How can this indicate rotation?

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SUMMARY

The discussion centers on the mathematical representation of rotation using complex numbers and quaternions. The equation r² = (x - ia)² + (y - ib)² + (z - ic)² indicates rotation in three-dimensional space, where 'i' represents the imaginary unit. It is established that multiplication by 'i' corresponds to a 90-degree rotation in the complex plane. Furthermore, the use of quaternions allows for a more comprehensive representation of three-dimensional rotations, where multiplication with another quaternion results in a rotational transformation.

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yicong2011
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If we say

r2= (x-a)2 + (y-b)2 + (z-c)2

can generally represent displacement,

why can

r2= (x- ia)2 + (y- ib)2 + (z- ic)2

generally represent rotation?

(i2 = -1)


And why

r2= x2 + y2 + (z - ia)2

represents rotation with respect to z axis
 
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I'm not quite sure what you're saying here.

What I see is the equation of a sphere of which the center has been translated.
First over real coordinates, and then over imaginary coordinates.

What I can tell you, is that if you can for instance represent 2-dimensional coordinates by a number in the imaginary plane, that multiplication by i equals rotation over an angle of 90 degrees.

More generally if you have 3-dimensional coordinates, you can represent them by a number in quaternion space. Multiplication with an other quaternion number comes out as a rotation.
 

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