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HANDBOOK OF PHYSIOLOGY 



CIRCULATION I 



FIG. 3. Illustration of how a surface has, at any point in it, two principal radii of curvature. 

 A: a cylindrical surface, where one radius of curvature is infinite: B; a synclastic surface, where the 

 two centers of curvature lie on the same side of the surface; C: an anticlastic surface, with centers 

 of curvature on opposite sides of it. [From General Physics for Sludenls. Edwin Edser. London: 

 Macmillan, 191 1.] 



P= T(l/r,+ l/fjl 



FIG. 4. The two principal radii of curvature at a point on 

 the surface of a cardiac ventricle. NP — normal at this point. 

 APB, CPD, circular arcs, touching the surface, O and O' 

 centers of curvature. [From Burton (8).] 



work to a sphere rather than a cylinder gives the 

 result 



iT 



(7) 



Tiiis is famihar as the formula for liquid drops, 

 where T is the surface tension. (For soap bubbles, 

 P = i^T/R, as there are two air-liquid interfaces 

 of the soap film.) The two special cases, the cylinder 

 and the sphere, are part of the general law of Laplace 

 (c. 1 821) for a surface "membrane" that divides 

 two spaces, which we can call "inside" and "out- 

 side." The membrane may be of any shape at all, 

 hut, for equilibrium, we must have: 



p=T' (l/r-l/rg) 



FIG. 5. Illustration of how the wall at the top of the aortic 

 arch is synclastic, at the bottom anticlastic. The wall thickness 

 has to alter accordingly. 



- = ^tn) 



(8) 



where ri and r, are the "principal radii of curvature" 

 of the membrane at any given point (fig. 3). The 

 definition of these "principal radii" is as follows. 

 At any point on the surface (as of the heart, fig. 4) 

 we may draw the normal (NP) at that point, at 

 right angles to a "tangent plane" (like a sheet of 

 paper touching the surface at that point). With their 

 centers at two different points (O and O') somewhere 

 on this normal, we may describe two arcs which 

 touch the surface in two different planes, at right 

 angles to each other. The two arcs will have two 

 different radii of curvature r^ and r.>, and we could 

 find an infinite number of such pairs of arcs (at 

 right angles to each other, but oriented differently in 

 the surface). The "principal radii of curvature" are 



