Silly geologists, using the same term to mean(geometrically) different things:
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In areas undergoing isostatic adjustment (i.e., everywhere, due to the last Ice Age), the dividing line between the region where the Earth's crust is moving up and the region where it is moving down.
In North America, the hinge line (*very* roughly) approximates the border between the United States and Canada. North of this line, sea levels appear to fall because the crust is pushing up; south of the line, sea levels appear to rise because the crust is falling in compensation. The line is constantly moving around as the stresses on the Earth's crust change.
Other hinge lines arise from isostatic compensation due to tectonic forces: the boundary zones of between tectonic uplift and subsidence. on passive continental margins, the Continental shelves appear to receive a greater sedimentary load further out in the ocean; layers of sedimentary rock generally grow thicker the farther they are from the coast. The increased load presses down on the lithosphere below, causing subsidence at the edge of the continental shelf, and uplift inland. Such an uplift during the Eocene Epoch 40 million years ago gave us the current incarnation of the Applachian Mountains.
The line along which stressed layers of rocks begin to bend, resulting in the line of maximum curvature in the resulting folded rock layers. When such layers are brought above the Earth's surface to create folded mountains, the hinge lines at the tops of anticlines often coincide with the mountain's ridge line, and the hinge lines at the bottoms of synclines often coincide with the bottoms of the valleys, but not always.
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The line where the two valves (halves) of a bivalve mollusk's shell are articulated together.