The function of the heart

There is some holiday coming up. Can't quite think of what it is... I'll have to ask my wife. She knows these things. I hope it's not my anniversary, cuz she would get upset with me if I forgot that. I am pretty sure that our anniversary is in December or March or something, so I think I'm in the clear.

Anyway...

The half-wing function

Let's start with a polynomial equation, we'll call it p(x). I will define it as the sum of three polynomials:

The first of these polynomials, p1(x), is linear in x. I'll have it going from k1 at x = 0, down to 0 at x=pi .

The second polynomial, p2(x), will be 0 at both ends and a second degree parabola in between. In this way, no matter how big I scale it (by using my handy k2 parameter, it won't affect the value of p(x) at the ends. Whenever I get a chance to do this with a polynomial, I go for it.  

I have one more polynomial, we'll make this one a cubic parabola. This one will get nailed down at the ends (x = 0 and x=pi), and will also be zero halfway in between.

I can pick values for the three parameters to give me any cubic polynomial that goes through (0, k₁) and (pi, 0). How cool is that? Just for kicks, I chose k₁ = 2.0, k₂ = 0.3, and k₃ = 0.6. The plot below shows the very agreeable cubic polynomial that is generated this way.

The half-wing function

I could have just played with the standard (canonical) form of the cubic parabola, but that gets confusing. with this formulation, k₁ allows me to adjust the starting height, k₂ allows me to adjust the amount of torsional wiggle, that is, how much the left half bows down and the right half bows the other way. Finally, the k₃ parameter allows me to adjust the midpoint up and down.

I did a little work with this idea of expressing polynomials differently a while ago. One particularly cool thing is that this leads to a way to do regression that always goes through some fixed points.

The full-wing function

I like that function, but I would like to see the other half of the wing. I can use the absolute value function to flip this function over. Now the variable along the abscissa is z. I use z to get x, and then plug x into the equation for p. Back when I was teaching algebra, I would say that the absolute value function is like a tavern. Whether you are negative or positive going in, you will always be positive coming out.

The flipping function

This flipping over function gives me a wonderful plot of a pair of sea gull wings, symbolizing peace, since sea gulls are very peaceful animals and never hurt another living thing. Or maybe the plot looks like a champagne glass. That kinda reminds me of some holiday, but I can't quite place it.

The full-wing plot

Going polar

Sea gulls sometimes go to the North Pole. Or maybe that's penguins. I dunno. Some bird flies up to there. Anyway, that whole polar thing gets me thinking about (what else) polar coordinates. The full-wing plot already goes from 0 to 2pi. Maybe I should plot it in polar coordinates? Here is how I get from one to the other.

Now it's just a matter of plugging it all into Excel. I entered all this stuff into a spreadsheet. I let z go from 0 to 2pi in kinda small steps. I computed x by using the flipping function. I computed p(x), with the values of the k parameters. And these all went into computing a sequence of (x, y) values. Just for grins, I decided to plot them.

Now that I have the blog written, I need to go find out what day I was supposed to be remembering!

I christen this function the Madeloid, in honor of my wife, the Gypsy Songstress, the Shopping Maven, the love of my life.

Would you like a copy of the spreadsheet that created this? Send me an email at john@johnthemathguy.com.