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Method to parameterize circles in R3 laying in a plane

  • Thread starter dispiriton
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  • #1

Homework Statement


In general how do i parametrize a circle of radius r at centre (a,b,c) laying on a plane? E.g. (x + y + z = 6)


Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
vela
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You could start with a circle of radius r centered at the origin that lies in the xy-plane. Then apply a rotation so that it lies in the plane parallel to the given plane. Finally, translate it so it's centered at (a,b,c).
 
  • #3
LCKurtz
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Vela has given you one method. Another is to find two unit vectors u and v that are perpendicular to the plane's normal vector N and perpendicular to each other. Then use

[tex]R(t) = \langle a,b,c\rangle +r \vec u \cos(t) + r\vec v \sin(t)[/tex]
 

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