A Study of Crofton Type Formulas for Surface Area in Three-Dimension

Abstract

Every three-dimensional object can be computed by a two-dimensional plane with the help of integration. Inspired by Crofton formulas and the Cavalieri's Principle, this work derives a general method for the surface area of a polygonal in three-dimensional space. In fact, the surface area of a three-dimensional object can be subdivided into a finite number of small rectangle. This research represents the area of a rectangle by the number of the intersection point between the rectangle and the line passing through the rectangle in all directions. Next, this research computes the proportionality constant D of integration. Eventually, this research extends the result to a boarder discussion on the application of the obtained result to a smooth surface in R^3. Within the process of integration , the volume of a four-dimensional object in TS2 is calculated. This research jumps out from the conventional representation of the surface area using the one-dimension integral geometry. The reader will realize another technique of representing the surface area with the integration of the number of intersection points in a sub-divided parallelograms. Moreover, not only does the research extent the concept of Cavalieri's principle to a three-dimensional application, but also the solution incites a possible way using the intersection point to explore the volume or the surface area of an object in a higher dimension world.