Understanding 3D circle parameterization

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In summary, the formula for a 3D circle parameterization includes three terms: the center point, the x-axis term, and the y-axis term. The x-axis term is determined by the radius and a unit vector, while the y-axis term is determined by the radius and a unit vector perpendicular to the normal vector of the circle's plane. To find the unit vector for the x-axis, start with any vector at right angles to the normal vector and normalize it to a unit length.
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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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  • #3
: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##.
 
  • #4
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
 
  • #5
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.
 
  • #6
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
 
  • #7
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}|##
 

1. What is 3D circle parameterization?

3D circle parameterization is a mathematical process used to describe the characteristics of a circle in three-dimensional space. It involves defining the center point, radius, and orientation of the circle in order to accurately represent its shape and position.

2. Why is understanding 3D circle parameterization important?

Understanding 3D circle parameterization is important in many fields, such as computer graphics, robotics, and engineering. It allows us to accurately model and manipulate circles in three-dimensional space, which is crucial for creating realistic and precise designs and simulations.

3. How is 3D circle parameterization different from 2D circle parameterization?

In 2D circle parameterization, we only need to define the center point and radius of a circle. In 3D, we also need to specify the orientation of the circle, which is defined by its normal vector. This extra dimension adds complexity but allows for more precise representation of circles in 3D space.

4. What are some common methods for parameterizing a 3D circle?

Some common methods for parameterizing a 3D circle include using Cartesian coordinates, spherical coordinates, and polar coordinates. Each method has its own advantages and may be more suitable for certain applications.

5. How is 3D circle parameterization used in computer graphics?

In computer graphics, 3D circle parameterization is used to create and manipulate objects in three-dimensional space. It is essential for creating realistic and visually appealing 3D models, as well as for performing transformations and animations on these objects.

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