How to use and rotate polar coordinates along the z-axis?

Question

I'm learning trigonometry in code. I know the rotate around a point, for the x-axis you use cos, for the y-axis you use sin. What about the z-axis? What if I wanted to rotate on both the z and x axis? Or y and z axis? How?
x-axis = radius * cos(angle)
y-axis = radius * sin(angle)
z-axis = ?

Answer 1

The simplest way to rotate points in 3D is to use the same math you do in 2D but apply them to different 2 dimensions. For example, you could rotate around the x-axis by rotating the y and z coordinates like you've learned to do with x and y in 2D. To rotate around the y axis, use the same math but with the 3D point's x and z coordinates. You could rotate around 1 axis and then by another.
It can be helpful to represent your rotations as matrices. A matrix can represent multiple rotate operations, rotations around an arbitrary axis, and many other kinds of transformations. There is more detail on rotation matrices at:
https://en.wikipedia.org/wiki/Rotation_matrix
You commented on the 2D polar coordinate system so I wanted to point out a couple related 3D coordinate systems.
The cylindrical coordinate system is basically a mixture of Cartesian in 1 dimension and polar in the other 2 dimensions. More detail at:
https://en.wikipedia.org/wiki/Cylindrical_coordinate_system
The spherical coordinate system is basically the most complete 3D analogue of the 2D polar coordinates. It is a 3D direction similar to GPS coordinates on Earth with a radius. More detail at:
https://en.wikipedia.org/wiki/Spherical_coordinate_system

Answer 2

What would you implement? The golden ratio is a number like PI.
The golden ratio is roughly 1.61803399 but the digits continue infinitely like PI.
This is what my Windows calculator gets: 1.6180339887498948482045868343656
Are you asking how to calculate it?
In JavaScript, you could calculate like this: (1 + Math.sqrt(5)) / 2
If you just want to save it somewhere in JavaScript, just copy it to a const and make it anywhere you want to use it.

Answer 3

Josh Greig Thanks. On the spherical coordinate system (I think it was that link) the formulas involved the phi symbol

Answer 4

Josh Greig Thanks so much for the detailed response! I'll check them links in the morning.

Answer 5

Josh Greig A question, how do you implement the golden ratio? I did some research and it isn't in the Math object

Answer 6

Mirielle Thanks for the example!