Lesson plan
Circular surface
In mathematics and, in particular, differential geometry a circular surface is the image of a map ƒ : I × S 1 → R 3 , where I ⊂ R is an open interval and S 1 is the unit circle, defined by : f(t,\theta) := \gamma(t) + r(t){\mathbf u}(t)\cos\theta + r(t){\mathbf v}(t)\sin\theta, \, where γ, u, v : I → R 3 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 R 3 , i.e. u and v are unit length and mutually perpendicular. The map γ : I → R 3 is called the base curve for the circular surface and the two maps u, v : I → R 3 are called the direction frame for the circular surface. For a fixed t 0 ∈ I the image of ƒ(t 0 , θ) is called a generating circle of the circular surface. 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.
Article summary
In mathematics and, in particular, differential geometry a circular surface is the image of a map ƒ : I × S 1 → R 3 , where I ⊂ R is an open interval and S 1 is the unit circle, defined by : f(t,\theta) := \gamma(t) + r(t){\mathbf u}(t)\cos\theta + r(t){\mathbf v}(t)\sin\theta, \, where γ, u, v : I → R 3 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 R 3 , i.e. u and v are unit length and mutually perpendicular. The map γ : I → R 3 is called the base curve for the circular surface and the two maps u, v : I → R 3 are called the direction frame for the circular surface. For a fixed t 0 ∈ I the image of ƒ(t 0 , θ) is called a generating circle of the circular surface. 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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Discussion questions
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