Intersection of complex sphere and cone

In summary, the intersection of a complex sphere and cone is the set of points that satisfy both the equations of the sphere and cone. It can be calculated by setting the equations equal to each other and solving for the variables. The resulting shapes can vary and have applications in fields such as computer graphics and physics. The use of complex numbers allows for a more precise representation of the intersection.
  • #1
huyichen
29
0
Show the intersection of complex sphere (|z1|^2+|z2|^2+|z3|^2=1) in C^3 and the complex cone (z1^2+z2^2+z3^3=1) in C^3 is a smooth submanifold of C^3.

I am trying to do it using regular level set, but I am not sure which one of (1,0) or (1,1,0) should be set to be the regular value?
 
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  • #2
You have a map from C^3 to R x C. The intersection is the preimage of (1,1,0).
 

1. What is the intersection of a complex sphere and cone?

The intersection of a complex sphere and cone is the set of points that lie on both the sphere and the cone. In other words, it is the points where the equations of the sphere and cone are satisfied simultaneously.

2. How is the intersection of a complex sphere and cone calculated?

The intersection of a complex sphere and cone can be calculated by setting the equations of the sphere and cone equal to each other and solving for the variables. This will give the coordinates of the points where the two surfaces intersect.

3. What does the intersection of a complex sphere and cone look like?

The intersection of a complex sphere and cone can have various shapes depending on the equations of the sphere and cone. It can be a point, a line, a circle, an ellipse, or a parabola.

4. What are some real-world applications of the intersection of a complex sphere and cone?

The intersection of a complex sphere and cone has applications in areas such as computer graphics, engineering, and physics. It can be used to model objects and their interactions, calculate trajectories, and create 3D images.

5. How does the intersection of a complex sphere and cone relate to complex numbers?

The equations of a complex sphere and cone involve complex numbers, and the intersection points also have complex coordinates. The use of complex numbers allows for a more accurate and complete representation of the intersection.

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