Implicit function for arbitrary cylinder

  • Thread starter intel4004
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Hello,

This is my first post here on Physics Forums. Hope to have a good time here. I am currently a phd-student in Denmark within a project covering applied research into applications of computer vision and physics to the seed cleaning industry. This is the only post that will include this small introduction of my self.

Currently in my research, I am looking for a way to represent the contour of an arbitrary cylinder; i.e. a cylinder having been rotated in 3D (atleast) about an arbitrary 3D vector; using an implicit function or level curve in 3D. The trivial cases are the three degenerate quadrics where the cylinder is a circle in either of the three individual planes projected on the third axis (for instance [tex]x^2+y^2=r^2[/tex]).

I am on the look out for a function:
[tex]F(x,y,z;r,\theta,\mathbf{v})[/tex], that equals some constant [tex]k[/tex] (zero perhaps) at the contour (surface) of a cylinder with radius [tex]r[/tex], with rotation angle [tex]\theta[/tex] around arbitrary 3D vector [tex]\mathbf{v}[/tex].

Is this possible? It would also be quite useful if [tex]F < k[/tex] for points [tex](x,y,z)[/tex] inside the cylinder, and [tex]F > k[/tex] for points outside the cylinder. I have tried for some time but I cannot seem to get my head around it.

I am aware of the geometrical construction using a parametric description, but I need the level curve representation.

Regards,
Intel4004
 

fresh_42

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There is not enough information here.
$$
F(x,y,z;r,\theta,\mathbf{v}) = \begin{cases} \frac{1}{2}k & \text{ if } x^2+y^2<r \\ k & \text{ if } x^2+y^2=r \\2k & \text{ if } x^2+y^2>r\end{cases}
$$
does the job.
 

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