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Home » Magic Final Fantasy: The Hidden Magic You Never Knew Game Needed! - DNSFLEX

Magic Final Fantasy: The Hidden Magic You Never Knew Game Needed! - DNSFLEX

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 This is the equation of a **circle** in the \( (r,z) \)-plane, rotated about the \(z\)-axis, forming a **torus**? No â actually, since \(r\) is radial, this describes a **sphere** of radius \(c/2\) centered at \((x, y, z) = (0, 0, 0, c/2)\) in cylindrical coordinates.

📌 Wait: \(\left(r - rac{c}{2}

📌 ight)^2\) is a circle in the \(r\)-axis plane, centered at \(r = c/2, z = 0\), radius \(c/2\). Rotated about the \(z\)-axis, this forms a **torus**? No â it forms a **sphere**.

📌 Indeed, expand:

📌 r^2 - c r + rac{c^2}{4} + z^2 = rac{c^2}{4} \Rightarrow r^2 + z^2 = c r

📌 But \(r^2 + z^2 = x^2 + y^2 + z^2\), and \(r = \sqrt{x^2 + y^2}\), so:

📌 Now square both sides carefully:

📌 This is not a sphere â it's a **quartic surface**. But wait: earlier we had:

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