Vesica Piscis
**VESICA PISCIS**

*Balmoral Software*

The Vesica Piscis
is the convex intersection of two congruent circles that pass through each
other's center. Here we take the common radius as 2 and the centers on the
y-axis at (0,±1). Since each circle from which S is constructed is
symmetric with respect to the y-axis and each is reflected across the x-axis
from the other, S is bisymmetric. In order for S to be traversed in a
counterclockwise direction around the origin starting from its right-hand
corner, its parametric equations are:

The abscissa extrema of S are at
and the ordinate extrema at the
two circle centers (0,±1), so the width x height of the bounding
rectangle of S is .
### Metrics

The perimeter of the Vesica Piscis is 8π/3 and its area is
### Boundary Circles & Circumellipse

The squared distance function x^{2}(t) + y^{2}(t)
of S ranges from a minimum of 1 at (0,±1) to a maximum of 3 at
, so the inradius is 1 and the
circumradius is . Since these
extrema occur on the coordinate axes, a candidate for the circumellipse is one
enclosed by the annulus between the two boundary circles, with
a = and
b = 1. For verification, we have

### Inellipse

In Lemma B, we have for the upper arch
x(t)y(t) = 2cos(2t/3 + π/6)[2sin(2t/3 + π/6) - 1]

This expression is maximized in the first quadrant at the complicated value

The circumellipse dimensions are

For verification, we have

### Summary Table

**Figure** | **Parameters** | Perimeter | Area | Centroid |

Incircle | R = 1 | 6.283185 | 3.141593 | |

Inellipse | | 7.923165 | 4.637100 |

Vesica Piscis | | 8.377580 | 4.913479 |

Circumellipse | | 8.737753 | 5.441399 |

Circumcircle | R = | 10.882796 | 9.424778 |

| |

*"The CBS Television Network"* | |

The Vesica Piscis (red) is a member of a group of similarly-shaped figures
described on these pages, including (inside to outside) the
mouth curve and the
cycloid:

Top Page

Home

Copyright © 2021 Balmoral Software (http://www.balmoralsoftware.com). All
rights reserved.