A tricurve is a circular triangle formed by reflecting circular arcs of
a common radius (Lexen 2018a, 2024). Begin with a circular
arc 
of radius 
and central angle 
, where 
. Reflect it in its chord 
to form a lens, and choose a point 
on the reflected arc
that divides its central angle into 
and 
.
Reflect each of the arcs 
and 
in its own chord. The original
arc 
and these last two arcs bound the tricurve. When 
, the boundary has a cusp
at 
.
With 
and 
measured in radians, the area
is
![A=R^2[sina+sinb-sin(a+b)].](/images/equations/Tricurve/NumberedEquation1.svg) |
For a short proof, a lens bounded by two equal circular arcs of radius 
and central
angle 
has area 
. The tricurve is obtained by subtracting the
two smaller lenses from the larger one, giving 
(Lexen 2018a, pp. 7-8).
Lexen (2018b, 2021) associates a tricurve with a congruent "phantom" obtained by a half-turn. The vertices of each tricurve are the centers of the other's boundary arcs.
Every tricurve admits a periodic monohedral tiling. The point 
completing the parallelogram 
lies on the original arc 
. Splitting that arc
at 
gives a four-arc boundary
whose opposite arcs are translates
of one another. Repeating this boundary pattern by translations gives a tiling
by congruent copies (Lexen 2018a, p. 5; Lexen
2024, p. 524).
