In the context of mathematics, particularly in topology and differential geometry, a **normal surface** typically refers to a type of surface that is embedded in a three-dimensional space and satisfies certain conditions regarding its curvature and other geometric properties. However, the term "normal surface" may also have specific meanings in different subfields of mathematics, such as in the study of 3-manifolds or algebraic geometry.
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Relative to a triangulation, a normal surface meets every tetrahedron in disjoint normal disks: four types of triangles cutting off vertices and three types of quadrilaterals separating opposite pairs of edges. A normal disk meets any edge at most once; its face arcs join distinct edges. Two different quadrilateral types cannot coexist in a tetrahedron.