$R_t$, $\phi$	$R_p$, $\theta$

Torus

Torus is an elementary mathematical object. Unlike a ball. it has one parameter more - radius of circular tube. It's called poloidal radius in the project, to follow common style of naming. So called toroidal radius of the torus describes a 2-D circle drawn by the line following through the tube center.

Here is formula that describes torus depicted on the right. It uses polar coordinate system ($\rho$,$\theta$,$\phi$). Unlike Cartesian coordinates, these equations are simple and convenient for manipulation. Only angles serve as unrestricted parameters for modelling and animation. Both radiuses are defined as preset parameters, or constants, at the time of animation.

It makes sense to scale angles to normalized coordinates, to match value of $\mathrm{2\pi }$ to plane $1$. As a result, the value of coordinate at start point of the circle will always become an integer value.

Below is equivalent description of a torus in Cartesian coordinates being used for parameters. Equation can be hardly manipulated in the process of modelling because it requires conformance to valid intervals of values strictly. Otherwise, the equation won’t be solved. For these reasons, this equation is presented for reference only. It was used early in the project.

Legend:
• $R_t$ - toroidal radius of the torus circle (line of the tube center)
• $0\le R_p\le R_t$ - poloidal radius of the tube section from its center
• $0\le \theta \le \mathrm{2\pi }$ - poloidal angle (across section)
• $0\le \phi \le \mathrm{2\pi }$ - toroidal angle (along the tube)
• $x,y,z$ - Cartesian coordinates in 3-D

References:
• 📚 Torus

⏪ 🔼 ⏩