematical discovery happens in real life. We started with a conjecture that seemed plausible but wasn’t 100% certain, thought about it informally for a bit to see if it really stayed plausible, and finally got into more formal mathematical exploration. But even the formal mathematical part was exploratory, in that it consisted of trying some things (notably the distance formula) to see where they’d lead. At first, we found the a = b result, which was a lucky and interesting discovery (i.e., not something “bad”) but not what we hoped for. That led to finding a different way to go in our explorations (backing up and trying something different is really common in math), which eventually did lead to what we wanted.

Mathematica

Mathematica is a powerful computer algebra system. You can write computer programs in it like you would with, e.g., MatLab, R, Python, etc., although that’s not my goal for you in this course. Rather, we’ll use it interactively, as essentially an incredibly powerful graphing calculator.

Some general guidelines for interactively using Mathematica:

• Other than very basic arithmetic, everything is done by calling a built-in Mathematica command or “function.”
• Everything that’s built in to Mathematica has a name that starts with a capital letter (it’s easy and common to forget to capitalize that first letter, which makes Mathematica unhappy).
• Arguments to commands always go inside square brackets, i.e., “[” and “]”. (This is also easy to forget, but makes Mathematica unhappy if you do.)
• If you don’t want to type commands, Mathematica has command palettes that let you build some commands by clicking and filling in boxes.
• Press the Shift and Enter keys simultaneously to evaluate something you’ve typed.

Right now, I want to look at some of its features for plotting surfaces in 3 dimensions.

In particular, surfaces that are defined “implicitly” as solutions to equations. For example, x² + y² + z² = r² implicitly defines a sphere of radius r at the origin.

The Mathematica function that does this is ContourPlot3D.

You can download the notebook in which we explored all of the above and more from Canvas. Once downloaded, you can open it in your own Mathematica, experiment further with anything in it, etc.

Next

(“Next” is Tuesday, because Monday is a holiday. Happy Labor Day!)

Many of the surfaces you produced by changing the equation for a sphere (specifically all the ones that didn’t involve changing the exponents) are examples of so-called “quadric surfaces.”

The quadric surfaces are an interesting enough family of surfaces that I want to spend some time on them on Tuesday.

Please read section 1.6 of the textbook to get ready for this.

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