Parametric Equation of a Circle in 3D

A circle in 3D is parameterized by six numbers: two for the orientation of its unit normal vector, one for the radius, and three for the circle center .

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While a 2D circle is parameterized by only three numbers (two for the center and one for the radius), in 3D six are needed. One set of parametric equations for the circle in 2D is given by
for a circle of radius and center .
In 3D, a parametric equation is
,
for a circle of radius , center , and normal vector ( is the cross product). Here, is any unit vector perpendicular to . Since there are an infinite number of vectors perpendicular to , using a parametrized is helpful. If the orientation is specified by an azimuth angle and a zenith angle , then , and can have simple forms:
,
,
.
This notation follows a variable naming convention sometimes used in mathematics. Another convention labels the zenith angle and azimuth .
Reference
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