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**.
["# The Geometric Truth: Why ( \ ext{right}^2 + (z - c/2)^2 = (c/2)^2 ) Creates a Sphere — Not a Torus, When Rotated Properly", "When you explore 3D geometry in spherical coordinates and surface transformations, intriguing shapes emerge — and one often misunderstood object is the shape defined by the equation:", "[\n\ ext{right}^2 + (z - c/2)^2 = \left(\frac{c}{2}\right)^2\n]", "At first glance, this defines a circle aligned along the ( r )-axis in the ( r\ ext{-}z ) plane, centered at ( r = c/2 ), ( z = 0 ), with radius ( c/2 ). But what happens when we rotate this curve about the ( z )-axis — and how does that produce surprising results?", "## Understanding the Equation", "This equation describes a circle in the ( r )-( z ) plane — that’s a 2D circle lying flat at ( z = 0 ), centered at ( (r = c/2, z = 0) ), radius ( c/2 ). As we rotate every point on this circle around the ( z )-axis, we generate a surface of revolution.", "However, simply rotating a single circle around the ( z )-axis doesn’t create a torus — a torus requires points at different radii to trace around, with a range of distances from the axis. Here, rotation of this specific circle around ( z ) doesn’t vary the ( r )-coordinate: since all points already lie at fixed ( r ), the rotation causes each point to sweep out a full circle in the ( r )-( \ heta ) plane.", "But wait — this is where geometry gets crucial.", "### Correct Interpretation: The Circle Is Not Rotated within Its Plane — Rather, It’s Already Defined by ( r ), and Rotation Around ( z ) Generates Full Spherical Symmetry", "Actually, when centered on the ( z )-axis at ( r = c/2 ), the circle lies in the radial direction — but to form a surface, we're rotating this circle about the ( z )-axis, not lifting it off the plane. So the true transformation is:", "> Rotate the circle ( r^2 + (z - c/2)^2 = (c/2)^2 ) around the ( z )-axis.", "However, since the entire circle lies in a plane perpendicular to the radial direction (the ( r )-axis), rotating it about ( z ) produces all points whose distance from the ( z )-axis varies between ( 0 ) (at center ( r = c/2 \pm c/2 )) — wait: let’s resolve the confusion.", "### Clarifying the Geometry: Is This Circle Actually In the ( r )-( z ) Plane?", "Yes: In cylindrical coordinates, ( r ) is radial distance from ( z )-axis, ( z ) vertical. So the circle roughly means:", "[\nr = \frac{c}{2} + \frac{c}{2} \cos \phi \quad \ ext{(not exact)}\n]", "Wait — more precisely, the equation:", "[\nr^2 + (z - c/2)^2 = (c/2)^2\n]", "describes a circle of radius ( c/2 ) located in the ( r\ ext{-}z ) plane at ( z = 0 ), offset along the radial direction by ( r = c/2 ). But this is not a vertical circle — it’s a circle lying in the plane ( z = 0 ), centered at ( (r = c/2, z = 0) ), rotating in the radial-azimuthal plane.", "### The Rotation That Forms a Sphere", "To form a sphere, we rotate a circle passing through the origin about an axis through its center — or, in 3D, when the center lies on the axis of rotation, rotation generates a sphere.", "But here, the center is not on the ( z )-axis — it’s at ( r = c/2 ), ( z = 0 ). So if we rotate this circle around ( z ), every point sweeps a circle of radius ( r ) — but since ( r ) varies from ( 0 ) to ( c ) (because the circle extends from ( r = 0 ) to ( r = c )), rotating it sweeps out all points at distance ( \sqrt{r^2 + z^2} ) such that ( r^2 + (z - c/2)^2 = (c/2)^2 ).", "Let’s square both sides of the original equation:", "[\nr^2 + (z - c/2)^2 = (c/2)^2\n\Rightarrow r^2 + z^2 - c z + \frac{c^2}{4} = \frac{c^2}{4}\n\Rightarrow r^2 + z^2 = c z\n]", "This is a known surface in cylindrical coordinates. Convert to Cartesian:", "( r^2 = x^2 + y^2 ), ( z = z ), so:", "[\nx^2 + y^2 + z^2 = c z\n\Rightarrow x^2 + y^2 + z^2 - c z = 0\n\Rightarrow x^2 + y^2 + \left(z - \frac{c}{2}\right)^2 = \left(\frac{c}{2}\right)^2\n]", "### This IS A SPHERE!", "Yes — completing the square gives a sphere of radius ( c/2 ) centered at ( (0, 0, c/2) ), not a torus.", "But why the confusion about torus?", "### Why It’s Not a Torus", "A torus arises when you rotate a circle in a plane perpendicular to the axis of rotation, where the center of the circle lies on the axis — for example, a circle in the ( x )-( z ) plane, centered at ( (R, 0, 0) ), rotated about ( z ), produces a torus of major radius ( R ) and minor radius ( R ).", "But here, although we're rotating a circle along the ( r )-axis, the circle is not in the ( x )-( z ) plane — it’s in a plane containing the ( z )-axis and offset radially. The rotation sweeps out a sphere — not a doughnut shape.", "### When Does Rotation Give a Torus?", "A torus forms when a circle lying in a plane intersecting two radial planes is rotated about an axis through its center but not intersecting it — like a circle centered at ( r = R ) on the ( z )-axis, rotated about ( z ), forming a torus of major radius ( R ) and minor radius ( R ).", "Our circle is centered at ( r = c/2 ) on the ( z \)-plane — but unless we interpret the surface differently, rotation produces a sphere, not a torus.", "Why then say “rotated about the ( z )-axis forms a sphere”? Because rotating that particular circle — which lies in a plane (the ( r\ ext{-}z ) plane), and whose center lies on the axis of rotation — generates a sphere if it describes a circle symmetrically around the axis.", "But wait — this circle has radius ( c/2 ), center at ( (r = c/2, z = 0) ). When rotated about ( z ), each point traces a circle of radius ( r ) — and since ( r ) goes from 0 to ( c ) (from origin to edge), every angular position sweeps a full circle. The total surface is the set of all ( (r, \ heta, z) ) such that ( \sqrt{r^2 + (z - c/2)^2} = c/2 ) — which as shown, simplifies to a sphere centered at ( (0, 0, c/2) ) with radius ( c/2 ).", "### Visualization and Takeaway", "- The surface defined by ( r^2 + (z - c/2)^2 = (c/2)^2 ) is a sphere of radius ( c/2 ), centered at ( (r=0, z=c/2) ) in cylindrical coordinates.\n- Rotation about the ( z )-axis does not deform the shape — it already lies in a fixed radial plane, and rotation produces full spherical symmetry.\n- The familiar torus arises from a different geometric setup: a circle not lying on an axis-boundary plane, rotated around an axis through its center but not on it.", "### Summary: The Misconception Clarified", "While rotating a circle in the ( r\ ext{-}z ) plane about the ( z )-axis seems like it might form a torus, the specific case here — a circle centered offset from the axis and lying in a plane containing the axis — generates a sphere when rotated. This exemplifies how subtle changes in geometry lead to vastly different surfaces.", "> Conclusion: The equation ( r^2 + (z - c/2)^2 = (c/2)^2 ) defines a sphere of radius ( \frac{c}{2} ), not a torus, when interpreted in 3D space and rotated properly about the ( z )-axis. Understanding the role of the center and orientation of the circle is key to avoiding confusion.", "---", "### Related Topics:\n- Surface of revolution in cylindrical coordinates\n- Spherical vs. toroidal symmetry\n- Geometric interpretation of quadratic surfaces", "Optimize this with schema for rich results — markup the sphere, link to 3D geometry guides, include a proof sketch of the final sphere equation."]
