An implicit equation in Cartesian coordinates for a torus radially symmetric about the z-axis is

or the solution of f(x, y, z) = 0, where

Algebraically eliminating the square root gives a quartic equation,

The three classes of standard tori correspond to the three possible aspect ratios between R and r:

When R > r, the surface will be the familiar ring torus or anchor ring. R = r corresponds to the horn torus, which in effect is a torus with no "hole". R < r describes the self-intersecting spindle torus; its inner shell is a lemon and its outer shell is an apple
When R = 0, the torus degenerates to the sphere.When R ≥ r, the interior

of this torus is diffeomorphic (and, hence, homeomorphic) to a product of a Euclidean open disk and a circle. The volume of this solid torus and the surface area of its torus are easily computed using Pappuss centroid theorem, giving:
These formulas are the same as for a cylinder of length 2πR and radius r, obtained from cutting the tube along the plane of a small circle, and unrolling it by straightening out (rectifying) the line running around the center of the tube. The losses in surface area and volume on the inner side of the tube exactly cancel out the gains on the outer side.
This injury to the epithelial basal lamina results in inflammatory cells and plasma proteins leaking into the alveolar space and forming fibrin, resulting in an initial fibroblast driven intra-alveolar fibroproliferation. The fibroblasts differentiate into myofibroblasts and continue to form fibrosis resulting in intra-alveolar fibroinflammatory buds (Massons Bodies) that are characteristic of organizing pneumonia. These Massons bodies consist of inflammatory cells contained in an extracellular matrix consisting of type I collagen, fibronectin, procollagen type III, tenascin C and proteoglycans. Angiogenesis , or the formation of blood vessels, occurs in the Massons bodies and this is driven by vascular endothelial growth factor. Remodeling occurs, resulting in the intra-alveolar fibroinflammatory buds (Massons Bodies) moving into the interstitial space and forming collagen globules that are then covered by type 1 alveolar epithelial cells with well developed basement membranes. These type 1 alveolar epithelial cells (pneumocytes) then proliferate, restoring the continuity and function of the alveolar unit.