## Circles, Squares, and Parabolas

If you take two circles, you can position them before your right eye in such a way that one will perfectly overlap the other.

In other words, a circle is geometrically similar to all other circles.

The same is true of squares.

However, the same is not true of rectangles, as you may have noticed if you ever tried to perfectly frame a painting with your camera and found that no matter how you positioned yourself, you’d either crop it or include some of its surroundings.  (If you’ve done this, but didn’t have a problem, count yourself lucky!) The non-similarity of rectangles caused trouble when people wanted to watch movies on old-fashioned TVs.  Do you squeeze everyone onto the screen, crop things off, or letterbox it?

So rectangles aren’t like circles and squares with respect to similarity.

But, parabolas are!

If this surprises you, it’s easy to understand why, especially if you’ve seen some parabolas, like the three below.

These three parabolas sure don’t look similar to each other.

Well, looks can be deceiving, because it’s true: All parabolas are similar to each other! Let’s prove this. (And by the way, when I say “parabola,” I mean the entire parabola…not just a section of it, just as earlier I meant the whole circle, the whole square, and the whole rectangle.)

Consider two parabolas.  If we can use translation, rotation, and dilation to superimpose one precisely over the other, then we’re done.  Let’s first translate both parabolas so that they have the same apex.  We’ll introduce xy coordinate axes and use the apex of these two parabolas as our origin.  Next, rotate the parabolas so that their axes are both aligned with the y-axis and so that they both open upward.  At this point, the equations of the two parabolas are given by $y = ax^2$ and $y = bx^2$, where $a$ and $b$ are positive constants.

Now we’re going to dilate the parabola corresponding to $y = ax^2$.  If we dilate it by a factor of $d$ (and with center of dilation at the origin), the resulting curve will consist of the set of points $(dx, dy)$ where $x$ and $y$ satisfy $y = ax^2$.  Let $x' = dx$ and $y' = dy$.  Notice that $y' = \frac{a}{d}x'^2$. This shows that when we dilate the parabola $y = ax^2$, the result is also a parabola.

Since $b$ is positive, we can pick $d$ to be $\frac{a}{b}$.  For this value of $d$, the parabola $y = ax^2$ dilates to the parabola $y' = bx'^2$, which is what we wanted to show!

And so we can deduce that the shallow green parabola in the illustration above is the view you would get if you looked really closely at the apex of the blue parabola (and it changed color).  Try it and see for yourself.