Circular surface

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

${\displaystyle f(t,\theta ):=\gamma (t)+r(t){\mathbf {u} }(t)\cos \theta +r(t){\mathbf {v} }(t)\sin \theta ,\,}$

where γ, u, v : I → R³ and r : I → R_>0, when R_>0 := { x ∈ R : x > 0 }. 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.^[1]

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.