Using a Natural Constraint to Approximate Area and Volume

This paper presents methods to calculate the 2D area (or the 3D volume) enclosed by a plane curve (or space surface) without analytical expression. One property of our methods is that the error of the measurement is proportional to the square of the measuring unit when the unit is small enough. Finally, w e p o i n t out a property common to calculating area and volume using the corresponding algorithm. This property can be used as a constraint to increase the accuracy of the estimation. 1. Background In many engineering elds, we often need to calculate the area of a plane region or the volume of a 3D space region. Generally speaking, if we k n o w the analytical expression of the curve bounding the plane region, we can easily calculate the area of the region by i n tegration. The same applies to the calculation of the volume of a 3D region. But in many elds, such as computer vision or surveying and mapping, we often do not know the analytical expressions of the bounding curve for the plane region or the bounding surface for the 3D region. In this situation, we m ust use other methods to approximately calculate the area and volume. One simple method for calculating the plane area is the \square grid method" 4] ] 7 ]. The \square grid method" has the following property: the error of the \square grid" method is bounded by the length of the boundary of the region. Let l 1 be the length of the curve bounding the region in the number plane, A be the true area of the enclosed region, N be the total number of integer points within the region (which is the approximate area obtained from the \square grid" method), we h a ve: j A;N j< l. Hua and Wang 4] also described three practical ways of measuring the volume: the Baymah Formula, the truncated-cone method, and the trapezoid-method. These volumetric methods are simple but the errors are large and not evaluated. In this paper, we propose much more accurate and eecient methods of measuring the area and volume. One property of our methods is that the error of the measurement is proportional to the square of the measuring unit when the unit is small enough. Another property is that our methods can be arbitrarily accurate. That is, we can set …