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 \def\righthead{\sl\hfil SOLSTICE }
 \def\lefthead{\sl Summer, 1992 \hfil}
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 \centerline{\big SOLSTICE:}
 \centerline{\bf SUMMER, 1992--11:14p.m., E.D.T., June 20}
 \centerline{\bf Volume III, Number 1}
 \centerline{\bf Institute of Mathematical Geography}
 \centerline{\bf Ann Arbor, Michigan}
 \centerline{\bf SOLSTICE}
 \line{Founding Editor--in--Chief:  {\bf Sandra Lach Arlinghaus}. \hfil}
 \centerline{\bf EDITORIAL BOARD}
 \line{{\bf Geography} \hfil}
 \line{{\bf Michael Goodchild}, University of California, Santa Barbara. 
 \line{{\bf Daniel A. Griffith}, Syracuse University. \hfil}
 \line{{\bf Jonathan D. Mayer}, University of Washington; joint appointment
  in School of Medicine.\hfil}
 \line{{\bf John D. Nystuen}, University of Michigan (College of
  Architecture and Urban Planning).}
 \line{{\bf Mathematics} \hfil}
 \line{{\bf William C. Arlinghaus}, Lawrence Technological University. \hfil}
 \line{{\bf Neal Brand}, University of North Texas. \hfil}
 \line{{\bf Kenneth H. Rosen}, A. T. \& T. Bell Laboratories.
 \line{{\bf Business} \hfil}
 \line{{\bf Robert F. Austin, Ph.D.} \hfil}
 \line{President, Austin Communications Education Services \hfil}
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 Copyright, June, 1992, Institute of Mathematical Geography.
 All rights reserved.
 {\bf ISBN: }
 {\bf ISSN: 1059-5325} 
 \centerline{\bf SUMMARY OF CONTENT}
 \noindent{\bf 1.  ARTICLES.}
 {\bf Harry L. Stern}. 
 {\bf Computing Areas of Regions With Discretely Defined Boundaries}.
 1. Introduction 2. General Formulation 3. The Plane 4.  The Sphere
 5.  Numerical Example and Remarks.  Appendix--Fortran Program.
 \noindent{\bf 2.  NOTE }
 {\bf Sandra L. Arlinghaus, John D. Nystuen, Michael J. Woldenberg}.  
 {\bf  The Quadratic World of Kinematic Waves}
 \noindent{\bf 3.  SOFTWARE REVIEW}
 RangeMapper$^{\hbox{TM}}$  ---  version 1.4.
 Created  by {\bf Kenelm W. Philip},  Tundra Vole Software,
 Fairbanks, Alaska.  Program and Manual by  {\bf Kenelm W. Philip}.
 Reviewed by {\bf Yung-Jaan Lee}, University of Michigan.
 \noindent{\bf 4.  PRESS CLIPPINGS}
 \noindent{\bf 5.  INDEX to Volumes I (1990) and II (1991) of
             {\sl Solstice}.}
            GEOGRAPHY }
 \centerline{\bf Computing Areas of Regions With Discretely Defined 
 \centerline{\bf Harry L. Stern}
 \centerline{\bf Polar Science Center}
 \centerline{\bf 1013 N.E. 40th Street}
 \centerline{\bf Seattle, WA 98195}
 \noindent{\bf 1. Introduction}
 It is well known that the area of a region in the plane can be computed
 by an appropriate integration around the boundary of the region [e.g.
 Hildebrand, page 306].  If the boundary is defined by a sequence of
 points connected by straight lines (a polygon), the parametric
 representation of the boundary is particularly simple, and an explicit
 formula for the area can be derived.  Using Stokes' Theorem, this idea
 can be extended to derive area formulas for regions on non-planar
 surfaces whose boundaries are defined by a sequence of points
 connected by appropriate curves.  In this note we present exact area
 formulas for regions in the plane and regions on the sphere whose
 boundaries are defined by such discrete sets of points.
 An application of these formulas arises in computing the area of
 a region on a map.  Suppose that the boundary of the region
 of interest is traced by an encoding device that records its
 coordinates, relative to some user-defined $(x,y)$ system,
 in a computer file.  Such a file may contain hundreds or thousands
 of coordinate pairs.  If the map covers a relatively small region,
 the surface of the earth can be approximated locally by a plane,
 and the area computed directly from the $(x,y)$ coordinate pairs. 
 If the map covers a large region, the earth can be approximated by
 a sphere.  The $(x,y)$ coordinate pairs are then converted to
 latitude and longitude using the appropriate map projection equations,
 and the area on the sphere is computed.
 The usual method for computing area is to divide up the
 two dimensional surface into a large number of small cells,
 and to add up the areas of those cells that lie inside the
 boundary of the region.  This method is computationally slow,
 because every cell must be tested for inclusion in the region,
 and because high accuracy requires a small cell size.  In contrast,
 the formulas derived here, besides being exact, are quickly evaluated
 on a computer because the computation is proportional to the
 number of boundary points.  The two dimensional area calculation
 is reduced to a one dimensional boundary calculation.
 The next section outlines the general mathematical formulation.
 Sections 3 and 4 give explicit results for the plane and sphere.
 A numerical example and concluding remarks are presented in the
 last section.
 \noindent{\bf 2. General Formulation}
 Stokes' theorem says
 \int\!\!\!\int_S ({\bf \nabla} \times \hbox{\bf F}){\bf \cdot}
 {\bf \hat{\hbox{\bf n}}}\, dA
 =\oint_C \hbox{\bf F}{\bf \cdot } {{d\hbox{\bf R}}\over {dt}} \, dt \eqno(1)
 where $S$ is the region of a surface bounded by the curve $C$,
 ${\bf \hat{\hbox{\bf n}}} $ is the unit outward normal on the surface,
 $ \hbox{\bf R}(t) $ is a parametric representation of $C$,
 and ${\bf F} $ is an arbitrary vector field.
 We suppose that the surface is specified in some way
 (e.g. $x^2 + y^2 + z^2 = 1 $ for the unit sphere),
 so that the unit outward normal
 ${ {\bf \hat{\hbox{\bf n}}}} $ can be determined
 (e.g. ${\bf \hat{\hbox{\bf n}}} =
 x  {\bf \hat{\hbox{\bf \i}}} +
 y  {\bf \hat{\hbox{\bf \j}}} +
 z  {\bf \hat{\hbox{\bf k}}} $ for the unit sphere).
 We then choose any vector field ${\hbox{\bf F}} $ such that the integrand
 on the left hand side of (1) is unity in $S$:
 ({\bf \nabla} \times \hbox{\bf F}){\bf \cdot }{\bf \hat{\hbox{n}}} = 1.
 With $\hbox{\bf F} $ determined (though not uniquely) by equation (2),
 the left hand side of (1) simply reduces to the area of $S$,
 A =\oint_C \hbox{\bf F}\cdot {{d\hbox{\bf R}}\over {dt}} \, dt. \eqno(3)
 In order to evaluate the integrand on the right hand side
 of (3), we need a description of $C$.
 Suppose that $N$ points on the surface are given,
 $ \hbox{\bf P}_1$, $\hbox{\bf P}_2$, $\ldots $, $\hbox{\bf P}_N$,
 and that $C$ is defined by connecting these points in sequence,
 returning to $\hbox{\bf P}_1 $
 (define $\hbox{\bf P}_{N+1} \equiv \hbox{\bf P}_1$).
 On each segment, from $\hbox{\bf P}_k $ to $\hbox{\bf P}_{k+1} $,
 let $\hbox{\bf R}_k(t) $ be a parametric representation
 of the connecting curve.
 There are many possible connecting curves to choose from,
 but the most natural choice is the geodesic, the curve of
 minimum length
 (e.g. a straight line in the plane, a great circle
 on the sphere).
 The geodesics can be found in principle from a description
 of the surface (for example, Weinstock pages 61-62).
 The collection of the $N$ geodesics
 $\hbox{\bf R}_k (t) $ connecting the $N$ points
 $ \hbox{\bf P}_1$, $\hbox{\bf P}_2$, $\ldots $, $\hbox{\bf P}_N$,
 constitutes the parametric description $\hbox{\bf R} (t) $ of $C$
 on the right hand side of (3).
 Now that we have specified how to construct the integral in (3)
 as a sum of integrals along the $N$ connecting geodesics,
 the area formula can be written more explicitly as
 A =\sum_{k=1}^N
 \hbox{\bf F}(s)\cdot {{d\hbox{\bf R}_k}\over {ds}} \, ds \eqno(4)
 where $s$ is the arc length parameter along the geodesic
 $\hbox{\bf R}_k (s) $, and $L_k$ is the
 total arc length of the $k$-th segment.  The geodesics need not
 necessarily be parameterized by arc length, but this is what
 we have used in the sections that follow.
 The determination in principle of all quantities is now complete.
 To summarize the steps: Given a surface and a set of points
 $\hbox{\bf P}_k,  k=1,2,\ldots ,N$ that defines the boundary of
 a region on the surface,
 \noindent(1) Find the unit outward normal on the surface,
 ${{\bf \hat{\hbox{\bf n}}}} $;
 \noindent(2) Find a vector field ${\hbox{\bf F}}$ that satisfies 
equation (2):
 ${\bf (\nabla \times \hbox{\bf F}) \cdot {\bf \hat{\hbox{\bf n}}}} = 1 $;
 \noindent(3) Find a parameterization ${\hbox{\bf R}}_k (s) $ of the geodesic
 from point ${\hbox{\bf P}}_k $ to ${\hbox{\bf P}}_{ k+1 } $;
 \noindent(4) Form the integrand in equation (4) and do the integration;
 \noindent(5) Sum the contributions in (4) to get the area of the region.
 \noindent Some specific cases follow.
 \noindent{\bf 3. The Plane}
 In the plane $z=0$, the unit outward normal is
 ${{\bf \hat{\hbox{\bf n}}}} = (0,0,1) $ and the condition (2) on the
 components $ (F_1 ,F_2 ,F_3 ) $ of ${\hbox{\bf F}}$ is
 {\partial {F_2} \over \partial x} - {\partial {F_1} \over \partial y} = 1.
 We choose $F_1 = -y/2$ and $F_2 = x/2$.
 The geodesics
 ${\hbox{\bf R}} (s) = (x(s),y(s),0) $ are straight lines, and the
 integral in equation (4) becomes
 I_k = \int_0^{L_k} {1\over 2} \left( x {{dy} \over {ds}}
                                    - y {{dx} \over {ds}}\right)\, ds.
 Let the boundary points
 ${\hbox{\bf P}}_k$ have coordinates $(x_k ,y_k )$.
 The parametric equations for the boundary segment connecting
 ${\hbox{\bf P}}_k $ and ${\hbox{\bf P}}_{k+1} $
 (of length $ L_k $) are
 x(s)=x_k+{s\over {L_k}} (x_{k+1} - x_k)
 \qquad y(s)=y_k+{s\over {L_k}}(y_{k+1}-y_k).  \eqno(7)
 Substituting these expressions into equation (6) with
 $ \Delta x = x_{k+1} - x_k $ and
 $ \Delta y = y_{k+1} - y_k $ gives
 I_k &= {1 \over 2}\int_0^{L_k}
                   \left\{\left(x_k+{{s\,\,\Delta x}\over{L_k}}\right)
                   \left({{\Delta y}\over {L_k}}\right)
                  -\left(y_k + {{s\,\,\Delta y}\over{L_k}}\right)
  \left({{\Delta x}\over {L_k}}\right)\right\}\,ds \cr
 &= {1\over 2}\int_0^{L_k}\left\{{{x_k\Delta y}\over {L_k}}-
                           {{y_k\Delta x}\over {L_k}}\right\}\, ds \cr
 &= {1 \over 2}(x_k\Delta y - y_k \Delta x) \cr
 &= {1\over 2}(x_ky_{k+1}-y_kx_{k+1}). 
 It follows that the area of the polygon in the plane whose
 vertexes are the points $ ( x_k , y_k ) $ is
 A={1\over 2}\sum_{k=1}^N(x_ky_{k+1}-y_kx_{k+1})
 where $ x_{N+1} \equiv  x_1$, $y_{N+1} \equiv  y_1 $, and
 the points $ ( x_k , y_k ) $ trace
 the boundary in a counter-clockwise sense.  If the
 order of the points is reversed, the negative of the area will
 \noindent{\bf 4. The Sphere}
 Without loss of generality we consider the unit sphere.
 It will be convenient to use both rectangular and spherical
 coordinates.  The longitude $\theta$, measured positive eastward,
 and latitude $\phi$, measured positive northward, are related to
 $x$, $y$, $z$ via
 x=\hbox{cos}\,\phi\,\,\hbox{cos}\,\theta \quad
 y=\hbox{cos}\,\phi\,\,\hbox{sin}\,\theta \quad
 and the unit vectors in the $\theta$, $\phi$, and radial directions
 are related to the rectangular unit vectors
 ${\bf \hat{\hbox{\bf \i}}}$,  ${\bf \hat{\hbox{\bf \j}}}$, 
 ${\bf \hat{\hbox{\bf k}}}$ via
   {\bf \hat{\hbox{\bf u}}}_{\theta}
  = (-\hbox{sin}\theta ){\bf \hat{\hbox{\bf \i}}}
   +( \hbox{cos}\theta ){\bf \hat{\hbox{\bf \j}}}
  = {{-y}\over{\sqrt{1-z^2}}}{\bf \hat{\hbox{\bf \i}}}
   +{{ x}\over{\sqrt{1-z^2}}}{\bf \hat{\hbox{\bf \j}}}  
 {\bf \hat{\hbox{\bf u}}}_{\phi}
   &=(\hbox{sin}\phi\,\hbox{cos}\theta ){\bf \hat{\hbox{\bf \i}}}
   +(\hbox{sin}\phi\,\hbox{sin}\theta ){\bf \hat{\hbox{\bf \j}}}
   +(-\hbox{cos}\phi){\bf \hat{\hbox{\bf k}}}\cr
  &={{xz}\over{\sqrt{1-z^2}}}\,{\bf \hat{\hbox{\bf \i}}}
  +{{yz}\over{\sqrt{1-z^2}}}\,{\bf \hat{\hbox{\bf \j}}}
   -{\sqrt{1-z^2}}\,{\bf \hat{\hbox{\bf k}}}\cr
 {\bf \hat{\hbox{\bf u}}}_r
      =(\hbox{cos}\phi\,\hbox{cos}\theta ){\bf \hat{\hbox{\bf \i}}}
      +(\hbox{cos}\phi\,\hbox{sin}\theta ){\bf \hat{\hbox{\bf \j}}}
      +(\hbox{sin}\phi ){\bf \hat{\hbox{\bf k}}}
     =x{\bf \hat{\hbox{\bf \i}}}+y{\bf \hat{\hbox{\bf \j}}}
     +z{\bf \hat{\hbox{\bf k}}}.
 The unit outward normal on the sphere is just the unit
 radial vector ${\bf \hat{\hbox{\bf u}}}_r $.  With the vector
 ${\hbox{\bf F}}$ written in terms of its spherical components
 ${\hbox{\bf F}}= F_{\theta}{\bf \hat{\hbox{\bf u}}}_{\theta} +
              F_{\phi}{\bf \hat{\hbox{\bf u}}}_{\phi} +
              F_r{\bf \hat{\hbox{\bf u}}}_r $,
 the condition (2) becomes [Hildebrand]
 (\nabla \times \hbox{\bf F})\cdot {\bf \hat{\hbox{\bf u}}}_r 
 {1\over {\hbox{cos}\,\phi}}
 \left[{{\partial} \over {\partial \theta}}(F_{\phi})
  {{\partial} \over {\partial \phi}}(\hbox{cos}\,\phi\,\,F_{\theta})
 \right] = 1.
 This is most naturally satisfied if we take
 {{\partial} \over {\partial \phi}}(\hbox{cos}\,\phi\,\,F_{\theta})
 {{\partial} \over {\partial \theta}}(F_{\phi})=0
 F_{\theta}=-\hbox{tan}\,\phi + {{g({\theta})} \over {\hbox{cos}\,\phi}}
 where $g$ is an arbitrary function of $\theta$, and $h$ is an
 arbitrary function of $\phi$.  No radial dependence has been
 introduced into $g$ and $h$ because we are only interested in
 the values of ${\hbox{\bf F}}$ on the surface $r = \hbox{\it constant}$.
 Also, the radial component of ${\hbox{\bf F}}$,  $F_r$,  is of
 no consequence: any tangent vector to the sphere,
 $ { d{\hbox{\bf R}} } / dt $, has no radial component, so the
 dot product ${\hbox{\bf F}} {\bf \cdot }\,  { d{\hbox{\bf R}} } / dt $
 annihilates any radial contribution from ${\hbox{\bf F}}$.  Therefore
 we take $ F_r = 0 $.
 Now that ${\hbox{\bf F}}$ is determined (up to two arbitrary functions),
 we turn to the parameterization of the boundary.  We suppose that
 $N$ pairs of longitude/latitude coordinates are given, namely
 $ \theta_k ,  \phi_k $ for $ k=1,2,\ldots ,N$
 (with $ \theta_{N+1} \equiv  \theta_1 $ and $ \phi_{N+1} \equiv 
 \phi_1 $), that form the boundary of the region
 when the points are connected in the
 given order.  The boundary points will also be denoted by
 ${\hbox{\bf P}}_k $, and by their rectangular coordinates
 $ ( x_k , y_k , z_k ) $.  We can use equation (10)
 to go from spherical to rectangular coordinates.
 To simplify the notation a bit, let $ k=1 $ and
 consider the great circular arc
 from ${\hbox{\bf P}}_1 $ to ${\hbox{\bf P}}_2 $.  Let $\Delta$
 represent the angle subtended at the center of the sphere
 by ${\hbox{\bf P}}_1 $ and ${\hbox{\bf P}}_2 $.  Then $\Delta$
 satisfies $ \hbox{cos}\, \Delta ={\hbox{\bf P}}_1 {\bf \cdot }\,
 {\hbox{\bf P}}_2 $
 since all the ${\hbox{\bf P}}_k $ are unit vectors.
 Note that $\Delta$ is also the length of the arc from ${\hbox{\bf P}}_1 $
 to ${\hbox{\bf P}}_2 $.  Let $\alpha$ be the arc length parameter
 along the great circle from ${\hbox{\bf P}}_1 $ to ${\hbox{\bf P}}_2 $,
 and let ${\hbox{\bf R}} ( \alpha ) $ be the position vector along the
 great circle.  Since ${\hbox{\bf R}} ( \alpha ) $ lies in the plane
 spanned by ${\hbox{\bf P}}_1 $ and ${\hbox{\bf P}}_2 $, we can write
 \hbox{\bf R}(\alpha)=A(\alpha)\hbox{\bf P}_1+B(\alpha)\hbox{\bf P}_2
 where $ A ( \alpha ) $ and $ B ( \alpha ) $ are determined from
 the following two conditions:
 \noindent (1) ${\hbox{\bf R}} ( \alpha ) $ lies on the unit sphere: $ 
{\hbox{\bf R}}
 {\bf \cdot }\,{\hbox{\bf R}} = 1 $;
 \noindent (2) The angle between ${\hbox{\bf P}}_1 $ and
 ${\hbox{\bf R}} ( \alpha ) $
 is $\alpha$: $ {\hbox{\bf P}}_1 {\bf \cdot }{\hbox{\bf R}}
 = \hbox{cos}\, \alpha $.
 Using equation (15) for ${\hbox{\bf R}} $ and the fact that
 ${\hbox{\bf P}}_1 {\bf \cdot }\,{\hbox{\bf P}}_2 = \hbox{cos}\, \Delta $,
 these conditions translate into
 A^2+B^2+2AB \hbox{cos} \Delta = 1
 \qquad A+B\hbox{cos}\Delta=\hbox{cos}\,\alpha
 respectively.  Solving for $A$ and $B$, we find
 \hbox{\bf R}(\alpha)=
 {{\hbox{sin}\,(\Delta - \alpha )}\over {\hbox{sin}\,\Delta}}
 \hbox{\bf P}_1
 {{\hbox{sin}\,(\alpha)} \over {\hbox{sin}\,\Delta}}
 \hbox{\bf P}_2.
 This is the arc length parameterization for the great circle
 through ${\hbox{\bf P}}_1 $ and ${\hbox{\bf P}}_2 $.
 With ${\hbox{\bf R}} ( \alpha ) $ determined, the next step is to compute
 $ { d{\hbox{\bf R}} } / { d \alpha } $ and then
 ${\hbox{\bf F}}{\bf \cdot }\, { d{\hbox{\bf R}} } / { d \alpha } $.
 Computation of $ { d{\hbox{\bf R}} } / { d \alpha } $ is simple,
 but we want to express the result in terms of the unit vectors
 ${\bf \hat{\hbox{\bf u}}}_{\theta} $ and ${\bf \hat{\hbox{\bf 
u}}}_{\phi} $,
 to facilitate taking the dot product with ${\hbox{\bf F}}$.
 Toward this end, write
 {{d\hbox{\bf R}}\over{d\alpha }}=G(\alpha){\bf \hat{\hbox{\bf u}}}_{\theta}
                                 +H(\alpha){\bf \hat{\hbox{\bf u}}}_{\phi}
 where $ G ( \alpha ) $ and $ H ( \alpha ) $ are determined as follows.
 Let $'$ denote $ d / d \alpha $ and write
 ${\hbox{\bf R}} ( \alpha ) = ( x ( \alpha ) ,  y ( \alpha ) ,
 z ( \alpha )) $ where the functions $ x$, $y$, $z$ are given explicitly
 by the components of equation (17).  Then the dot product of
 equation (18) with ${\bf \hat{\hbox{\bf u}}}_{\theta} $ and
 ${\bf \hat{\hbox{\bf u}}}_{\phi} $ gives, respectively,
 $ G ( \alpha ) $ and $ H ( \alpha ) $.  Using equations (11a,b)
 to express ${\bf \hat{\hbox{\bf u}}}_{\theta} $ and
 ${\bf \hat{\hbox{\bf u}}}_{\phi} $
 in terms of ${{\bf \hat{\hbox{\bf \i}}}, {\bf \hat{\hbox{\bf \j}}}, 
 {\bf \hat{\hbox{\bf k}}} } $ we have
 G(\alpha)&=\hbox{\bf R}'{\bf \cdot}{\bf \hat{\hbox{\bf u}}}_{\theta} \cr
          &=(x'{\bf \hat{\hbox{\bf \i}}}+y'{\bf \hat{\hbox{\bf \j}}}
            +z'{\bf \hat{\hbox{\bf k}}})
             {\bf \cdot}
 \left[{{-y}\over{\sqrt{1-z^2}}}\,{\bf \hat{\hbox{\bf \i}}}
 +{{x} \over{\sqrt{1-z^2}}}\,{\bf \hat{\hbox{\bf \j}}}\right]\cr
 H(\alpha )&=\hbox{\bf R}'{\bf \cdot}{\bf \hat{\hbox{\bf u}}}_{\phi} \cr
           &=(x'{\bf \hat{\hbox{\bf \i}}}+y'{\bf \hat{\hbox{\bf \j}}}
           +z'{\bf \hat{\hbox{\bf k}}})
             {\bf \cdot}
 \left[{{xz}\over{\sqrt{1-z^2}}}\,{\bf \hat{\hbox{\bf \i}}}
 +{{yz}\over{\sqrt{1-z^2}}}\,{\bf \hat{\hbox{\bf \j}}}
 -{\sqrt{1-z^2}}\,{\bf \hat{\hbox{\bf k}}}\right] \cr
 where the last step follows because $ ( x' x + y' y +
 z' z ) $ is the derivative of the constant $ ( x^2 +
 y^2 + z^2 ) / 2 $.

 Using equations (14)
 for the components of ${\hbox{\bf F}}$
 and converting from $ \theta ,  \phi $ to $ x,y,z $ gives
 \hbox{\bf F}=\left[{{-z}\over {\sqrt{1-z^2}}}
            +{{g(\theta )}\over {\sqrt{1-z^2}}}\right]
 {\bf \hat{\hbox{\bf u}}}_{\theta} 
            +h(\phi ){\bf \hat{\hbox{\bf u}}}_{\phi}.
 Using the components of
 $ d{\hbox{\bf R}} / d \alpha $ from equations (19) and (20), we have
 \hbox{\bf F}{\bf \cdot}{{d\hbox{\bf R}}\over{d \alpha }} =
 \left[{{-z}\over {\sqrt{1-z^2}}}+{{g(\theta )}\over {\sqrt{1-z^2}}}\right]
 -{{z'h(\phi )}\over {\sqrt{1-z^2}}}
 This is the integrand for the segment of the boundary integral from
 ${\hbox{\bf P}}_1 $ to ${\hbox{\bf P}}_2 $.  Integration is with respect
 to $\alpha$, from $ \alpha = 0 $ to $ \alpha = \Delta $.  The
 variables $ x,y,z $ and their derivatives (with respect to
 $\alpha$ ) $ x' ,y' ,z' $ are all functions of
 $\alpha$, as given by the components of equation (17).
 We can choose the functions $g$ and $h$ to simplify equation (22).
 Nothing is gained by retaining the last term, so
 we take $h\equiv 0$. This simplifies the
 integrand to
 \hbox{\bf F}{\bf \cdot}{{d\hbox{\bf R}}\over{d \alpha }} =
 {{(xy'-yx')(g(\theta )-z)}\over{1-z^2}}
 Notice the potential singularities at $ z= \pm 1 $, i.e. the North Pole
 and the South Pole.  Writing the denominator as $ 1 - z^2
 = ( 1 - z ) ( 1 + z ) $, we see that if $ g \equiv  1 $ we remove
 the singularity at $z=1$, and if $ g \equiv  -1 $ we remove the
 singularity at $z=-1$.
 We must not put $g=0$, since then ${\hbox{\bf F}}$ would vanish
 everywhere on the equator, violating equation (2) there.
 This would lead to a value of zero for the areas of
 the northern and southern hemispheres.
 In the following development we take $ g \equiv  1 $.
 In case one of the ${\hbox{\bf P}}_k $ is the South
 Pole, $g$ should be replaced by $-1$.
 We can now write the first term in the area summation of equation (4)
 I_1=\int_0^{\Delta } {{xy'-yx'}\over{1+z}}\,d\alpha .
 Notice the similarity to the expression for the plane, equation (6).
 We have explicit expressions for $x,y,z,x' ,y' $
 from the components of equation (17) and its derivatives,
  x={{\hbox{sin}\,(\Delta - \alpha )}\over{\hbox{sin}\,\Delta}}x_1
   +{{\hbox{sin}\,(\alpha )}\over{\hbox{sin}\,\Delta}}x_2 \eqno(25a) 
 x'={-{\hbox{cos}\,(\Delta - \alpha )}\over{\hbox{sin}\,\Delta}}x_1
   +{{\hbox{cos}\,(\alpha )}\over{\hbox{sin}\,\Delta}}x_2
 and similar equations for $ y, y' $ and $ z, z' $.
 Substituting these expressions into equation (24) and using
 standard trigonometric identities leads to
 I_1=(x_1y_2-y_1x_2)\int_0^{\Delta} {{d\alpha }\over
 {\hbox{sin}\,\Delta +z_1\hbox{sin}\,(\Delta -\alpha ) 
 Recalling that this is the contribution to the area summation from
 the segment $k=1$ between ${\hbox{\bf P}}_1 $ and ${\hbox{\bf P}}_2 $,
 we can write the total area as
 where the terms $ J_k $ are the integrals
 J_k=\int_0^{{\Delta}_k} {{d\alpha }\over
 {\hbox{sin}\,({\Delta}_k) +z_k\hbox{sin}\,({\Delta}_k -\alpha )
 and $ {\Delta}_k $ comes
 from $ \hbox{cos} ( {\Delta}_k ) ={\hbox{\bf P}}_k {\bf \cdot }
 {\hbox{\bf P}}_{k+1} $.
 The integral can be put into a standard form and explicitly
 integrated with the substitution $ w = e^{ i \alpha } $.
 Under this transformation, $ d \alpha = dw / (iw) $, $ \hbox{sin} \alpha
 = ( w - w^{-1} ) / 2i $, and the integral becomes
 J_k=\int_1^{e^{i \Delta }} {{2\,\,dw}\over{aw^2+2bw+c}}
 a=z_{k+1}-z_ke^{-i \Delta} \quad b=i\,\hbox{sin}\,\Delta
 \quad c=z_ke^{i \Delta }-z_{k+1}.
 The subscript $k$ on $ \Delta $ has been dropped to reduce
 notational clutter.
 The value of $ J_k $
 depends on the sign of the discriminant $ D = b^2 - ac $, or
 D=z_k^2+z_{k+1}^2-2z_kz_{k+1}\,\hbox{cos}\,\Delta - \hbox{sin}^2\Delta .
 The three cases are [Marsden, Appendix A]
 {{1}\over {\sqrt{D}}}\,\hbox{ln}\,
  \left[{{aw+b-\sqrt{D}}\over{aw+b+\sqrt{D}}}\right] &($D>0$)\cr
 {{2}\over {\sqrt{-D}}}\,\hbox{arctan}\,
  \left[{{aw+b}\over{\sqrt{-D}}}\right] &($D<0$)\cr
 {{-2}\over{aw+b}} &($D=0$)\cr
 where the expressions must be evaluated between the upper
 and lower limits of $ w = e^{ i \Delta } $ and $ w = 1 $.
 The imaginary parts of the resulting complex expressions
 are zero, as they must be since the original integrand and limits
 are real.  Algebraic simplification leads us to define
 in terms of which the expressions for $ J_k $ become
 {{1}\over {\sqrt{D}}}\,\hbox{ln}\,
  \left[{{Q+\sqrt{D}}\over{Q-\sqrt{D}}}\right] &($D>0$)\cr
 {{2}\over {\sqrt{-D}}}\,\hbox{arctan}\,
  \left[{{\sqrt{-D}}\over{Q}}\right] &($D<0$)\cr
 {{Q}\over{(1+z_k)(1+z_{k+1})(1+\hbox{cos}\,\Delta )}}
 This completes the determination of the terms
 in the area formula (27).
 We will now summarize the steps and put them in an algorithmic
 \noindent Given a sequence of (longitude,latitude) coordinates
 on the unit sphere, $ ( {\theta}_k , {\phi}_k )$,
 $k = 1,2,\dots ,N $, find the area of the region that is
 enclosed when the points are connected in sequence by arcs
 of great circles.


 \noindent(1) Set the running sum to $0$ and set $k$ to $1$.
 \noindent(2) Compute
 $ \hbox{cos}\, \Delta$ $=$ ${\hbox{\bf P}}_k {\bf \cdot }{\hbox{\bf 
 either from
 $ x_k x_{k+1}$ + $y_k y_{k+1}$ + $z_k z_{k+1} $
 or from $\,\, $
 $ \hbox{cos}{\phi}_k$ $ \hbox{cos}{\phi}_{k+1}$ 
 $\hbox{cos}$ $( {\theta}_{k+1} - {\theta}_k )$ +
 $\hbox{sin} {\phi}_k$ $ \hbox{sin} {\phi}_{k+1}$. 
 Notice that we won't ever need $ \Delta $ by itself, just its cosine.
 \noindent(3) Compute $Q$ from (33): $  Q
 = z_k + z_{k+1} + 1 + \hbox{cos}\, \Delta $ or $  Q
 = \hbox{sin}\, {\phi}_k + \hbox{sin}\, {\phi}_{k+1}+1+\hbox{cos}\, 
\Delta $.
 \noindent(4) Compute the discriminant $D$ from (31): $  D
 = z_k^2 + z_{k+1}^2 - 2 z_k z_{k+1}
 \hbox{cos}\, \Delta - \hbox{sin}^2 \Delta $ or $  D
 = ( \hbox{sin}\, \phi_k + \hbox{sin}\, \phi_{k+1} )^2 -
 ( 1 + \hbox{cos}\, \Delta )  ( 1 - \hbox{cos}\, \Delta +
 2 \hbox{sin}\, \phi_k \hbox{sin}\, \phi_{k+1} ) $.
 \noindent(5) Compute the integral contribution $ J_k $ in the area
 formula (27), using the appropriate form of equation (34).
 \noindent(6) Compute the first factor in the area formula (27), 
 x_ky_{k+1} - y_k x_{k+1}
 $ \hbox{cos} \phi_k $  $\hbox{cos} \phi_{k+1}$
 $\hbox{sin}$ $( \theta_{k+1} - \theta_k ). $
 \noindent(7) Multiply together the results of steps 5 and 6 to get the 
 term in the summation of (27), and add this to the running sum.
 \noindent(8) If $k$ is less than $N$ then increment $k$ and go to step 2.
 A computer program that implements the above algorithm
 is given in the appendix.
 \noindent {\bf 5. Numerical Example and Remarks}
 It is of interest in Arctic oceanography to calculate the areas of the
 watersheds that drain into the Arctic Ocean.  The
 boundary of the Asian watershed that drains into the Arctic Ocean was
 digitized from a Mercator map of the world
 by tracing its circumference with an encoding device.
 This produced a computer file with 672 $ (x,y) $ coordinate pairs,
 in which the $x$ axis coincided with the equator, the $y$ axis coincided
 with the Greenwich Meridian, and the unit of length was chosen to be
 one degree of longitude on the equator.  These $ (x,y) $ map
 coordinates are related to longitude $ \theta $ and latitude $ \phi $
 by [Snyder]
 x={{180}\over{\pi }}\theta \qquad
 y={{180}\over{\pi }}\hbox{ln}\,\left[\hbox{arctan}\,\left({{\phi}\over{2}}
 where $ \theta $ and $ \phi $ are in radians.  Inverting these relations
 and substituting the $ (x,y) $ map coordinates gives
 a sequence $ ( {\theta}_k ,  {\phi}_k ), \,\,\,  k = $ 1 to 672, of
 points on the sphere that defines the boundary of the watershed.
 At first a simple integration program was written in which the
 region lying between the minimum and maximum latitudes and
 longitudes of the watershed was divided into differential elements
 of size $ \Delta \phi $ by $ \Delta \theta $.  The area of the watershed
 was calculated as $ \sum  \hbox{cos}\, \phi\,  \Delta \phi \,  \Delta 
\theta $
 where the summation was taken over all elements inside the watershed
 boundary.  With each degree of latitude and longitude divided into
 32 parts, this amounted to 5,918,720 elements, of which 2,516,738
 were found to lie within the watershed.  The program required
 more than 51 hours of elapsed time on a Sun workstation to arrive
 at the area, $ 1.424 \times 10^7\,\, \hbox{km}^2 $.
 This dismal performance led to the derivation of the formulas in this
 work.  Using the same 672 coordinates for input, the program in the
 appendix arrived at the same answer in about two seconds.  The
 5.9 million complicated comparisons in the first program were
 replaced by 672 iterations of simple calculations.
 Of course
 in any real physical problem such as the one described here, there are
 sources of error such as uncertainty in the exact location of the
 boundary, inadequate representation of the boundary by too few points,
 and the non-sphericity of the earth.  These problems can be dealt with
 by acquiring better maps, digitizing the boundary with more points, and
 modifying the formulas here to take into account the flattening of the
 earth at the poles, which introduces a correction on the order of three
 parts per thousand.
 \noindent{ \bf Acknowledgment}
 \noindent This work was supported by NASA Grant NAGW 2513.
 Thanks also to Erika Dade for bringing this problem to my attention
 and doing the original watershed calculations.
 \noindent{\bf Appendix -- Fortran Program}
 \line{\phantom{c}\qquad program area \hfil}
 \line{\phantom{c}\qquad implicit undefined (a-z) \hfil}
 \line{c\qquad \hfil}
 \line{c\hrulefill }
 \line{c\qquad \hfil}
 \line{c\qquad Read a sequence of (longitude,latitude) coordinates. \hfil}
 \line{c\qquad Compute the area on the unit sphere that is enclosed
               by connecting \hfil}
 \line{c\qquad these points in sequence with arcs of great circles. \hfil}
 \line{c \hfil}
 \line{c\qquad Refer to ``Computing Areas of Regions with Discretely
                          Defined \hfil}
 \line{c\qquad Boundaries". \hfil}
 \line{c \hfil}
 \line{c \hfil}
 \line{c\qquad Constants. \hfil}
 \line{c \hfil}
 \line{\phantom{c}\qquad real pi, piOver180 \hfil}
 \line{\phantom{c}\qquad parameter (pi = 3.14159265358979,
                         piOver180 = pi / 180.0) \hfil}
 \line{c \hfil}
 \line{c\qquad Parameters. \hfil}
 \line{c \hfil}
 \line{\phantom{c}\qquad integer maxPoints \hfil}
 \line{\phantom{c}\qquad parameter (maxPoints = 1000) \hfil}
 \line{c \hfil}
 \line{c\qquad Mean radius of earth in kilometers. \hfil}
 \line{c \hfil}
 \line{\phantom{c}\qquad real Rearth \hfil}
 \line{\phantom{c}\qquad parameter (Rearth = 6371.2) \hfil}
 \line{c\qquad Variables.\hfil}
 \line{\phantom{c}\qquad integer n, k \hfil}
 \line{\phantom{c}\qquad real    sum, first, integral, cosDelta, D, Q, R 
 \line{\phantom{c}\qquad real    cosPhiK, cosPhiK1, sinPhiK, sinPhiK1 \hfil}
 \line{\phantom{c}\qquad real    phi(maxPoints), theta(maxPoints) \hfil}
 \line{\phantom{c}\qquad character*14 filename \hfil}
 \line{c\qquad Read number of lon/lat coordinate pairs, and \hfil}
 \line{c\qquad the name of the file containing those coordinates. \hfil}
 \line{\phantom{c}\qquad read(5,*) n, filename \hfil}
 \line{c\qquad Read the coordinates. Longitude is first. Both in 
 \line{\phantom{c}\qquad open(1, file=filename)\hfil}
 \line{\phantom{c}\qquad read(1,*) (theta(k),phi(k), k=1,n)\hfil}
 \line{\phantom{c}\qquad close(1)\hfil}
 \line{c\qquad Convert to radians.\hfil}
 \line{\phantom{c}\qquad do 10 k=1,n \hfil}
 \line{\phantom{c}\qquad \quad phi(k)   = phi(k)   * piOver180 \hfil}
 \line{\phantom{c}\qquad \quad theta(k) = theta(k) * piOver180 \hfil}
 \line{\phantom{c}$\,$10 continue \hfil}
 \line{c \hfil}
 \line{c\qquad Make the sequence of coordinates cyclic. \hfil}
 \line{\phantom{c}\qquad phi(n+1)   = phi(1) \hfil}
 \line{\phantom{c}\qquad theta(n+1) = theta(1) \hfil}
 \line{c\qquad Initialize for the summation. \hfil}
 \line{\phantom{c}\qquad sum = 0.0 \hfil}
 \line{\phantom{c}\qquad cosPhiK1 = cos(phi(1)) \hfil}
 \line{\phantom{c}\qquad sinPhiK1 = sin(phi(1)) \hfil}
 \line{\phantom{c}\qquad do 20 k=1,n \hfil}
 \line{c\qquad \quad Previous "k+1" values
                               become new "k" values.\hfil}
 \line{\phantom{c}\qquad \quad cosPhiK  = cosPhiK1 \hfil}
 \line{\phantom{c}\qquad \quad sinPhiK  = sinPhiK1 \hfil}
 \line{c\qquad \quad Get new "k+1" values.\hfil}
 \line{\phantom{c}\qquad \quad cosPhiK1 = cos(phi(k+1))\hfil}
 \line{\phantom{c}\qquad \quad sinPhiK1 = sin(phi(k+1))\hfil}
 \line{c\qquad \quad Compute first factor in
                               k-th term of summation.\hfil}
 \line{\phantom{c}\qquad \quad first = cosPhiK * cosPhiK1
                                       * sin(theta(k+1)-theta(k))\hfil}
 \line{c\qquad \quad Compute integral in k-th term of summation. \hfil}
 \line{c\qquad \quad First get cosine of delta, then discriminant,
                     then Q.\hfil}
 \line{\phantom{c}\qquad \quad cosDelta = cosPhiK * cosPhiK1 
                                          * cos(theta(k+1)-theta(k))\hfil}
 \line{\phantom{c}\qquad      . \qquad+ sinPhiK * sinPhiK1\hfil}
 \line{\phantom{c}\qquad \quad  D = (sinPhiK + sinPhiK1)**2\hfil}
 \line{\phantom{c}\qquad      . \qquad - (1.0+cosDelta)*(1.0-cosDelta
 \line{\phantom{c}\qquad \quad Q =  sinPhiK + sinPhiK1 + 1.0 + cosDelta\hfil}
 \line{\phantom{c}\qquad \quad if (D .gt. 0.0) then\hfil}
 \line{\phantom{c}\qquad \quad \quad R = sqrt (D)\hfil}
 \line{\phantom{c}\qquad \quad \quad integral = alog ( (Q+R)/(Q-R) ) / 
 \line{\phantom{c}\qquad \quad else if (D .lt. 0.0) then\hfil}
 \line{\phantom{c}\qquad \quad \quad R = sqrt (-D)\hfil}
 \line{\phantom{c}\qquad \quad \quad integral = 2.0 * atan ( R/Q ) / R\hfil}
 \line{\phantom{c}\qquad \quad else \hfil}
 \line{\phantom{c}\qquad \quad \quad integral = Q / ((1.0+sinPhiK)
 \line{\phantom{c}\qquad \quad endif\hfil}
 \line{c\qquad \quad Accumulate sum and go on to next segment.\hfil}
 \line{\phantom{c}\qquad \quad sum = sum + first * integral\hfil}
 \line{\phantom{c}$\,$ 20 continue\hfil}
 \line{c\qquad Write results and stop.\hfil}
 \line{\phantom{c}\qquad write(6,90) sum, sum/(4.0*pi), 
 \line{\phantom{c}\qquad stop\hfil}
             format(1x, 'area (on unit sphere) = ', e14.6,\hfil}
 \line{\phantom{c}\qquad .\qquad /1x, 'area / (4*pi) = ', e14.6,\hfil}
 \line{\phantom{c}\qquad .\qquad /1x, 'area (km**2 on earth) = ', 
 \line{\phantom{c}\qquad end\hfil}
 \noindent{\bf References}
 \ref (1) Francis B. Hildebrand, ``Advanced Calculus for Applications",
 Prentice-Hall, 1976.
 \ref (2) J. E. Marsden and A. J. Tromba, ``Vector Calculus",
 W. H. Freeman \& Co., 1976.
 \ref (3) John P. Snyder, ``Map Projections - A Working Manual",
 U. S. Geological Survey Professional Paper 1395,
 U. S. Government Printing Office, 1987.
 \ref (4) Robert Weinstock, ``Calculus of Variations",
 Dover Publications, 1974.
 \centerline{\bf The Quadratic World of Kinematic Waves}
 \centerline{Sandra L. Arlinghaus, John D. Nystuen, Michael J. Woldenberg
 $^{\star }$}

      Kinematic waves differ from ``ordinary" waves insofar
 as it is the kinematics--the dynamic aspects of motion other than mass
 and force--that are the focus.  Thus, Langbein and Leopold [1968, p. 1]
 define a kinematic wave as ``a grouping of moving objects in zones along
 a flow path and through which the objects pass.  These concentrations
 may be characterized by a simple relation between the speed of the
 moving objects and their spacing as a result of interaction between them."
 Flow in a channel is characteristically expressed as a function of 
 concentration, be that as cars per hour as a function of cars 
 per mile or as transport in cubic feet per minute of sand in a one inch
 tube as a function of linear concentration of sand in pounds per square
 foot [Langbein and Leopold 1968; Haight 1963; Lighthill and Whitham I and
 II 1955].  Examples of kinematic waves are abundant in physical and
 urban settings alike--in realms as disparate as sand transport in a flume
 or car movement on an Interstate Highway [Langbein and Leopold, 1968].
 When empirical data are graphed, they often trace out a parabola
 (or a curve close to a parabola); thus, the relationship between
 concentration and flow is often a quadratic one [Langbein and Leopold, 

     The classical analysis of the parabolic graphs of these waves 
 rests on considering what happens to flow as a result of minor
 perturbations in local concentrations--techniques are based in
 notions from the calculus [Langbein and Leopold 1968].  Consider
 a concave down parabola with its maximum in the first quadrant
 that passes through the origin.  Flow is a function of concentration;
 thus, concentration appears on the $x$-axis and flow on the $y$-axis.  
 Choose two points on the curve, one with coordinates $(x_1, y)$ and
 the other with coordinates $(x_2, y)$--the $x$-coordinates are
 different and lead to the same $y$-coordinate.  They are placed
 symmetrically on the $x$-axis about a vertical line through the
 curve's maximum (Figure 1; for electronic readers only, please
 draw this curve and subsequent ones as per text).  Assuming that
 $x_1$ is to the left of the maximum, the traditional analysis
 notes that at $x_1$, a slight increase in concentration results in
 a slight increase in flow; a slight decrease in concentration at $x_1$
 results in a slight decrease in flow.  The channel is relatively sparsely
 congested--slight changes in concentration result in directly parallel
 changes in flow.  Further, the closer one is to the $x$-coordinate of
 the maximum, the less difference these slight changes cause.  On the
 other hand, at $x_2$ (to the right of the maximum) a slight increase
 in concentration results in a decrease in flow, suggesting a channel which
 cannot easily assimilate any extra traffic.  Further, a slight decrease
 in concentration at $x_2$ results in an increase in flow, again 
reflecting a
 relatively congested condition of this channel.  When the horizontal
 line suggested by $x_1$ and $x_2$ is tangent to the parabola, at its
 maximum, the kinematic wave is stationary relative to the channel;
 thus, as the distance of horizontal lines increases away from this
 tangent line, there is a corresponding increase in the amount of 
 change caused by local perturbations.  The origin, as a location for $x_1$,
 represents a completely uncrowded condition, while the second
 intersection of the curve with the $x$-axis represents the most crowded
 position within this interval [Langbein and Leopold 1968].

     The traditional analysis, based merely on considering what
 slight changes in $x_1$ and $x_2$ might suggest, fits well with
 real-world travel experience.  Consider the concentration on the
 $x$-axis to be density of automobiles as vehicles per mile;
 on the $y$-axis, consider flow to be vehicles per hour.  Practical
 evidence does suggest that an improvement in the maximum capacity
 of the road does result in improved transmission of flow, but only
 up to a point.  Thus, highway systems are widened around cities
 and endowed with limited access to increase the number of vehicles 
 per hour that can move from origin to destination.  Beyond about 1800 
 vehicles per hour, this ``improvement" is no longer useful [Nystuen
 1992]; indeed, congestion increases and flow per hour decreases toward
 the point of gridlock---the ultimate disaster that can affect millions of
 individuals.  This sort of ceaseless ``improvement," to the point of
 disaster, of what worked well in a less congested arena, appears in a
 variety of contexts; when an optical cable with the capacity to serve
 millions is cut, disaster comes to many rather than to few, and
 chaos in communication becomes a real possibility [Austin 1991].
     The traditional analysis also allows for computation of various other
 features associated with the kinematics of the phenomenon it describes.
 For example, the average speed of particles in the channel,
 or wave celerity, can be measured at any point on the curve, simply
 by finding the slope of the chord joining that point to the origin
 [Langbein and Leopold 1968].
 However, when a given density leads to a certain flow, which is then
 used to determine the next input to create a new density level,
 feedback occurs.  Feedback is not measured in the traditional
 analysis.  It also fits with travel experience and indeed is the
 sort of process that can get chaotic.  Thus, it seems plausible to
 consider graphical analysis of kinematic curves, based in
 Feigenbaum's Graphical Analysis from the mathematics of Chaos Theory, as a
 supplement to the traditional analysis.

     Consider the following set of parabolas as Figures 2 through 7:
 $y=1.5x(1-x)$; $y=2x(1-x)$; $y=3x(1-x)$; $y=3.75x(1-x)$; $y=4x(1-x)$; and,
 $y=5x(1-x)$.  The e-reader should draw each of these curves, noting
 that each parabola is of the sort described above---consider the units
 on the axes, ranging from 0 to less than 1.5, as percentages.  Thus,
 0.5 on the $x$-axis represents a concentration of 50\%.  Also include
 in each graph the line $y=x$.  Each parabola intersects this 45-degree
 line in two points--one at the origin and one that is either to the 
 left or to the right of the curve's maximum.  As the coefficient of
 the curve increases from 1.5 to 5, the curves become successively less
 flat, have a higher maximum, and have a second intersection with the
 line $y=x$ farther to the right.

     To represent geometric feedback visually on Figures 2 to 7, proceed
 as follows [based on material from Feigenbaum 1980; Gleick 1987;
 Devaney and Keen 1989].  Locate the point 0.1 on the x-axis of
 each figure.  Draw a vertical line from that point (as a ``seed"
 value for the graphical analysis) to the parabola.  Now draw a 
 horizontal line from the curve to the line $y=x$; next read
 vertically from this location to the parabola.  The effect here is
 to use output as input; for, 0.1 was the initial input.  When
 that value was mapped to the parabola, an output resulted---
 when that output was mapped horizontally to $y=x$, it was then used
 as input when it was next sent to the curve.  Successive iteration
 of this process should result in the following paths from the iteration
 (``orbits"):  Figure 2---a staircase with shallow rises; Figure 3---
 a staircase with sharper rises than in Figure 2; Figure 4---
 a tightly bounded cyclical orbit closing in on the second intersection
 of the line with the parabola; Figure 5---an unpredictable, bounded
 orbit; Figure 6---a chaotic, bounded orbit; Figure 7---an orbit that
 escapes to negative infinity (from a curve whose maximum is beyond
 the 100\% concentration level).  Geometrically, control over the
 dynamics of the orbit becomes less stable as one proceeds from Figures
 2 to 7.  It makes little difference which initial seed is
 chosen; the dynamics of the orbit are invariant with respect to these
 curves (parabolas).  Unlike the traditional analysis, in which there
 is considerable variation in the measures used, with respect to a single
 curve, the pattern of the orbit is constant throughout each
 figure---as a sort of a shape-invariant.  Indeed, any of these curves
 might be employed equally for the traditional, but not for the 
 graphical, analysis.

     What determines the extent of stability in the geometric dynamics
 noted in these figures are the height of the parabola and the position
 of the second intersection of $y=x$ with that parabola.  Higher
 parabolas have intersection point with $y=x$ farther to the right of
 the curve's maximum, producing more uncontrolled feedback.  This fits
 well with traffic observations; increase of a road's maximum capacity
 beyond some critical level leads to disastrous congestion.  The tool
 of graphical analysis looks promising as a tool in analyzing
 real-world phenomena [Feigenbaum 1980; Gleick 1987] 
 that follow kinematic waves as well as those that
 follow more complicated curves [Arlinghaus, Nystuen, and Woldenberg
 $^{\star }$  Author Woldenberg wishes to acknowledge input from
 M. Sonis regarding the analysis of kinematic waves---1981.
 \centerline{\bf References }

 \ref Arlinghaus, S. L., Nystuen, J. D., and Woldenberg, M. J. 1992
 (forthcoming in July, 1992).  An application of graphical analysis to
 semidesert soils.  {\sl Geographical Review\/}.

 Austin, Robert F.  Personal communication, 1991.

 Devaney, R. L. and Keen, L. 1989.  {\sl Chaos and fractals:  The 
 Mathematics behind the Computer Graphics\/}.  Proceedings of symposia
 in applied mathematics, vol. 39, American Mathematical Society,
 Providence, RI.

 Feigenbaum, M. J. 1980.  Universal behavior in non-linear systems.
 {\sl Los Alamos Science\/}, summer: 4-27.

 Gleick, J. 1987.  {\sl Chaos:  Making a New Science\/}. 
 New York:  Penguin Books.

 Haight, F. A., 1963.  {\sl Mathematical theories of traffic flow\/}.
 New York:  Academic Press.

 Langbein, W. B. and Leopold, L. B. 1968.  {\sl River Channel Bars and 
 Dunes---Theory of Kinematic Waves\/}, USGS, Professional Paper 4222,
 pp. 1-20. United States Government Printing Office, Washington.

 Lighthill, J. J., and Whitham, G. B., 1955.  On kinematic
 waves.  I.  Flood movement in long rivers.  {\sl Royal Soc. [London] 
 v. 229A, p. 281-316.

 Lighthill, J. J., and Whitham, G. B., 1955.  On kinematic waves II.  
 A theory of traffic flow on long crowded roads.  {\sl Royal Soc. [London]
 Proc.\/}, v. 229A, p. 317-345.

 Nystuen, J. D.  Seminar on ``Intelligent Vehicle Highway Systems."
 University of Michigan.
 \noindent{\bf REVIEW} of RangeMapper$^{\hbox{TM}}$ (version 1.4b).
 A utility for biological species range mapping, 
 and similar mapping tasks in other fields.
 Price:  \$350
 Program and manual written by {\bf Kenelm W. Philip}.
 Tundra Vole Software
 1590 North Becker Ridge Road
 Fairbanks, Alaska 99709
 (907) 479-2689
 Reviewed by {\bf Yung-Jaan Lee}, Ph.D. Candidate in Urban, Technological,
 and Environmental Planning, The University of Michigan, Ann Arbor, MI 48109.
 \noindent From the author's flyer:
 ``RangeMapper is a Macintosh mapping and data plotting utility.
 It allows rapid and accurate display of lat/long data on the 
 user's choice of maps."

 \centerline{``{\bf RangeMapper Features}"}

 ``Range Mapper can bring up low-resolution maps of the
 world, or portions thereof, in north polar azimuthal, simple
 cylindrical, Mercator, orthographic, stereographic, or
 Lambert azimuthal equal-area projections.

 Data may be plotted to maps from ASCII files of latitude,
 longitude, and site name in several different formats, in
 several sizes of open/filled circles and squares.  Program-readable
 data files can be dumped directly from a database or
 spreadsheet.  Lat/long coords may be read directly from
 the maps, and plotted points may be `verified' by clicking
 on them.  The Alaska map is based on the CIA World Data
 Bank file, and is usable down to 20-30 mile regions.

 The world map is derived from the Micro World Data Bank II file.
 It is usable down to regions of the order of
 500 miles or so in extent, which is adequate for species 
 range mapping on small-scale maps.

 Designed originally for biological species range mapping,
 the program has many other uses wherever data files need to
 be accurately plotted to maps.  In conjunction with the word
 processor `Nisus', RangeMapper may also be used as a
 visual interface to a text database, so you can open a text file
 on a site by clicking on that site on the displayed map.   
 The `verify' feature permits rapid checking of your ASCII
 data files for errors.  In conjunction with a DA text editor,
 your data files may be edited interactively from within the 
 program---making error correction a rapid and easy job.

 Points may also be placed on the maps by hand, either by
 eye or by reading coordinates off the map and dropping a
 dot at the correct coordinates.

 RangeMapper can save maps to disk, print them directly to
 an IMageWriter or LaserWriter, or export them as PICT files to 
 be imported into a drawing program (as MacDraw or Canvas)
 for enhancement and annotation.

 Maps produced by RangeMapper may have a user-designed
 latitude/longitude grid overlaid, and a title and caption
 may be added.  Data plotted to RangeMapper
 may be overlaid in up to 14 separate layers, each of which may
 be toggled on and off independently.  Data may be plotted as dots
 or as connected lines."

     The processing speed of this software is, to some extent, slow, 
 especially for a small-scale map or a map with filled area.  This may
 be due to the fact that this software involves a vast number of 

     Users accustomed to working with Geographic Information Systems
 should be aware that this software is, as it says, a mapping utility only.
 The spiral-bound documentation is adequate and contains
 samples of maps apparently made using RangeMapper;
 a couple of improvements seem in order.

 \item{1.}  On page 2, the author describes RangeMapper as needing at least
 1500KB of free memory, and that the "MultiFinder partition" should
 be set to that value in the Get Info dialog box.  This is confusing,
 as the user will probably select the MultiFinder icon and try
 to change the partition in Get Info.  In fact, the user
 should highlight the {\bf RangeMapper} icon, rather than the MultiFinder
 icon, and then go to Get Info dialog box to change the partition.

 \item{2.}  On page 6, the user is instructed to select the file `*MWDB3.All'
 under the File menu.  However, there is no such file in this software.
 Instead, the user should select the file `MWDB2.All' and then check the
 {\bf show state/provs} under the Mapping menu in order to display the
 circumpolar map demonstration.

 \item{3.}  The printing requirements should appear early in the first part
 of the manual.

 \item{4.}  An Index at the end of the manual would be helpful.

 Some other suggestions for improvement of the software are:

 \item{1.}  It would help to employ more of the standard Macintosh
 environment conventions, such as:
 \qquad a {\bf Close} selection under the File menu;
 \qquad a {\bf Window} sub-menu in the pull-down menu;
 \qquad the filename displayed at the top of the screen (different
 from the title of the map);
 \qquad a close box, zoom box, size box, and scroll bars displayed on
 the screen, as in a standard Macintosh window.

 \item{2.}  The ``Menus" section could be moved to the beginning of the
 manual, rather than in the middle.  If not, the author should 
 describe the difference between {\bf Map} and {\bf Open} function
 in the {\bf File} menu at the beginning.

 \item{3.}  After displaying a map, a selection box will automatically 
show up
 on the screen.  The author should explain why this box comes up.
 It only later becomes apparent that it is used to link a map to
 adjacent regions, if available.

 It may be more efficient to run this software using a Macintosh II
 or higher, or better, with a math co-processer because of very slow
 printing times.  If not, users must carefully follow the recommended
 printing procedure to reduce the size of the output file, such as
 turning off ``Graphics Smoothing" and checking ``Precision Bitmap
 Alignment" (in the ``Moving RangeMapper Output to Word Processors"
 section and the ``Printing: RangeMapper" section).  

 In addition to the two drawing programs (Canvas 3.0 and SuperPaint 2.0), 
 MacDraw II 1.1 and MacPaint 2.0 are capable of image size reduction.
 After exporting a map to MacDraw or MacPaint, one can still copy the 
 map to any word processor.

 Those needing only a mapping program will find this sortware useful,
 especially if working on high latitude areas.
 {\bf Note}: 
 Canvas is a trademark of Deneba Systems;
 SuperPaint is a trademark of Aldus Corporation;
 NISUS is a trademark of Paragon Concepts, Inc.;
 Apple and LaserWriter are registered trademarks of Apple Computer, Inc.;
 Macintosh is a trademark licensed to Apple Computer, Inc.; MacDraw
 is a trademark of Apple Computer, Inc.
 \noindent{\bf FEATURES}
 \noindent{\bf Press Clippings}
 \centerline{\bf FROM SCIENCE, AAAS}
 {\sl Science\/}, November 29, 1991, Vol.  254,  No. 5036,  copyright, 
 the   American   Association  for  the  Advancement of Science.  Many
 thanks to Joseph Palca at {\sl Science\/} for his continuing interest 
 in  online  journals.   The  citation  appeared  in  ``Briefings" and
 is entitled ``{\bf  Online Journals}," by Joseph Palca.

 NOTE:   Readers  wishing  to contact Richard Zander, Editor of {\sl
 Flora Online\/}, can do so at bitnet address: 

 \centerline{\bf FROM SCIENCE NEWS}
 {\bf Math for all seasons}
 by Ivars Peterson
 January 25, 1992, Vol. 141, No. 4.  Page 61.  Reprinted with permission
 of {\sl Science News\/}.
     When the American Association for the Advancement of Science 
 announced with considerable fanfare last year the 1992 debut of
 {\sl The Online Journal of Current Clinical Trials\/}, it was
 billed as the world's first peer-reviewed science journal available
 to subscribers electronically.  What the organizers of this effort
 didn't know was that several such electronic journals already existed.
 One of these concerns the application of mathematics to geography.

      {\sl Solstice:  An Electronic Journal of Geography and Mathematics\/}
 --- published by Sandra Lach Arlinghaus of the Institute of Mathematical
 Geography, a small, independent research organization in Ann Arbor, Mich.
 --- first appeared in 1990.  Its two issues per year, published 
 appropriately on the dates of the summer and winter solstices, go to about
 50 individuals, wwho receive the journal free.  Transmission costs for
 distributing the journal electronically over a computer network to all
 subscribers amount to less than \$5 per issue, with the cost of
 printing passed on to the user.  Libraries and other institutions that 
 prefer printed copies pay for each issue, and those copies
 are generated from computer files only when needed.

 ``It's all very cheap, all environmentally sound," Arlinghaus says.

 But getting the journal going wasn't easy, she remarks.  The biggest
 production problem involved photographs and figures, which can't be
 transmitted electronically in the same, compact way as 
 letters, numbers or even mathematical notation.  At present, 
 individuals wishing to see particular illustrations must obtain
 photocopies directly from the Institute of Mathematical Geography.
 Arlinghaus also admits that she has had trouble obtaining manuscripts
 for publication in this still-unconventional medium.  But individuals
 who might initially have been skeptics ``become more receptive when
 they see the actual product," she says.
 One from AAAS in reply to Peterson; one from IMaGe in reply to AAAS,
 during period from January through May, 1992.
 \centerline{\bf AAG NEWSLETTER}
 Volume 27, Number 6, June 1992.

 ``Online Geographical Journals," page 10.
 \noindent{\bf INDEX to Volumes I (1990) and II (1991) of
             {\sl Solstice}.}
 \noindent{\bf Volume I, Number 1, Summer, 1990}

 \noindent 1.  REPRINT

 William Kingdon Clifford, {\sl Postulates of the Science of Space\/}

      This reprint of a portion of  Clifford's  lectures  to  the
 Royal  Institution in the 1870's suggests many geographic topics
 of concern in the last half of the twentieth century.   Look for
 connections  to  boundary  issues,  to  scale problems, to self-
 similarity and fractals, and to non-Euclidean  geometries  (from
 those based on denial of Euclid's parallel  postulate  to  those
 based on a sort of mechanical ``polishing").  What else did,  or
 might, this classic essay foreshadow?

 \noindent 2.  ARTICLES.

 Sandra L. Arlinghaus, {\sl Beyond the Fractal.}  

     An original article.  The fractal notion of  self-similarity
 is  useful  for  characterizing  change  in  scale;  the  reason
 fractals are effective in the geometry of central  place  theory 
 is  because  that  geometry  is hierarchical in nature.  Thus, a
 natural place to look for other connections of this  sort  is to
 other geographical concepts that are also hierarchical.   Within
 this fractal context, this article examines the case of  spatial
     When the idea of diffusion is extended to see ``adopters" of
 an innovation as ``attractors" of new adopters,  a  Julia set is 
 introduced as a possible axis against which to measure one class
 of geographic phenomena.   Beyond the fractal  context,  fractal
 concepts,  such  as  ``compression"  and  ``space-filling"   are
 considered in a broader graph-theoretic setting.
 William C. Arlinghaus, {\sl Groups, Graphs, and God}

      An original article based on a talk given  before  a MIdwest
 GrapH TheorY (MIGHTY) meeting.  The author,  an  algebraic  graph
 theorist, ties his research interests to a broader  philosophical
 realm,  suggesting  the  breadth  of  range  to  which  algebraic
 structure might be applied.

     The  fact  that  almost  all  graphs  are rigid (have trivial
 automorphism groups) is exploited to argue probabilistically  for
 the  existence  of  God.  This  is  presented  with the idea that 
 applications  of  mathematics  need  not be limited to scientific
 \noindent 3.  FEATURES
 \item{i.}  Theorem Museum --- Desargues's  Two  Triangle  Theorem 
            from projective geometry.
 \item{ii.} Construction Zone --- a centrally symmetric hexagon is
            derived from an arbitrary convex hexagon.
 \item{iii.} Reference Corner --- Point set theory and topology.
 \item{iv.}  Educational Feature --- Crossward puzzle on spices.
 \item{v.}   Solution to crossword puzzle.
 \noindent{\bf Volume I, Number 2, Winter, 1990}
 \noindent 1.  REPRINT

 John D. Nystuen (1974), {\sl A City of Strangers:  Spatial Aspects
 of Alienation in the Detroit Metropolitan Region\/}.  

     This paper examines the urban shift from ``people space" to 
 ``machine space" (see R. Horvath,  {\sl Geographical Review\/},
 April, 1974) in the Detroit metropolitan  region  of 1974.   As
 with Clifford's {\sl Postulates\/}, reprinted in the last issue
 of {\sl Solstice\/}, note  the  timely  quality  of many of the 

 \noindent 2.  ARTICLES

 Sandra Lach Arlinghaus, {\sl Scale and Dimension: Their Logical

      Linkage  between  scale  and  dimension  is made using the 
 Fallacy of Division and the Fallacy of Composition in a fractal
 Sandra Lach Arlinghaus, {\sl Parallels between Parallels\/}.

      The earth's sun introduces a symmetry in the perception of 
 its trajectory in the sky that naturally partitions the earth's
 surface  into  zones  of  affine  and hyperbolic geometry.  The
 affine zones, with  single  geometric  parallels,  are  located 
 north and south of the  geographic  parallels.   The hyperbolic
 zone, with multiple geometric parallels, is located between the
 geographic  tropical  parallels.   Evidence  of  this geometric
 partition is suggested in the geographic environment --- in the
 design of houses and of gameboards.
 Sandra L. Arlinghaus, William C. Arlinghaus, and John D. Nystuen.
 {\sl The Hedetniemi Matrix Sum:  A Real-world Application\/}.

     In a recent paper, we presented an algorithm for finding the
 shortest distance between any two nodes in a network of $n$ nodes
 when  given  only  distances between adjacent nodes [Arlinghaus, 
 Arlinghaus, Nystuen,  {\sl Geographical  Analysis\/}, 1990].  In
 that  previous   research,  we  applied  the  algorithm  to  the
 generalized  road  network  graph surrounding San Francisco Bay.  
 Here,  we  examine consequent changes in matrix entires when the
 underlying  adjacency pattern of the road network was altered by 
 the  1989  earthquake  that closed the San Francisco --- Oakland
 Bay Bridge.
 Sandra Lach Arlinghaus, {\sl Fractal Geometry  of Infinite Pixel
 Sequences:  ``Su\-per\--def\-in\-i\-tion" Resolution\/}?

    Comparison of space-filling qualities of square and hexagonal
 \noindent 3.  FEATURES
 \item{i.}       Construction  Zone ---  Feigenbaum's  number;  a
 triangular coordinatization of the Euclidean plane.
 \item{ii.}  A three-axis coordinatization of the plane.
 \noindent {\bf Volume II, Number 1, Summer, 1991}
 \noindent 1.  ARTICLE

 Sandra L. Arlinghaus, David Barr, John D. Nystuen.
 {\sl The Spatial Shadow:  Light and Dark --- Whole and Part\/}

      This account of some of the projects of sculptor David Barr
 attempts to place them in a formal, systematic, spatial  setting
 based  on  the  postulates  of  the  science of space of William
 Kingdon Clifford (reprinted in {\sl Solstice\/}, Vol. I, No. 1.).
 \noindent 2.  FEATURES

 \item{i}  Construction Zone --- The logistic curve.
 \item{ii.} Educational feature --- Lectures on ``Spatial Theory"
 \noindent {\bf Volume II, Number 2, Winter, 1991}
 \noindent 1.  REPRINT

 Saunders Mac Lane, ``Proof, Truth, and Confusion."  Given as the
 Nora and Edward Ryerson Lecture at The University of Chicago in
 1982.  Reppublished with permission of The University of Chicago
 and of the author.

 I.  The Fit of Ideas.  II.  Truth and Proof.  III.  Ideas and Theorems.
 IV.  Sets and Functions.  V.  Confusion via Surveys.
 VI.  Cost-benefit and Regression.  VII.  Projection, Extrapolation,
 and Risk.  VIII.  Fuzzy Sets and Fuzzy Thoughts.  IX.  Compromise
 is Confusing.

 \noindent 2.  ARTICLE

 Robert F. Austin.  ``Digital Maps and Data Bases:  
 Aesthetics versus Accuracy."

 I.  Introduction.  II. Basic Issues.  III. Map Production.
 IV.  Digital Maps.  V.  Computerized Data Bases.  VI.  User

 \noindent 3.  FEATURES

 Press clipping; Word Search Puzzle; Software Briefs.
 \noindent{\bf OTHER publications of }
 \centerline{\it 2790 BRIARCLIFF}
 \centerline{\it ANN ARBOR, MI 48105-1429; U.S.A.}
 \centerline{(313) 761-1231; IMaGe@UMICHUM}
 \vskip 0.2cm
 \centerline{\it ``Imagination is more important than knowledge"}
 \centerline{\it A. Einstein}
 \centerline{\bf MONOGRAPH SERIES}
 \centerline{\sl Scholarly Monographs--Original Material}
 \centerline{Prices on request, exclusive of shipping and handling;}
 \centerline{payable in U.S. funds on a U.S. bank, only.}
 Monographs are printed by {\bf Digicopy} on 100\% recycled paper
 of archival quality; both hard and soft cover is available.
 \vskip 0.2cm
 1.  Sandra L. Arlinghaus and John D. Nystuen.  {\it Mathematical
 Geography and Global Art:  the Mathematics of David Barr's ``Four
 Corners Project\/},'' 1986. 
 \vskip 0.1cm
 This monograph contains Nystuen's calculations, actually used
 by Barr to position his abstract tetrahedral sculpture
 within the earth.  Placement of the sculpture vertices in Easter
 Island, South Africa, Greenland, and Indonesia was chronicled in
 film by The Archives of American Art for The Smithsonian
 Institution.  In addition to the archival material, this 
 monograph also contains Arlinghaus's solutions to broader theoretical
 questions--was Barr's choice of a tetrahedron unique within his
 initial constraints, and, within the set of Platonic solids?
 \vskip 0.2cm
 2.  Sandra L. Arlinghaus.  {\it Down the Mail Tubes:  the Pressured
 Postal Era, 1853-1984\/}, 1986. 
 \vskip 0.1cm

 The history of the pneumatic post, in Europe and in the
 United States, is examined for the lessons it might offer to the
 technological scenes of the late twentieth century.  As Sylvia L.
 Thrupp, Alice Freeman Palmer Professor Emeritus of History, The
 University of Michigan, commented in her review of this work
 ``Such brief comment does far less than justice to the 
 intelligence and the stimulating quality of the author's writing,
 or to the breadth of her reading.  The detail of her accounts of
 the interest of American private enterprise, in New York and
 other large cities on this continent, in pushing for construction
 of large tubes in systems to be leased to the government, brings
 out contrast between American and European views of how the new
 technology should be managed.  This and many other sections of
 the monograph will set readers on new tracks of thought.'' 
 \vskip 0.2cm
 3.  Sandra L. Arlinghaus.  {\it Essays on Mathematical Geography\/},
 \vskip 0.1cm

 A collection of essays intended to show the range of power
 in applying pure mathematics to human systems.  There are two types of 
essay:  those which employ traditional mathematical
 proof, and those which do not.  As mathematical proof may itself
 be regarded as art, the former style of essay might represent
 ``traditional'' art, and the latter, ``surrealist'' art.  Essay
 titles are:  ``The well-tempered map projection,'' ``Antipodal
 graphs,'' ``Analogue clocks,'' ``Steiner transformations,'' ``Concavity
 and urban settlement patterns,'' ``Measuring the vertical city,''
 ``Fad and permanence in human systems,'' ``Topological exploration
 in geography,'' ``A space for thought,'' and ``Chaos in human
 systems--the Heine-Borel Theorem.''
 \vskip 0.2cm
 4.  Robert F. Austin, {\it A Historical Gazetteer of Southeast Asia\/},
 \vskip 0.1cm

 Dr. Austin's Gazetteer draws geographic coordinates of Southeast
 Asian place-names together with references to these
 place-names as they have appeared in historical and literary
 documents.  This book is of obvious use to historians and to
 historical geographers specializing in Southeast Asia.  At a
 deeper level, it might serve as a valuable source in establishing
 place-name linkages which have remained previously unnoticed, in 
 documents describing trade or other communications connections,
 because of variation in place-name nomenclature.
 \vskip 0.2cm
 5.  Sandra L. Arlinghaus, {\it Essays on Mathematical Geography--II\/},
 \vskip 0.1cm

 Written in the same format as IMaGe Monograph \#3, that seeks to use
 ``pure'' mathematics in real-world settings, this volume
 contains the following material:  ``Frontispiece--the Atlantic
 Drainage Tree,'' ``Getting a Handel on Water-Graphs,'' ``Terror in Transit:
 A Graph Theoretic Approach to the Passive Defense of Urban Networks,''
 ``Terrae Antipodum,'' ``Urban Inversion,'' 
``Fractals:  Constructions, Speculations,
 and Concepts,'' ``Solar Woks,'' ``A Pneumatic Postal Plan:  The 
 Chambered Interchange and ZIPPR Code,'' ``Endpiece.''
 \vskip 0.2cm
 6.  Pierre Hanjoul, Hubert Beguin, and Jean-Claude Thill, {\it Theoretical
 Market Areas Under Euclidean Distance\/}, 1988. 
 (English language text; Abstracts written in French and in English.) 
 \vskip 0.1cm
 Though already initiated by Rau in 1841, the economic theory of the 
shape of 
 two-dimensional market areas has long remained concerned with a
 representation of transportation costs as linear in distance. 
 In the general gravity model, to which
 the theory also applies, this corresponds to a decreasing exponential
 function of distance deterrence.  Other transportation cost and
 distance deterrence functions also appear in the literature, however.
 They have not always been considered from the viewpoint of the shape
 of the market areas they generate, and their disparity asks the
 question whether other types of functions would not be worth
 being investigated.  There is thus a need for a general theory
 of market areas:  the present work aims at filling this gap, 
 in the case of a duopoly competing inside the Euclidean plane
 endowed with Euclidean distance. \vskip 0.1cm

 (Bien qu'\'ebauch\'ee par Rau d\`es 1841, la th\'eorie \'economique
 de la forme des
 aires de march\'e planaires s'est longtemps content\'ee de l'hypoth\`ese
 de co\^uts de transport proportionnels \`a la distance.  Dans le mod\`ele
 gravitaire g\'en\'eralis\'e, auquel on peut \'etendre cette th\'eorie, ceci
 correspond au choix d'une exponentielle d\'ecroissante comme fonction de
 dissuasion de la distance.  D'autres fonctions de co\^ut de transport
 ou de dissuasion de la distance apparaissent cependant dans la 
 La forme des aires de march\'e qu'elles engendrent n'a pas toujours \'et\'e
 \'etudi\'ee ; par ailleurs, leur vari\'et\'e am\`ene \`a se demander
 si d'autres fonctions encore ne m\'eriteraient pas d'\^etre examin\'ees. 
 Il para\^it donc utile
 de disposer d'une th\'eorie g\'en\'erale des aires de march\'e : ce \`a
 quoi s'attache ce travail en cas de duopole, dans le cadre
 du plan euclidien muni d'une distance euclidienne.)
 \vskip 0.2cm
 7.  Keith J. Tinkler, Editor, {\it Nystuen---Dacey Nodal Analysis\/}, 1988.

 Professor Tinkler's volume displays the use of this graph theoretical
 tool in geography, from the original Nystuen---Dacey article, to a 
 bibliography of uses, to original uses by Tinkler.  Some reprinted 
 material is included, but by far the larger part is of previously 
 material.  (Unless otherwise noted, all items listed below are previously
 unpublished.)  Contents:  `` `Foreward' " by Nystuen, 1988; 
 ``Preface" by Tinkler,
 1988; ``Statistics for Nystuen---Dacey Nodal Analysis," by Tinkler, 1979;
 Review of Nodal Analysis literature by Tinkler (pre--1979, reprinted 
with permission;
 new as of 1988); FORTRAN program listing for Nodal Analysis by Tinkler;
 ``A graph theory
 interpretation of nodal regions'' by John D. Nystuen and Michael F. Dacey,
 reprinted with permission, 1961; Nystuen---Dacey data concerning
 telephone flows in Washington and Missouri, 
 1958, 1959 with comment by Nystuen, 1988; 
 ``The expected distribution of nodality in random (p, q) 
 graphs and multigraphs,'' by Tinkler, 1976.
 8.  James W. Fonseca, {\it The Urban Rank--size Hierarchy:  A Mathematical
 Interpretation\/}, 1989.

 The urban rank--size hierarchy can be characterized as an equiangular spiral
 of the form $r=ae^{\theta \, \hbox{cot}\alpha}$.  An equiangular spiral
 can also be constructed from a Fibonacci sequence.  The urban rank--size
 hierarchy is thus shown to mirror the properties derived from Fibonacci
 characteristics such as rank--additive properties.  A new method of
 structuring the urban rank--size hierarchy is explored which essentially
 parallels that of the traditional rank--size hierarchy below rank 11.
 Above rank 11 this method may help explain the frequently noted
 concavity of the rank--size distribution at the upper levels.  The
 research suggests that the simple rank--size rule with the exponent equal
 to 1 is not merely a special case, but rather a theoretically justified norm
 against which deviant cases may be measured.  The spiral distribution model
 allows conceptualization of a new view of the urban rank--size hierarchy in
 which the three largest cities share functions in a Fibonacci hierarchy.
 9.  Sandra L. Arlinghaus, {\it An Atlas of Steiner Networks\/}, 1989. 

 A Steiner network is a tree of minimum total length joining a prescribed,
 finite, number of locations; often new locations are introduced into the 
 prescribed set to determine the minimum tree.  This Atlas explains the
 mathematical detail behind the Steiner construction for prescribed sets
 of n locations and displays the steps, visually, in a series of 
Figures.  The
 proof of the Steiner construction is by mathematical induction, and enough
 steps in the early part of the induction are displayed completely that the
 reader who is well--trained in Euclidean geometry, and familiar with 
 concepts from graph theory and elementary number theory, should be able to
 replicate the constructions for full as well as for degenerate Steiner 
 10.  Daniel A. Griffith, {\it Simulating $K=3$ Christaller Central Place
 Structures:  An Algorithm Using A Constant Elasticity of Substitution
 Consumption Function\/}, 1989.

 An algorithm is presented that uses BASICA or GWBASIC on IBM compatible
 machines.  This algorithm simulates Christaller $K=3$ central place
 structures, for a four--level hierarchy.  It is based upon earlier published
 work by the author.  A description of the spatial theory, mathematics, and
 sample output runs appears in the monograph.  A digital version is available
 from the author, free of charge, upon request; this request must be
 accompanied by a 5.5--inch formatted diskette.  This algorithm has been 
 developed for use in Social Science classroom laboratory situations,
 and is designed to (a) cultivate a deeper understanding of central place
 theory, (b) allow parameters of a central place system to be altered and
 then graphic and tabular results attributable to these changes viewed,
 without experiencing the tedium of massive calculations, and (c) help
 promote a better comprehension of the complex role distance
 plays in the space--economy.  The algorithm also should facilitate
 intensive numerical research on central place structures; it is expected
 that even the sample simulation results will reveal interesting insights 
 abstract central place theory.

 The background spatial theory concerns demand and competition in the
 space--economy; both linear and non--linear spatial demand functions are
 discussed.  The mathematics is concerned with (a) integration of non--linear
 spatial demand cones on a continuous demand surface, using a constant
 elasticity of substitution consumption function, (b) solving for roots of
 polynomials, (c) numerical approximations to integration and root
 extraction, and (d) multinomial discriminant function classification of 
 commodities into central place hierarchy levels.  Sample
 output is presented for contrived data sets, constructed from
 artificial and empirical information, with the wide range of all possible
 central place structures being generated.  These examples should facilitate
 implementation testing.  Students are able to vary single or multiple
 parameters of the problem, permitting a study of how certain changes 
 themselves within the context of a theoretical central place structure. 
 Hierarchical classification criteria may be changed, demand elasticities may
 or may not vary and can take on a wide range of non--negative values, the
 uniform transport cost may be set at any positive level, assorted fixed
 costs and variable costs may be introduced, again within a rich range of
 non--negative possibilities, and the number of commodities can be 
 altered.  Directions for algorithm execution are summarized.  An ASCII 
 version of the algorithm, written directly from GWBASIC, is included in 
an appendix; hence, it is free of typing errors.
 11.  Sandra L. Arlinghaus and John D. Nystuen,
      {\it Environmental Effects on Bus Durability\/}, 1990.  

  This monograph draws on the authors'
 previous publications on ``Climatic" and ``Terrain" effects on bus 
 Material on these two topics is selected, and reprinted, from
 three published papers that appeared in the {\sl Transportation
 Research Record\/} and in the {\sl Geographical Review\/}.  New
 material concerning ``congestion" effects is examined at the
 national level, to determine ``dense," ``intermediate," and
 ``sparse" classes of congestion, and at the local level of
 congestion in Ann Arbor (as suggestive of how one  might use local data).
 This material is drawn together in a single volume, along with a summary
 of the consequences of all three effects simultaneously, in order to suggest
 direction for more highly automated studies that should follow naturally
 with the release of the 1990 U. S. Census data.
 12.  Daniel A. Griffith, Editor.
 {\sl Spatial Statistics:  Past, Present, and Future\/},  1990. 

     Proceedings of a Symposium of the same name held at Syracuse
 University in Summer, 1989.  Content includes a Preface by Griffith and 
 following papers:  

 \line{{\sl Brian Ripley}, ``Gibbsian interaction models"; \hfil}
 \line{{\sl J. Keith Ord}, ``Statistical methods for point pattern data"; 
 \line{{\sl Luc Anselin}, ``What is special about spatial data"; \hfil}
 \line{{\sl Robert P. Haining}, ``Models in human geography: \hfil}
 \line{\qquad problems in specifying,
 estimating, and validating models for spatial data"; \hfil}
 \line{{\sl R. J. Martin},
 ``The role of spatial statistics in geographic modelling"; \hfil}
 \line{{\sl Daniel Wartenberg}, \hfil }
 \line{``Exploratory spatial analyses:  outliers,
 leverage points, and influence functions"; \hfil}
 \line{{\sl J. H. P. Paelinck},
 ``Some new estimators in spatial econometrics"; \hfil}
 \line{{\sl Daniel A. Griffith}, \hfil }
 \line{``A numerical simplification for estimating parameters of 
 spatial autoregressive models"; \hfil}
 \line{{\sl Kanti V. Mardia}
 ``Maximum likelihood estimation for spatial models"; \hfil}
 \line{{\sl Ashish Sen}, ``Distribution of spatial correlation 
 \line{{\sl Sylvia Richardson},  
 ``Some remarks on the testing of association between spatial 
 \line{{\sl Graham J. G. Upton}, ``Information from regional data";\hfil}
 \line{{\sl Patrick Doreian},
 ``Network autocorrelation models:  problems and prospects." \hfil}

 Each chapter is preceded by an ``Editor's Preface" and followed by a 
Discussion and, in some cases, by an author's Rejoinder to the Discussion.
 13.  Sandra L. Arlinghaus, Editor.
 {\sl Solstice---I\/},  1990. 
 14.  Sandra L. Arlinghaus, {\sl Essays on Mathematical Geography--III\/},
 15.  Sandra L. Arlinghaus, Editor, {\sl Solstice---II\/}, 1991.
 \centerline{\it Editor, Daniel A. Griffith}
 \centerline{\it Professor of Geography}
 \centerline{\it Syracuse University}
 \centerline{Founder as an IMaGe series:  Sandra L. Arlinghaus}
 \noindent 1.  {\sl Spatial Regression Analysis on the PC:
 Spatial Statistics Using Minitab}.  1989.  
 \centerline{\it Editor of MICMG Series, John D. Nystuen}
 \centerline{\it Professor of Geography and Urban Planning}
 \centerline{\it The University of Michigan}
 \noindent 1.  {\sl Reprint of the Papers of the Michigan InterUniversity
 Community of Mathematical Geographers.}  Editor, John D. Nystuen.
 Contents--original editor:  John D. Nystuen.
 \noindent 1.  Arthur Getis, ``Temporal land use pattern analysis with the
 use of nearest neighbor and quadrat methods."  July, 1963
 \noindent 2.  Marc Anderson, ``A working bibliography of mathematical
 geography."  September, 1963.
 \noindent 3.  William Bunge, ``Patterns of location."  February, 1964.
 \noindent 4.  Michael F. Dacey, ``Imperfections in the uniform plane."
 June, 1964.
 \noindent 5.  Robert S. Yuill, A simulation study of barrier effects
 in spatial diffusion problems."  April, 1965.
 \noindent 6.  William Warntz, ``A note on surfaces and paths and
 applications to geographical problems."  May, 1965.
 \noindent 7.  Stig Nordbeck, ``The law of allometric growth."
 June, 1965.
 \noindent 8.  Waldo R. Tobler, ``Numerical map generalization;"
 and Waldo R. Tobler, ``Notes on the analysis of geographical
 distributions."  January, 1966.
 \noindent 9.  Peter R. Gould, ``On mental maps."  September, 1966.
 \noindent 10.  John D. Nystuen, ``Effects of boundary shape and the
 concept of local convexity;"  Julian Perkal, ``On the length of
 empirical curves;" and Julian Perkal, ``An attempt at
 objective generalization."  December, 1966.
 \noindent 11. E. Casetti and R. K. Semple, ``A method for the
 stepwise separation of spatial terends."  April, 1968.
 \noindent 12.  W. Bunge, R. Guyot, A. Karlin, R. Martin, W. Pattison,
 W. Tobler, S. Toulmin, and W. Warntz, ``The philosophy of maps."
 June, 1968.
 \centerline{\bf Reprints of out-of-print textbooks.}
 \centerline{\bf Printer and obtainer of copyright permission:  Digicopy}
 \centerline{Inquire for cost of reproduction---include class size}
 1.  Allen K. Philbrick.  {\sl This Human World}.
 Publications of the Institute of Mathematical Geography have
 been reviewed in 
 \item{1.} {\sl The Professional Geographer\/} published
 by the Association of American Geographers;
 \item{2.}  {\sl The Urban Specialty Group Newsletter\/}
 of the Association of American Geographers;
 \item{3.}  {\sl Mathematical Reviews\/} published by the
 American Mathematical Society;
 \item{4.}  {\sl The American Mathematical Monthly\/} published
 by the Mathematical Association of America;
 \item{5.}  {\sl Zentralblatt\/}  Springer-Verlag, Berlin
 \item{6.}  {\sl Mathematics Magazine \/}, published by the Mathematical
 Association of America.