We port classical straightedge and compass constructions to manifold surfaces under the geodesic metric. We propose two complementary approaches: one working on the tangent plane; and another working directly on the surface. In both cases, many constructions lack some of the geometric properties they have in the Euclidean case. We devise alternative constructions that guarantee at least a subset of such properties. We integrate our constructions in the context of a prototype system supporting the interactive drawing of primitives of vector graphics.
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