In mathematics and, in particular, differential geometry a circular surface is the image of a map ƒ : I × S1 → R3, where I ⊂ R is an open interval and S1 is the unit circle, defined by
where γ, u, v : I → R3 and r : I → R>0, when Moreover, it is usually assumed that u · u = v · v = 1 and u · v = 0, where dot denotes the canonical scalar product on R3, i.e. u and v are unit length and mutually perpendicular. The map γ : I → R3 is called the base curve for the circular surface and the two maps u, v : I → R3 are called the direction frame for the circular surface. For a fixed t0 ∈ I the image of ƒ(t0, θ) is called a generating circle of the circular surface.S. Izumiya, K. Saji, and N. Takeuchi, "Circular Surfaces", Advances in Geometry, de Gruyter, Vol 7, 2007, 295–313.
Circular surfaces are an analogue of ruled surfaces. In the case of circular surfaces the generators are circles; called the generating circles. In the case of ruled surface the generators are straight lines; called rulings.
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