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