I Understanding 3D circle parameterization

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The discussion focuses on understanding the parameterization of a circle in 3D space. Key points include the roles of vectors u and n, where u is any unit vector and n is a perpendicular vector defining the circle's plane. The center of the circle is represented by vector c, while the terms involving cos(t) and sin(t) relate to the circle's projection in the XY plane. To calculate vector u from the normal vector n, one can use the dot product to find a vector perpendicular to n and then normalize it. The conversation emphasizes the flexibility in choosing u and n to define a specific circle in 3D.
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Understanding 3D circle parameterization
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1) The third term, ##\overrightarrow{c}## just locates the center.
2) The first term, ##R\cos(t)\overrightarrow{u}## is like the u vector is the X-axis of a simple circle of radius ##R## in the XY plane.
3) The second term, ##R\sin(t)\overrightarrow{n}\times\overrightarrow{u}## is like the Y-axis of a simple circle of radius ##R## in the XY plane.
( ##\overrightarrow{n}\times\overrightarrow{u}## is a unit vector at right angles to both ##n## and ##u##. Since ##n## is normal to the plane of the circle, that puts ##\overrightarrow{n}\times\overrightarrow{u}## in the plane of the circle at right angles to ##u##.)

How you can calculate ##\overrightarrow{n}## and ##\overrightarrow{u}## would depend on what information you have in the problem you are working on.
 
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:welcome:

I think that ##\vec u## is any unit vector, and ##\vec n## is any vector perpendicular to ##\vec u##. Any choice defines a circle. For any given circle, you have a choice of ##\vec u## and two options for ##\vec n##.
 
FactChecker said:
1) The third term, ##\overrightarrow{c}## just locates the center.
2) The first term, ##R\cos(t)\overrightarrow{u}## is like the u vector is the X-axis of a simple circle of radius ##R## in the XY plane.
3) The third term, ##R\sin(t)\overrightarrow{n}\times\overrightarrow{u}## is like the Y-axis of a simple circle of radius ##R## in the XY plane.
( ##\overrightarrow{n}\times\overrightarrow{u}## is a unit vector at right angles to both ##n## and ##u##. Since ##n## is normal to the plane of the circle, that puts ##\overrightarrow{n}\times\overrightarrow{u}## in the plane of the circle at right angles to ##u##.)

How you can calculate ##\overrightarrow{n}## and ##\overrightarrow{u}## would depend on what information you have in the problem you are working on.
Thanks!
I have the normal of the circle's plane, center point and radius
 
user1003 said:
Thanks!
I have the normal of the circle's plane, center point and radius
I had to correct a mistake for the third line. It is the second term, not the third.
 
FactChecker said:
1) The third term, ##\overrightarrow{c}## just locates the center.
2) The first term, ##R\cos(t)\overrightarrow{u}## is like the u vector is the X-axis of a simple circle of radius ##R## in the XY plane.
3) The second term, ##R\sin(t)\overrightarrow{n}\times\overrightarrow{u}## is like the Y-axis of a simple circle of radius ##R## in the XY plane.
( ##\overrightarrow{n}\times\overrightarrow{u}## is a unit vector at right angles to both ##n## and ##u##. Since ##n## is normal to the plane of the circle, that puts ##\overrightarrow{n}\times\overrightarrow{u}## in the plane of the circle at right angles to ##u##.)

How you can calculate ##\overrightarrow{n}## and ##\overrightarrow{u}## would depend on what information you have in the problem you are working on.
Hey

so my question is how do I get the vector u if I have the normal of the circle's plane, center point and radius?
thanks
 
user1003 said:
Hey

so my question is how do I get the vector u if I have the normal of the circle's plane, center point and radius?
thanks
Start with any vector at right angles to the normal vector ##\overrightarrow{n} = (n_x,n_y,n_z)##,
The general equation for a non-zero vector, ##\overrightarrow{v} = (v_x,v_y,v_z) \ne (0,0,0)##, at right angles is to say that the dot product with ##\overrightarrow{n}## is 0.
## 0 = n_x v_x + n_y v_y + n_z v_z##.
We know that ##\overrightarrow{n}## is not the zero vector. Assuming that ##n_z\ne 0##, we can set ##v_x=1, v_y=1, v_z = (-n_x - n_y)/n_z ##.
That will give a vector in the plane at right angles to ##\overrightarrow{n}##.
Then we want to normalize ##\overrightarrow{v}## to a unit length to get ##\overrightarrow{u}##:
##\overrightarrow{u} = \overrightarrow{v}/|\overrightarrow{v}|##
 
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