Laplace and Poisson

One of the enjoyable experiences of scientists today is solving new problems with mathematical tools that were developed hundreds of years ago. These tools are typically generic and were originally developed to solve problems in physics. In this chapter we make use of the well-known equations of Laplace (Pierre-Simon, Marquis de Laplace 1749-1827) and Poisson (Simon-Denis Poisson 17811840). (See Figure 7.1.) The two equations, respectively, have an extremely simple form:∆f = 0 for some function f (7.1)and∆f = div g for some function f and vector field g . (7.2)These two equations are partial differential equations and they have a broad use in diverse branches of mathematical physics. They are widely used in electromagnetism, astronomy and fluid dynamics, but in this chapter we will interpret them in the context of image and geometry processing. In the following, we will show some interesting image editing and geometric problems and how they can be solved by simple means. We will make use of these equations, but without using the terminology of differential equations or physics. The relationship of our basic terminology to these equations will be made clear later, toward the end of the chapter.