My Mathematical Valentines Message

Here’s a little mathematical exercise with a Valentines theme:

Sketch the curve in the x-y plane described by the equation

\left(x^2 +y^2  + 2ay \right)^2 = 4a^2 \left( x^2 + y^2 \right)

for

x<3.

Geddit?

Answer: the equation is that of a cardioid:

cardioid-2

5 Responses to “My Mathematical Valentines Message”

  1. Oh, dear. Oh, dear. Oh, dear.

    (Or have I got it wrong?)

  2. Anton Garrett Says:

    Rewrite in polar coordinates and it reduces to

    r = 2a (1 – sin \theta )

    at which point I recognised it from prior knowledge. Nice!

  3. Ah. Now Peter has changed the sign in front of the 2ay term. The original version had a different sign – and that didn’t produce anything like a heart for a>0.

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