The conductance of a microscopic constriction in a two-dimensional electron gas is obtained as a function of both the constriction width and curvature. When the quantized conductance G at plateaus is given by the channel number Nc times the quantum unit 2e2/h, Nc is found to be a function of not only the width and but also the curvature. At a given W, Nc increases by one whenever the constriction curvature decreases by about a certain value. Until the shape smoothness becomes comparable to the two parallel boundaries, there exist more channels avaliable for conduction in a smaller-curvature constriction than in a larger-curvature one. This result is very interesting because Nc has been considered to depend on the width W only. this reflects that the number of the quantized transverse levels depend o both the constriction width and curvature in a two-dimensional electron gas.