Cye Waldman has
found a plane curve,
previously unknown to the NCB, that renders a credible heart shape and
spiral. We present this to the mathematics community for
comment. The "heart" is defined by a polar equation and is also
shown in the complex form. At the NCB we have dubbed this as
"Cœur d'Cye" or "Waldman's Heart." We solicit your comments.
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| The angular range was selected to
position the cusp of the heart at the origin. The figure below
shows the heart. |
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The arc
length, area and centroid of this heart can be found
analytically. To wit,
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| In order to turn this into
a spiral, consider |
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| The animations below show
that the heart is imperfect as the spiral
begins, but approaches the desired shape after a few
turns. |
The figure below
shows the application of the heart model in a three-dimensional
rendering.
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Naturally, we were curious
as to the effect of switching the sine and cosine in the above
equation. To that end, let
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The result is a
leaf-shaped figure that is
shown in the
animation on the right where it is overlaid on the heart for
comparison. Also note the difference in the angular range.
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| The equations for
the leaf
are given as follows: |
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Notice the leaf's
area is just one-eighth of the heart. |
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