  We study conchoids to algebraic curve from the perspective of algebraic
geometry, analyzing their main algebraic properties. We introduce the formal
definition of conchoid of an algebraic curve by means of incidence diagrams. We
prove that, with the exception of a circle centered at the focus and taking $d$
as its radius, the conchoid is an algebraic curve having at most two
irreducible components. In addition, we introduce the notions of special and
simple components of a conchoid. Moreover we state that, with the exception of
lines passing through the focus, the conchoid always has at least one simple
component and that, for almost every distance, all the components of the
conchoid are simple. We state that, in the reducible case, simple conchoid
components are birationally equivalent to the initial curve, and we show how
special components can be used to decide whether a given algebraic curve is the
conchoid of another curve.
