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A particle constrained to a stationary smooth surface of revolution $z=f(r)$ under uniform gravity conserves $h=r^2\dot{\theta }$ and mechanical energy per unit mass. Substitute $\dot{z}=f^{\prime }(r)\dot{r}$ and $\dot{\theta }=h/r^2$ into that energy to obtainThis proves the radial reduction. Simple roots of $E-U_h$ are turning points; a stationary point $U_h^{\prime }(r_0)=0$ supports an exact circular orbit. At a local maximum of $U_h$, another trajectory at the same energy can approach the orbit asymptotically as a separatrix. Existence of roots elsewhere is insufficient for confinement: their dynamically accessible ordering relative to the launch radius matters.

In unwrapped upper-half-plane coordinates, the annulus $b\le y\le a$ on a tractoid identifies $x$ modulo $2\pi$. Connect $(x_0,b)$ and $(x_0+2\pi ,b)$ by the upper circular hyperbolic geodesic with center $(x_0+\pi ,0)$. Its largest height is $\sqrt{b^2+{\pi }^2}$. The displayed condition puts the whole arc in the annulus. Under the angular identification its endpoints coincide, and the resulting geodesic segment winds once before returning. Its terminal tangent generally differs from the initial tangent, so this construction does not give a smooth periodic closed geodesic. It explains how local hyperbolic geometry creates a returning segment on a surface of revolution.

Let a simple closed regular planar curve be invariant under point reflection through a center and remain strictly on one side of a coplanar axis of rotation, at distance $d$ from its center. If $L$ is its arc length, its surface of revolution has surface area $2\pi dL$. To prove this, choose coordinates with the center at the origin and the axis of rotation at $y=-d$. The surface area formula is $2\pi \oint (d+y)ds$. Pair each point with its reflected point: the arc length element $ds$ is unchanged and $y$ changes sign. Thus $\oint yds=0$, proving the result. The one-side condition prevents the surface from passing through the axis of rotation or requiring an absolute value with changing sign.

A toroid is a ring-shaped surface of revolution obtained by rotating a closed planar profile about an axis in its plane that stays outside the profile. A circular profile gives a torus; a rectangular profile gives a ring with rectangular cross-section. The enclosed solid supplies the core geometry in a toroidal winding, while the mathematical surface is its boundary. The distinction matters when computing a surface area versus a volume.

The tractoid is the surface of revolution of a unit tractrix. For $X(\theta ,v)=(h(\theta ),\sin \theta \cos v,\sin \theta \sin v)$ with $0<\theta <\pi /2$, its coordinate tangent vectors are orthogonal and have squared lengths ${\cot }^2\theta ,{\sin }^2\theta$. The coordinates $x=v$, $y=\csc \theta$ transform this first fundamental form into $(d{x}^2+d{y}^2)/y^2$, giving a local isometry to the hyperbolic plane. Thus its Gaussian curvature is minus one. The angular coordinate is periodic globally, while its unwrapped coordinate is a real variable; this distinguishes the local metric identification from a global embedding of the whole hyperbolic plane.