riggsby@h-sc1.UUCP (andrew riggsby) (11/15/85)
Is there a general method for determining the surface area of an arbitrary sur- face or solid similar to (or not similar, for that matter) using displacement
to determine the volume of a solid. Note that this includes 2-d surfaces in
E^3.
Thanks.
Andrew Riggsby
riggsby@harvunxu
gwyn@brl-tgr.ARPA (Doug Gwyn <gwyn>) (11/18/85)
> Is there a general method for determining the surface area of an arbitrary sur- face or solid similar to (or not similar, for that matter) using displacement > to determine the volume of a solid. Note that this includes 2-d surfaces in > E^3. Well, you could try electroplating or adsorption under some circumstances, but I don't think there's anything as elegant for this as the use of immersion displacement to determine volume. Of course, in theory the surface area of a bounded solid of finite volume could be infinite..
ken@turtlevax.UUCP (Ken Turkowski) (11/18/85)
In article <749@h-sc1.UUCP> riggsby@h-sc1.UUCP (andrew riggsby) writes: >Is there a general method for determining the surface area of an arbitrary >surface or solid similar to (or not similar, for that matter) using >displacement to determine the volume of a solid. Note that this includes >2-d surfaces in E^3. If you can determine the normal to the surface, you can use the divergence theorem: __ integral \/ . A dV = integral A . n dS V S Where V is the entire volume of the object, S is the surface of the object, __ \/ is the vector differential operator "del", . is the dot product, dV is an incremental volume of integration, dS is an incremental surface of integration, n is the normal to the surface, and A is a vector function. For your purposes, you would want to pick a function A such that A.n == 1 In some applications, doing a surface integral is easier than a volume integral; hopefully, for yours, a volume integral is easier. -- Ken Turkowski @ CIMLINC (formerly CADLINC), Menlo Park, CA UUCP: {amd,decwrl,hplabs,seismo,spar}!turtlevax!ken ARPA: turtlevax!ken@DECWRL.DEC.COM
robertj@garfield.UUCP (11/20/85)
In article <749@h-sc1.UUCP> riggsby@h-sc1.UUCP (andrew riggsby) writes: >Is there a general method for determining the surface area of an arbitrary sur- face or solid similar to (or not similar, for that matter) using displacement >to determine the volume of a solid. Note that this includes 2-d surfaces in >E^3. > >Thanks. > > Andrew Riggsby > riggsby@harvunxu > If the surface is given in parametric form and is of the class C^2 then there is a method that I know of which is good for arbitrary dimensions. It involves the (n-1)th integral (i.e. in E^3 this is the double integral) of the determinant of the metric tensor of the surface. In more detail, let the surface be parametrized by: R = Xi = Xi(U1,U2,...,U(n-1)) let Ri designate the partial derivative of R with respect to the ith coordinate,Ui. Then Gij = Ri.Rj where Ri.Rj is the usual scalar product or dot product. All such Gij define a matrix with dterminant G. If we let I denote the the (n-1)th integral (that is double integrals for E^3, triple integrals for E^4) then if S is the surface area of the surface in question and is given by: S = I{ (G)^(1/2) dU1dU2dU3...dU(n-1) } over the region in question (that is the set on which each Ui is defined ). This I think is what you are looking for and I think that it is correct inwhat it is saying. If not humblest apologies. Reference: Manfredo P. do Carmo, The Differential Geometry of Curves and Surfaces. Cheers Robert Janes
riggsby@h-sc1.UUCP (andrew riggsby) (11/25/85)
This is a clarification of my original question (seeking a method to find the
surface area on an object). What I was really interested in was a "practical"
method which would find the area of a tooth, a rock or some other irregular
but real object.
Thanks
Andrew Riggsby