1. What is the Torus Equation?

The torus parametric equations describe the surface of a doughnut-shaped object in 3D space. A torus is defined by two radii: the major radius (R) from the center of the hole to the center of the tube, and the minor radius (r) of the tube itself.

2. How Does the Calculator Work?

The calculator uses the parametric equations:

Where:

• \( R \) — Major radius (distance from center to tube center)
• \( r \) — Minor radius (radius of the tube)
• \( \theta \) — Angle around the tube (0 to 2π)
• \( \phi \) — Angle around the central axis (0 to 2π)

Explanation: The equations generate 3D coordinates for every point on the torus surface by combining circular motion around the tube with circular motion around the central axis.

3. Importance of Torus Calculations

Details: Torus calculations are essential in computer graphics, physics simulations, engineering design, and mathematical modeling of various natural and man-made structures.

4. Using the Calculator

Tips: Enter both radii (must be positive), and angles in radians. The calculator will output the corresponding (x,y,z) coordinates on the torus surface.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between R and r?
A: R is the distance from the center of the torus to the center of the tube, while r is the radius of the tube itself.

Q2: What angle range should I use?
A: Typically θ and φ range from 0 to 2π (360°) to cover the entire surface, but any real number is valid due to trigonometric periodicity.

Q3: Can I use degrees instead of radians?
A: The calculator requires radians. Convert degrees to radians by multiplying by π/180.

Q4: What if R < r?
A: When R < r, the torus intersects itself, creating a spindle torus. When R = r, it's called a horn torus.

Q5: How is this used in real applications?
A: Torus equations are used in CAD modeling, physics (magnetic confinement devices), computer graphics, and more.