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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

This entry was posted on February 14, 2015 at 2:55 pm and is filed under Cute Problems with tags , . You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

5 Responses to “My Mathematical Valentines Message”

  1. Bryn Jones's avatar
    Bryn Jones Says:
    February 14, 2015 at 10:34 pm

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

    (Or have I got it wrong?)

    Reply
  2. Anton Garrett's avatar
    Anton Garrett Says:
    February 14, 2015 at 11:34 pm

    Rewrite in polar coordinates and it reduces to

    r = 2a (1 – sin \theta )

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

    Reply
  3. Bryn Jones's avatar
    Bryn Jones Says:
    February 15, 2015 at 12:05 am

    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.

    Reply

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