How can this indicate rotation?

[yicong2011]

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

 

[I like Serena]

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.