While doing a survey of the various equations that generate the universal love symbol, a heart curve, I find that many fit into this parametrised form:

$$(y - a|x|^b)^2 + (cx)^2 = d $$

Where   a b c d   are parameters.

Is there an existing name to this equation?
Or is this equation being used in some fields of mathematics or physics?
If not, I'll name it Paul Ma's Heart Equation   :-)

Examples of a heart curve for the parameters   a b c d   are:
Set   a=0.75   b=1   c=0.75   d=1
Set   a=0.6   b=(2/3)   c=0.8   d=0.9

And if you set
    a=(4/5)   b=0.5   c=(4/5)   d=(16/25)
it is identical to the equation in the math.stackexchange forum:
An equation that generates a beautiful or unique shape for motivating students in mathematics

My Heart Equation
    $(y - a|x|^b)^2 + (cx)^2 = d$
can also be used to deform the heart curve. Examples:
For boomerang: Set   a=0.5   b=1   c=0.13   d=1
For circle: Set   a=0   b=any   c=1   d=1

I have written a light hearted blog on this Heart Equation:
where the heart curves are graphed for the above mentioned parameters.


1 Answer 1


there are equation $a^2 (abs(x)-y)^2 + b^2 (abs(x)+y)^2 = 2(ab)^2$ its make love symbol

if ratio $abs(a/b)>1$ :its make the heart

if ratio $abs(a/b)$ too lagre : its make the boomerang

if ratio $abs(a/b) =1$ : its circle

if ratio $abs(a/b) < 1$ : making it upside down

where $a,b$ is in equation $x^2/a^2+y^2/b^2 =1$


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