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Thursday, 30 March 2017

From curves to cylinders

We know that from a planar curve α:IR2 we can define the generalized cylinder as S={(α(s),t):sI,tR}. This can also view as follows. 

Consider a smooth function f:OR2R and aR a regular value. Then we know that C=f1({a}) is a regular curve: the 1-dimensional case than a surface. With this function f, we define a new function F as F:O×RR, F(x,y,z)=f(x,y),
which is differentiable. We prove that a is a regular value of F. Indeed, F=(Fx,Fy,Fz)=(f,0). Consequently, (x,y,z) is a critical point of F if and only if (x,y) is a critical point of f. In particular, if F(x,y,z)=a, then f(x,y)=a and (x,y) is not a critical point of f, and thus, (x,y,z) is not a critical point of F. This proves that a is a regular value of F
As a consequence, S=F1({a}) is a surface. We describe this surface: S={(x,y,z)O×R:f(x,y)=a}=C×R. This shows that S is the generalized cylinder over the planar curve C.

The next picture is a generalized cylinder based on the cardioid.

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