# Cassini Ovals, Red Blood Cells and Cell Division

- Author:
- Irina Boyadzhiev

A Cassini oval is defined as the set of all points in the plane for which the product of the distances to two fixed points is constant. The equation of this curve is , where and are the x-coordinates of the fixed points and is the constant. Depending on the parameters and the ovals can be a single loop, a lemniscate or two disjoined loops.

- If we rotate the ovals about the y-Axis and change the parameter
from values of less than to we observe a biconcave disc which closely resembles the shape of a red blood cell. There are different theories why the red blood cells have this shape. Some suggest that this shape optimizes the surface area to volume ratio for gas exchange [2]. Another explanation suggests that in the biconcave shape a lot of the mass is distributed in the periphery and this limits the rotation during flow in the large vessels [3]. - If we rotate Cassini ovals about the x-Axis and then change the values of the parameter
from less than to greater than we observe a process that resembles the division of a cell [1].

- D.McKenney, J.A.Nickel,
*Mathematical model for cell division*, Mathematical and Computer Modelling, Volume 25, Issue 2, January 1997, Pages 49-52. - Architecture of the cytoskeleton in red blood cells
- Uzoigwe,C.,
*The human erythrocyte has developed the biconcave disc shape to optimise the flow properties of the blood in the large vessels*.Med Hypotheses. 2006;67(5):1159-63. Epub 2006 Jun 23. - Angelov, B., Mladenov, I.M.
*On the geometry of red blood cell. International Conference on Geometry, Integrability and Quantization*, Coral Press, Sofia, pp.27-46