Utilizing polar coordinates, complex numbers, spherical coordinates, and vector calculus, it's demonstrated that a circle in the complex plane corresponds to a circle on the sphere, showcasing their interrelation through various mathematical frameworks.
A) By Using Polar Coordinates:
In the complex plane, a circle with center (
) and radius r can be represented by
, where
is the polar angle. Mapping this circle onto a sphere involves projecting each point
onto the unit sphere. Polar coordinates facilitate this mapping, providing a direct link between the complex plane and the sphere.
B) By Applying Complex Numbers:
Complex numbers
can be associated with points on the complex plane. Mapping these points onto a sphere involves extending the imaginary axis to form a three-dimensional space. The sphere is then generated by projecting each point onto the unit sphere. Complex numbers aid in this transformation, demonstrating the correspondence between circles in the complex plane and spheres.
C) By Using Spherical Coordinates:
Spherical coordinates
provide a natural representation of points on a sphere. A circle in the complex plane, expressed in polar coordinates, can be seamlessly translated into spherical coordinates, highlighting the connection between the two geometries.
D) By Applying Vector Calculus:
Representing the sphere using vector calculus involves considering parametric equations for the sphere. A circle in the complex plane can be associated with a parametric representation, allowing for a direct connection to the sphere through vector calculus operations.