This is a surface of revolution about the $ z $-axis (since $ \phi $ is the polar angle). To identify the conic, we convert to Cartesian coordinates. Recall:

["Understanding Surfaces of Revolution About the $ z $-Axis: A Guide Using Cartesian Coordinates", "When studying surfaces of revolution, one of the most fundamental and visually revealing approaches is to analyze the surface generated by revolving a two-dimensional curve around the $ z $-axis. Since the axis of symmetry is the $ z $-axis, the generating curve lies in a plane containing this axis—typically the $ xz $- or $ yz $-plane—and is defined as a function of the polar angle $ \phi $, an essential parameter in cylindrical coordinates.", "### What Is a Surface of Revolution About the $ z $-Axis?", "A surface of revolution about the $ z $-axis arises when a plane curve, bounded and defined by $ r(\phi) $ and $ z(\phi) $ in cylindrical coordinates ($ r = \sqrt{x^2 + y^2} $, $ \phi $ is the polar angle), is rotated around the $ z $-axis. This means every point on the original curve traces a circle parallel to the $ xy $-plane at height $ z $, with radius equal to its cyclic distance from the $ z $-axis.", "The key idea is that such surfaces are symmetric about the $ z $-axis, a property that simplifies analysis, classification, and parametrization.", "### Role of the Polar Angle $ \phi $ in Defining the Curve", "The polar angle $ \phi $ describes the orientation of points on the generating curve relative to the positive $ z $-axis. However, in standard setups for surfaces of revolution, $ \phi $ typically parameterizes the generating curve in cylindrical coordinates as $ (r(\phi), \phi, z(\phi)) $. This curve traces a trajectory where, for each angular position (e.g., $ \phi = \ ext{constant} $), a cylindrical slice is defined.", "Since the surface is formed by rotating this entire curve, $ \phi $ may directly govern the radial distance $ r(\phi) $ from the $ z $-axis—forming a shape like a cone, hyperboloid, or paraboloid, depending on how $ r(\phi) $ and $ z(\phi) $ vary.", "### Converting to Cartesian Coordinates to Identify the Conic", "To determine what conic section or three-dimensional surface we obtain, converting from cylindrical to Cartesian coordinates is invaluable. Recall:", "$$\nx = r(\phi) \cos\ heta,\quad y = r(\phi) \sin\ heta,\quad z = z(\phi)\n$$", "Given that $ r(\phi) = \sqrt{x^2 + y^2} $, substituting into the parametric equations for the surface of revolution yields:", "$$\nx(\phi, \ heta) = \sqrt{x(\phi)^2 + y(\phi)^2} \cdot \cos\ heta,\quad z(\phi, \ heta) = z(\phi)\n$$", "But more directly, eliminating $ \phi $ and parameterizing purely by $ \ heta $ and $ r $, we can analyze cross-sections.", "For example, fixing $ \phi = \ ext{constant} $ gives a circle of radius $ r(\phi) $ at height $ z(\phi) $. As $ \phi $ varies, points sweep out surfaces depending on $ r(\phi) $ and $ z(\phi) $. Using the relation:", "$$\nx^2 + y^2 = r(\phi)^2,\quad z = z(\phi)\n$$", "we can express the surface equation in Cartesian form by eliminating $ \phi $, revealing whether the surface embodies an ellipsoid, hyperboloid, or paraboloid.", "### Classifying the Surface: Conic Sections in Cartesian Form", "Depending on the functional forms of $ r(\phi) $ and $ z(\phi) $, the surface may match standard quadric surfaces. For instance:", "- If $ r(\phi) \propto \cos\phi $ or $ \sin\phi $, and $ z(\phi) $ linear, the surface could be a hyperboloid of one sheet or two sheets.\n- If $ r^2 \propto z $, or $ z \propto r^2 $, then the surface is a paraboloid.\n- Circular cross-sections scaling smoothly with $ z $ yield elliptic or hyperbolic ellipsoids.", "By translating the polar-coordinate description into Cartesian implicit or parametric equations, we can identify the exact conic or surface geometry.", "### Practical Example: Revolution of a Line", "Suppose $ r(\phi) = k\phi $ and $ z(\phi) = m\phi $ for constants $ k, m $—this traces a linear function in cylindrical coordinates. Rotating this line ($ \phi = \ ext{const} $) about the $ z $-axis gives a cone, whose Cartesian equation is:", "$$\nx^2 + y^2 = \left( \frac{z}{m/k} \right)^2 = C z^2\n$$", "or $ x^2 + y^2 = a^2 z^2 $, which defines a circular cone opening along $ z $.", "### Summary", "- A surface of revolution about the $ z $-axis is generated by rotating a planar curve parameterized by polar angle $ \phi $.\n- Converting to Cartesian coordinates removes angular dependence and enables explicit surface equations.\n- Identifying the surface type relies on analyzing how $ r(\phi) $ and $ z(\phi) $ define $ x, y, z $.\n- Whether elliptic, hyperbolic, or hyperboloidal, the symmetry about $ z $-axis and rotational invariance simplify classification.", "Understanding this connection between polar angular parameterization and Cartesian geometry empowers precise modeling in physics, engineering, and computer graphics—especially in simulating realistic 3D forms around central axes.", "---", "Source: Advanced analytical geometry and surface modeling, leveraging cylindrical/spherical coordinates.\nKeywords: surface of revolution, $ z $-axis, conic sections, cylindrical coordinates, Cartesian conversion, $ \phi $ angle, parametric surfaces."]