# How can generate a point using rotation from other point in n-dimensional space? [closed]

Suppose we have a point $$P_1 = [X_1,X_2, X_3, ... X_n]$$ in n-dimension space, How does the rotation of point $$P_1$$ get another point $$P_2= [Y_1, Y_2, Y_3, ...Y_n]$$? There is a method to do multiple rotations to form a well-known curve starting from $$P_1$$, for example figure

• Are you talking about $\mathbb{R}^{n}$? Commented Aug 15, 2021 at 5:02
• Yes, where each point has a coordinate in the n axis. Commented Aug 15, 2021 at 12:02
• " There is a method to do multiple rotations to form a well-known curve starting from", how is it named? Commented Aug 16, 2021 at 5:43
• Anyway, in the 2 dimensional case you can use complex numbers. For example, powers (positive and negative) of $1 + 0.2i$ give a beautiful spiral around the origin. Commented Aug 16, 2021 at 18:08
• @Elmex80s I ask for that I forgot to make the " ? ". I didn't understand your method of using complex numbers, can you give an example? Commented Aug 17, 2021 at 13:34

From Powers of complex numbers I take these pictures

and

which spiral a point around $$O$$. By taking negative powers you can go other way around as well.

For dimensions higher than 3 I think almost every invertible matrix will do.

• Thank you for this explanation, I am not a mathematician therefore i have difficulty understanding. So, given one point like in your figure blue ones, how i can find the coordinate of the others points? Commented Aug 18, 2021 at 14:13
• It are all powers of the complex number $z$. $z$ can be every number you want as long $0 < |z| < 1$ or $1 < |z|$. Note in your example the angles increase with every step, in mine their are constant. Commented Aug 18, 2021 at 16:36
• ok good, there is only the version of a complex number? because i want the same things with real numbers. If you have any suggestions please make them as ref or explain them. Commented Aug 18, 2021 at 17:49
• Complex numbers are pairs of real numbers. Lookup how their multiplication works and you should be able to continue. I could write it down but it is searchable everyhwere. Commented Aug 19, 2021 at 10:58
• Ok thank you brother Commented Aug 19, 2021 at 11:34