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.
["Understanding the Geometry: The Torus-Like Sphere in the (r,z) Plane Rotated About the z-Axis", "In the world of coordinate geometry, cylindrical coordinates express points using radial distance, height, and angular position. The equation describing a circle in the ( (r,z) )-plane—rotated about the ( z )-axis—generates a familiar and elegant 3D surface: the torus. However, an important geometric insight reveals that this surface, under rotation, actually produces a sphere when the circle is centered appropriately.", "Let’s explore what this means.", "---", "### Defining the Equation in Cylindrical Coordinates", "In cylindrical coordinates, a point is defined by ( (r, \ heta, z) ), where:", "- ( r ) is the radial distance from the ( z )-axis,\n- ( \ heta ) is the azimuthal angle,\n- ( z ) is height along the ( z )-axis.", "Consider a circle in the ( (r, z) )-plane centered at ( (r = 0, z = c/2) ) with radius ( c/2 ). Its Cartesian equation is:", "[\nr^2 + \left(z - \frac{c}{2}\right)^2 = \left(\frac{c}{2}\right)^2\n]", "Expanding this:", "[\nr^2 + z^2 - c z + \frac{c^2}{4} = \frac{c^2}{4}\n]", "[\nr^2 + z^2 - c z = 0\n]", "Now, because the shape is independent of ( \ heta ), we can rotate this 2D circle about the ( z )-axis. In cylindrical coordinates, every point at radial distance ( r ) from the ( z )-axis traces a circular ring of radius ( r ) located at height ( z ).", "Thus, rotating the circle centered at ( (0, 0, c/2) ), radius ( c/2 ) about the ( z )-axis yields a torus — a surface with a circular cross-section swept around an axis.", "---", "### From Torus to Sphere: When Is This a Sphere?", "The key observation lies in where the center of the original circle lies. Our circle is centered at ( (r = 0, z = c/2) ). If this center lay on the ( z )-axis — that is, at ( r = 0 ) and ( z = 0 ) — then rotating about the ( z )-axis would not create a torus, but instead a sphere.", "But here, the center is at ( z = c/2 ), so rotating forms a torus unless the center is deviated precisely in a way that balances the radial coordinate.", "Wait — consider modifying the initial assumption:", "Suppose instead the original circle lies in the ( (r, z) )-plane not centered on the ( z )-axis, but displaced vertically so that its center is exactly at ( z = c/2 ), with radial displacement zero — then, by symmetry, rotational symmetry about ( z )-axis produces a sphere.", "Indeed, if a circle of radius ( \frac{c}{2} ) in the ( (r,z) )-plane is centered at ( (r, z) = (0, c/2) ), then rotating it about the ( z )-axis yields every point at distance ( r ) from the ( z )-axis where:", "[\nr^2 + \left(z - \frac{c}{2}\right)^2 = \left(\frac{c}{2}\right)^2\n]", "But this expansion gives:", "[\nr^2 + z^2 - c z + \frac{c^2}{4} = \frac{c^2}{4} \implies r^2 + z^2 = c z\n]", "Now switch to Cartesian coordinates: recall ( r^2 = x^2 + y^2 ), and ( z ) remains ( z ). Substituting:", "[\nx^2 + y^2 + z^2 = c z\n]", "Rewriting:", "[\nx^2 + y^2 + \left(z - \frac{c}{2}\right)^2 = \left(\frac{c}{2}\right)^2\n]", "This is the equation of a sphere of radius ( \frac{c}{2} ), centered at ( (x, y, z) = \left(0, 0, \frac{c}{2}\right) ). The full surface consists of all points equidistant from this central point — confirming the result.", "---", "### Why This Matters: Geometry of Rotations", "Even though the standard torus forms by rotating a circle around the ( z )-axis with nonzero radial offset, a subtle shift changes the surface entirely. When the circle’s center lies on the axis of rotation — particularly at a height of ( c/2 ) — the resulting surface becomes a sphere, not a torus.", "This highlights a powerful idea: the symmetry axis and center of a generating curve determine the global topology of the rotated surface.", "---", "### Summary", "- A circle in the ( (r,z) )-plane, centered at ( (0, c/2) ), radius ( c/2 ), generates a torus when rotated about the ( z )-axis.\n- But when the center is on the ( z )-axis — especially at ( z = c/2 ) — and the circle has radial extent, rotation yields a sphere.\n- The full equation becomes ( x^2 + y^2 + \left(z - \frac{c}{2}\right)^2 = \left(\frac{c}{2}\right)^2 ), a sphere of radius ( c/2 ) centered at ( (0, 0, c/2) ) in Cartesian coordinates.\n- This illustrates how rotational symmetry creates symmetric surfaces — and how small geometric tweaks alter topology dramatically.", "---", "Double-Check Your Understanding:\nThe equation describing this surface is not a torus unless the center is off-axis. When centered on the ( z )-axis at height ( c/2 ), rotation delivers a sphere — a cornerstone concept in 3D geometry and global analysis.", "🔍 Useful Terms for SEO Optimization:\n- Circle rotation around z-axis\n- Generating circle in cylindrical coordinates\n- Sphere from rotational symmetry\n- Torus vs sphere geometry\n- Parametric surfaces in cylindrical coordinates\n- 3D surface equations in cylindrical system", "---", "Understand this elegant transformation: rotating the right circle creates a sphere — nature’s symmetry made visible."]
