Fast Polygon Area and Newell Normal Computation

Abstract The textbook formula for the area of an n-vertex two-dimensional polygon uses 2n + 1 multiplicat ions and 2n − 1 additions. We give an improved formula that uses n + 1 multiplicat ions and 2n − 1 additions. A similar formula is derived for a three-dimens ional planar polygon where, given the unit normal, the textbook equation cost of 6n − 4 multiplications and 4n + 1 additions is reduced to n − 2 multiplications and 2n − 1 additions. Our formula also speeds up Newell's method to compute a robust approximate normal for a nearly planar three-dimensional polygon, using 3n fewer addit ions than the textbook formula. Further, when using this method, one can get the polygon 's planar area as equal to the length of Newell 's normal for a small addi tional fixed cost.