Vesica Piscis

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 x²(t) + y²(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

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:

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