Filled complex triangles, their oriented three-edge boundary contours, diameter, and perimeter

For $u,v\in \mathbb{C}$, write

${\mathrm{\ell }}_{uv}(t)=(1-t)u+tv(0\le t\le 1)$

for the directed line segment from $u$ to $v$. It is a piecewise-$C^1$ complex contour, and its length is $\mathrm{\mid }v-u\mathrm{\mid }$ by A continuous piecewise-$C^1$ path is rectifiable and its length is the sum of the speed integrals over its pieces.

For an ordered triple $a,b,c\in \mathbb{C}$, the filled complex triangle is

$\mathrm{\Delta }[a,b,c]=\{(1-s-t)a+sb+tc:s,t\ge 0,s+t\le 1\}\mathrm{.}$

Its positively oriented boundary contour is

$\mathrm{\partial }\mathrm{\Delta }[a,b,c]={\mathrm{\ell }}_{ab}\ast {\mathrm{\ell }}_{bc}\ast {\mathrm{\ell }}_{ca},$

with concatenation understood up to increasing reparametrization as in Rectifiable complex contours, reversal, concatenation, closedness, and orientation. Reversing the order of two vertices reverses the orientation. Repeated or collinear vertices are allowed; thus this notation also covers degenerate triangles.

The perimeter and diameter are

$P(\mathrm{\Delta }[a,b,c])=\mathrm{\mid }b-a\mathrm{\mid }+\mathrm{\mid }c-b\mathrm{\mid }+\mathrm{\mid }a-c\mathrm{\mid }$

and

$\mathrm{diam}(\mathrm{\Delta }[a,b,c])=\sup \{\mathrm{\mid }z-w\mathrm{\mid }:z,w\in \mathrm{\Delta }[a,b,c]\},$

where the latter is the metric diameter of Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space under the standard identification of $\mathbb{C}=\mathbb{R}[x]\mathrm{/}(x^2+1)$ as the Euclidean plane and as a normed real algebra: what the identification preserves. These quantities are defined for every ordered triple: the filled triangle is nonempty because it contains $a$, and if $z=(1-s-t)a+sb+tc$ then the modulus laws of Conjugation is an involutive real-field automorphism, $z\bar{z}=\mathrm{\mid }z{\mathrm{\mid }}^2$, and modulus is definite, multiplicative, and subadditive give $\mathrm{\mid }z-a\mathrm{\mid }\le \mathrm{\mid }b-a\mathrm{\mid }+\mathrm{\mid }c-a\mathrm{\mid }$. Hence it lies in a ball of finite positive radius and is bounded. Its perimeter is the length of its boundary contour, including in the degenerate cases.

If $f$ is continuous on the trace of the boundary, abbreviate

$I_f[a,b,c]={\int }_{\mathrm{\partial }\mathrm{\Delta }[a,b,c]}f(z)dz\mathrm{.}$