The geometry of a constriction, which plays a crucial role in quasi-one-dimensional (quasi-D) ballistic transport, is investigated by performing calculations of the conductance. If the constriction becomes smoothly narrower inside, the current-carrying states evolve adiabatically leading to quantized conductance without a resonance structure. In contrast, quasi-0D (confined) states can form in a local widening inside the constriction and give rise to resonant tunneling. The effects of an obstacle at the entrance and the roughening along the constriction are also studied. © 1989 The American Physical Society.