  We study $ \mathcal{H}ol(\Gamma\cup\text{Int}(\Gamma)) $, the normed algebra
of all holomorphic functions defined on some simply connected neighbourhood of
a simple closed curve $\Gamma$ in $\mathbb{C} $, equipped with the supremum
norm on $ \Gamma $. We explore the geometry of nowhere vanishing, point
separating sub-algebras of $ \mathcal{H}ol(\Gamma \cup \text{Int}(\Gamma)) $.
We characterize the extreme points and the exposed points of the unit balls of
the said sub-algebras for $\Gamma$ analytic. We also characterize the
smoothness of an element in these sub-algebras by using Birkhoff-James
orthogonality techniques without any restriction on $\Gamma$. As a culmination
of our study, we assimilate the geometry of the aforesaid sub-algebras with
some classical concepts of complex analysis and establish a connection between
Birkhoff-James orthogonality and zeros of holomorphic functions.
