Back to . . . .  NCB Deposit  # 166 Dr. Cye Waldman cye@att.net "A Possible New Heart Curve and Spiral" with both polar and complex equations "Cœur de Cye" or "Waldman's Heart" A Leaf Spiral

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.

The angular range was selected to position the cusp of the heart at the origin.  The figure below shows the heart.

The arc length, area and centroid of this heart can be found analytically.  To wit,

In order to turn this into a spiral, consider

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.

Naturally, we were curious as to the effect of switching the sine and cosine in the above equation.  To that end, let
 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.
 The equations for the leaf are given as follows: Notice the leaf's area is just one-eighth of the heart.

 2018