Some properties of the complex sphere

The complex sphere is also called the Riemann sphere. It is a complex number representation method that makes each point on the sphere uniquely correspond to a complex number on the complex plane. A one-to-one mapping is formed between the two. As shown in the figure below, according to the similar properties of elementary triangles, the transformation relationship between coordinate points on the complex sphere and the corresponding coordinate points on the complex plane can be derived. This article attempts to deduce this relationship .

SpRsQ coordinates are

$(x', y', z')$

Suppose the corresponding point Q coordinate on the complex plane is

$(x, y)$

Then, according to the similar triangle ratio formula:

$\\ \frac{2}{2-z'}=\frac{x}{x'}\\ \frac{2}{2-z'}=\frac{y}{y'}$

In addition

$x'^2+y'^2+(z'-1)^2 = 1$

and so

$\\x'=\frac{x(2-z')}{2}\\ y'=\frac{y(2-z')}{2}$

and so

$\ frac {x ^ 2 (2-z ') ^ 2} {4} + \ frac {y ^ 2 (2-z') ^ 2} {4} + (z'-1) ^ 2 = 1$

and so:

$\ frac {(2-z ') ^ 2 (x ^ 2 + y ^ 2)} {4} = 1-z' ^ 2 + 2z'-1 = 2z'-z '^ 2$

and so:

$(z '^ 2-4z' + 4) (x ^ 2 + y ^ 2) = 8z'-4z '^ 2$

Finished up:

$(x ^ 2 + y ^ 2 + 4) z '^ 2- (4x ^ 2 + 4y ^ 2 + 8) z' + 4x ^ 2 + 4y ^ 2 = 0$

$\\\ Delta = b ^ 2-4ac = (4x ^ 2 + 4y ^ 2 + 8) ^ 2-4 (x ^ 2 + y ^ 2 + 4) (4x ^ 2 + 4y ^ 2) \\ = ( 4x ^ 2 + 4y ^ 2) ^ 2 + 16 (4x ^ 2 + 4y ^ 2) + 64-16 (x ^ 4 + x ^ 2y ^ 2 + y ^ 2x ^ 2 + y ^ 4 + 4x ^ 2 + 4y ^ 2) \\ = (16x ^ 4 + 16y ^ 4 + 32x ^ 2y ^ 2) +64 (x ^ 2 + y ^ 2) + 64-16 (x ^ 4 + 2x ^ 2y ^ 2 + y ^ 4) -64 (x ^ 2 + y ^ 2) \\ = 16 (x ^ 4 + 2x ^ 2y ^ 2 + y ^ 4) +64 (x ^ 2 + y ^ 2) + 64-16 (x ^ 4 + 2x ^ 2y ^ 2 + y ^ 4) -64 (x ^ 2 + y ^ 2) = 64$

and so:

$\\ z '= \ frac {4x ^ 2 + 4y ^ 2 + 8 \ pm \ sqrt {\ Delta}} {2 (x ^ 2 + y ^ 2 + 4)} = \ frac {4x ^ 2 + 4y ^ 2 + 8 \ pm 8} {2 (x ^ 2 + y ^ 2 + 4)} \\ z'_1 = \ frac {2 (x ^ 2 + y ^ 2)} {(x ^ 2 + y ^ 2 +4)} \\ z'_2 = \ frac {4x ^ 2 + 4y ^ 2 + 16} {2 (x ^ 2 + y ^ 2 + 4)} = \ frac {4 (x ^ 2 + y ^ 2 +4)} {2 (x ^ 2 + y ^ 2 + 4)} = 2$

Obviously only $z'_1$ meet the requirements.

Therefore, the $(x, y)$ coordinates $(x', y', z')$ of the complex spherical surface corresponding to a point on the complex plane are :

$\\ x '= \ frac {x (2-z')} {2} = \ frac {4x} {x ^ 2 + y ^ 2 + 4} \\ y '= \ frac {y (2-z' )} {2} = \ frac {4y} {x ^ 2 + y ^ 2 + 4} \\ z '= \ frac {2 (x ^ 2 + y ^ 2)} {(x ^ 2 + y ^ 2 +4)}$

The correlation formula is obtained, let’s take $with'$ a look at the rules

Change the form:

$z '= \ frac {2 (x ^ 2 + y ^ 2)} {x ^ 2 + y ^ 2 + 4} = \ frac {2} {1+ \ frac {4} {x ^ 2 + y ^ 2 }}$

among them

$x^2+y^2=r^2$

r is the argument in the complex plane

and so:

$z'=\frac{2(x^2+y^2)}{x^2+y^2+4} = \frac{2}{1+\frac{4}{x^2+y^2}}= \frac{2}{1+\frac{4}{r^2}}$

The graphic is:

Corresponding complex sphere:

It can be seen that, taking the radius of the complex plane great circle corresponding to the great circle parallel to the complex plane on the complex sphere as the boundary (that is, r=2), z is >1 and <1, respectively

There should be a better way to solve the combination of numbers and shapes, but I haven't found it yet.

Some properties of the complex sphere

end!

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