| The first problem addressed is the so-called developable surface problem: Given two curves contained in parallel but distinct planes, develop an algorithm to connect these curves (or portions of them) with a developable surface. An algorithm is derived and several examples are run using it. Then several examples in which singularities occur are examined, and the algorithm is adapted to run these examples.;Estimation of surface curvature is the second problem examined. Four different algorithms are proposed, and each is run on several test cases. The results of each are compared.;The third problem discussed is a variant of the Plateau problem: Given a closed continuous curve which has a rectangular projection in the xy plane, compute a minimal surface which contains it. The problem is approached using the differential geometry of the surface itself, rather than the calculus of variations of the function that defines the surface. Four different algorithms are derived, and various test cases are run.;The last problem addressed is that of prescribed constant mean curvature: Given a closed continuous curve which has a rectangular projection in the xy plane, estimate a surface which has predetermined constant mean curvature everywhere and contains the curve as its boundary. An algorithm is proposed, and run on several test cases which have known solutions. Finally, an example is run to find the largest mean curvature attainable with the method on a unit square. |
