The Teichmüller space of the torus is the moduli space of marked Euclidean metrics on a torus modulo volume. This space can be identified with the upper half-plane model of the hyperbolic plane. Drag the locator to see the corresponding Euclidean metric on the torus obtained by gluing parallel like-colored edges of the parallelogram together. The cylinder is the result of gluing the red edges together. Further identifying the blue circles, matching up the black dots, produces the corresponding Euclidean metric. With "automatically update cylinder mesh" turned off, the locator moves smoothly.