Curve, arc, bow, arch mean a line or something which follows a line that is neither straight nor angular but rounded.
Curve is the general term and the most widely applicable. It may be used in reference to a line, edge, outline, turn, or formation that keeps changing its direction without interruption or angle.
Arc is used specifically to denote a part or section of the circumference of a circle; in more general use it is applied to things that have or assume a strongly curved form.
Bow, unlike the preceding words, has always designated concrete things that are curved, and draws its implications especially from its reference to the archer's bow with its long, gradually curving strip of wood that may be bent almost into a U. Hence many things which resemble an archer's bent bow are describable or designatable as a bow.
Arch, though once equivalent to arc in denotation, is now basically applied to a supporting structure built up of wedge-shaped pieces of stone or other substance in such a way that they form a semicircular curve with a keystone at the apex or two opposite curves with a joint at the apex and provide an opening underneath (as for a window, a door, or a passageway).
Hence arch is applied to any similarly curved structure.