Parametric Equation of a Circle in 3D

​
orientation
radius
center
(
C
x
,
C
z
)
C
y
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
{
C
x
,
C
y
,
C
z
}
.

Details

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
x(t)=rcos(t)+
C
x
​​y(t)=rsin(t)+
C
y
for a circle of radius
r
and center
(
C
x
,
C
y
)
.
In 3D, a parametric equation is
P(t)=rcos(t)u+rsin(t)nu+C
,
for a circle of radius
r
, center
C=(
C
x
,
C
y
,
C
z
)
, and normal vector
n
(

is the cross product). Here,
u
is any unit vector perpendicular to
n
. Since there are an infinite number of vectors perpendicular to
n
, using a parametrized
n
is helpful. If the orientation is specified by an azimuth angle
ϕ∈[-π,π]
and a zenith angle
θ∈[0,π]
, then
n
,
u
and
nu
can have simple forms:
n=
cos(ϕ)sin(θ)
sin(θ)sin(ϕ)
cos(θ)
,
u=
-sin(ϕ)
cos(ϕ)
0
,
nu=
cos(θ)cos(ϕ)
cos(θ)sin(ϕ)
-sin(θ)
.
This notation follows a variable naming convention sometimes used in mathematics. Another convention labels the zenith angle
ϕ
and azimuth
θ
.

References

[1] http://en.wikipedia.org/wiki/Spherical_coordinate_system

External Links

Parametric Equations (Wolfram MathWorld)
Circle (Wolfram MathWorld)
Ellipse (Wolfram MathWorld)
Circle (Wolfram Documentation Center)
Orthogonal Group (Wolfram MathWorld)
Azimuth (Wolfram MathWorld)
Zenith Angle (Wolfram MathWorld)
Spherical Coordinates (Wolfram MathWorld)
Perpendicular Vector (Wolfram MathWorld)

Permanent Citation

Aaron Becker
​
​"Parametric Equation of a Circle in 3D"​
​http://demonstrations.wolfram.com/ParametricEquationOfACircleIn3D/​
​Wolfram Demonstrations Project​
​Published: February 19, 2014