From: utzoo!decvax!decwrl!turtleva!ken Newsgroups: net.math Title: Re: Squares - (nf) Article-I.D.: turtleva.140 Posted: Tue Feb 1 22:29:15 1983 Received: Thu Feb 3 02:01:51 1983 References: yale-com.747 I suspect that Andy Hickmott really means a distance measurement based on D = | x | + | y |, where | . | is the magnitude or absolute value. Such a measure is known as the l1 norm. The Euclidean or l2 norm is the one we normally use to interpret geometrical relationships. The other norm widely used in mathematics is the l(infinity) norm, which is D = max ( | x |, | y | ) In general, we can have the lp norm, defined as p p 1/p D = ( | x | + | y | ) but only the l1, l2, and l(infinity) norms are used in practice. Such norms do indeed have numerous applications in areas such as optimization and approximation. They are used to indicate closeness of two points in space whether that space be 2-space, 3-space, n-space, or function space. The choice of which norm to use usually depends on which is easiest to compute. Computer graphics uses a modification of the l1 norm quite frequently. It is D = max(|x|, |y|) + 1/2 * min(|x|, |y|) and is much cheaper computationally than the l2 norm, but looks more like an octagon than a circle. If using this to approximate the l1 norm, the RMS and maximum errors are about 10%, whereas if the 1/2 above is replaced by 5/16 or 3/8, the errors are more around 5%. Yes, Andy, such funny distance measurements do have their place in applied mathematics. Ken Turkowski {ucbvax,decvax}!decwrl!turtlevax!ken