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Curved shapes are frequent subjects of maneuvers by the human hand. In robotics, it is well known that antipodal grasps exist on curved objects and guarantee force closure under proper finger contact conditions. This paper presents an efficient algorithm that computes, up to numerical resolution, all pairs of antipodal points on a simple, closed, and twice continuously differentiable plane curve. Dissecting the curve into segments everywhere convex or everywhere concave, the algorithm marches simultaneously on a pair of such segments with provable convergence and interleaves marching with numerical bisection recursively. It makes use of new insights into the differential geometry at two antipodal points. We have avoided resorting to traditional nonlinear programming, which would neither be quite as efficient nor guarantee to find all antipodal points. A byproduct of our result is a procedure that constructs all common tangent lines of two curves, achieving quadratic convergence rate. Dissection and the coupling of marching with bisection constitute an algorithm design scheme potentially applicable to computational problems involving curves and curved shapes. [ABSTRACT FROM AUTHOR]