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,

*p*1(*x*), is linear in*x*. I'll have it going from*k*1_{ }at*x*= 0, down to 0 at*x=pi*.
The second polynomial,

*p*2(*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*k*2 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,

*k*_{1}) and (*pi*, 0). How cool is that? Just for kicks, I chose*k*_{1}= 2.0,*k*_{2}= 0.3, and*k*_{3}= 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,

*k*_{1}allows me to adjust the starting height,*k*_{2}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*k*_{3}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 2

*pi*. 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 2*pi*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.*

They say that "Love hurts"...well, my head hurts from all this math. LOL Nice blog, sir!

ReplyDeleteThank you Joel! Next up... the equation for Barry White.

ReplyDelete