# Method to parameterize circles in R3 laying in a plane

## 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)

## The Attempt at a Solution

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vela
Staff Emeritus
Homework Helper
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
Homework Helper
Gold Member
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

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