Mathematical equations of love, heart, penis and the boomerang

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Love

Love is complicated. But the mathematics of it is very simple:
  - It starts with "I love you" where "I love" is a constant, and "you" is a variable.
  - Later on, it is:  1 + 1 = 1
  - And later still:  1 + 1 >= 3
Any questions?

Now, let us explore the mechanics of love.


Heart

Love comes from the heart. The mathematical equation of the heart is:

        To see the above graph, go to WolframAlpha website
            http://www.wolframalpha.com/
        At the input area, type in:
            (y - 0.75|x|)^2 + (0.75x)^2 = 1
        And you'll see the heart curve.

You get a slightly different shape of the heart by changing the value of 0.75. Have a try at 0.6 or 0.9 or other values. I find 0.75 more aethetically pleasing.

Actually, it just strikes me that the following parametrised equation
            (y - a|x|^b)^2 + (cx)^2 = d

(where you can set the values of the parameters  a, b, c and d)
can draw just about any heart shaped curve that one can imagine (and more) ... I'm claiming this parametrised equation as Paul Ma's Heart Equation.

So far, no one has disputed my claim in the math.stackexchange forum:
http://math.stackexchange.com/questions/902120/is-there-a-name-to-this-equation-y-axb2-cx2-d

By setting  a=0.75,  b=1,  c=0.75  d=1  into Paul's Heart Equation, it becomes the previously mentioned heart curve.

If you set  a=1,  b=0.5,  c=1  d=4,  you'll produce:
    (y - |x|^0.5)^2 + x^2 = 4

Type the above into the WolframAlpha input area and you'll see that its heart is pretty good looking too:

How about you have a go at various other values of the parameters  a, b, c, d?  Examples are:
    a=0.5,  b=0.5,  c=0.7  d=0.5
    a=0.6,  b=(2/3),  c=0.8  d=0.9

If you discover other good sets of values to use, I would be interested to hear from you.


Boomerang

Now, when you give out love, love always comes back to you. Hence you expect the mathematical equation of a boomerang to be similar to that of a heart, right?  Indeed it is.  Set  a=0.5,  b=1,  c=0.13,  d=1  in Paul's Heart Equation to produce:
    (y - 0.5|x|)^2 + (0.13x)^2 = 1

Type the above into WolframAlpha and you'll get:

Does love make the world go round?

Well, the heart certainly makes the world go round. Take a look at Paul's Heart Equation again:

Let  a=0, b=any, c=1, d=1  and it becomes a perfect circle !
    y^2 + x^2 = 1

But in a real world nothing is ever so perfect.  There are always pits and bumps.
Set  a=1, b=0.5, c=1, d=500  and you'll get:
    (y - |x|^0.5)^2 + x^2 = 500

Penis

Of course, you can't talk about love without mentioning the penis.

The mathematical equation of an aroused penis is:
    y = |sin(x)| + 5*exp(-x^100)*cos(x)  from -3 to 3

Type the above into Wolframalpha to produce

A limp penis:  (by ShmemicalShmengineer in Reddit)
    0 = 2.8x^2(x^2(2.5x^2+y^2-2)+1.2y^2(y(3y-0.75)-6.0311)+3.09)
          + 0.98y^2((y^2-3.01)y^2+3) - 1.005

A fat one in polar form:
    y = Cos(x) + Cos(2x)  polar

The Bum

This is covered in my blog
From Golden Ratio to golden arse
http://onemanadreaming.blogspot.com.au/2014/04/from-golden-ratio-to-golden-arse.html

Breast

This is covered in my blog
Mathematical equations for breasts
https://onemanadreaming.blogspot.com/2022/04/mathematical-equations-for-breasts.html

Here is one equation from the above link for a pair of breasts:

y =   sqrt(1 - (x+3/2)^2 / (1+(x+3/2)^10)^(1/5))
     + sqrt(1 - (10x+15)^2 / (1+(10x+15)^10)^(1/5)) / 10
     + sqrt(1 - (x-3/2)^2 / (1+(x-3/2)^10)^(1/5))
     + sqrt(1 - (10x-15)^2 / (1+(10x-15)^10)^(1/5)) / 10

The above equation was taken from somewhere on the internet few years ago.
Unfortunately, now I couldn't find the source nor the author anymore  :-(


Post Script

There are other equations for the heart.

    (y^2 + x^2 - 1)^3 - (x^2)*(y^3) = 0

Here is another one:
    x^2 + (y - (2(x^2+|x|-6)) / (3(x^2+|x|+2)))^2 = 36

Using 2 equations:
    y = (1-(|x|-1)^2)^0.5  and  y = -3(1-(|x|/2)^0.5)^0.5  from -2 to 2

In polar form:
    y = x  polar  (x from -1.5pi to 1.5pi)

Another one in polar form:
    y = (sin(x) sqrt(|cos(x)|) / (sin(x) + 1.4)) - 2sin(x) + 2  polar

And another one in polar form - this equation has a name, called a Cardioid:
    y = 1 - sin(x)  polar
Its corresponding Cartesian equation is:
    (x^2 + y^2 + y)^2 = x^2 + y^2
 

In 3D

(x^2 + 2.25y^2 + z^2 - 1)^3 - (x^2)(z^3) - 0.1125(y^2)(z^3) = 0
(called Taubin heart surface)

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