Method to parameterize circles in R3 laying in a plane

[dispiriton]

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

 

[vela]

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).

 

[LCKurtz]

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

$$R(t) = \langle a,b,c\rangle +r \vec u \cos(t) + r\vec v \sin(t)$$