SOLSTICE: AN ELECTRONIC JOURNAL OF GEOGRAPHY AND MATHEMATICS. Volume III, Number 1. Summer, 1992. \hsize = 6.5 true in %THE FIRST 14 LINES ARE TYPESETTING CODE. \input fontmac %delete to download, except on Univ. Mich. (MTS) equipment. \setpointsize{12}{9}{8}%same as previous line; set font for 12 point type. \parskip=3pt \baselineskip=14 pt \mathsurround=1pt \headline = {\ifnum\pageno=1 \hfil \else {\ifodd\pageno\righthead \else\lefthead\fi}\fi} \def\righthead{\sl\hfil SOLSTICE } \def\lefthead{\sl Summer, 1992 \hfil} \def\ref{\noindent\hang} \font\big = cmbx17%this may cause problems in some installations--replace %if it does with a different font. \font\tn = cmr10 \font\nn = cmr9 %The code has been kept simple to facilitate reading as e-mail \centerline{\big SOLSTICE:} \vskip.5cm \centerline{\bf AN ELECTRONIC JOURNAL OF GEOGRAPHY AND MATHEMATICS} \vskip5cm \centerline{\bf SUMMER, 1992--11:14p.m., E.D.T., June 20} \vskip12cm \centerline{\bf Volume III, Number 1} \smallskip \centerline{\bf Institute of Mathematical Geography} \vskip.1cm \centerline{\bf Ann Arbor, Michigan} \vfill\eject \hrule \smallskip \centerline{\bf SOLSTICE} \line{Founding Editor--in--Chief: {\bf Sandra Lach Arlinghaus}. \hfil} \smallskip \centerline{\bf EDITORIAL BOARD} \smallskip \line{{\bf Geography} \hfil} \line{{\bf Michael Goodchild}, University of California, Santa Barbara. \hfil} \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).} \smallskip \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. \hfil} \smallskip \line{{\bf Business} \hfil} \line{{\bf Robert F. Austin, Ph.D.} \hfil} \line{President, Austin Communications Education Services \hfil} \smallskip \hrule \smallskip The purpose of {\sl Solstice\/} is to promote interaction between geography and mathematics. Articles in which elements of one discipline are used to shed light on the other are particularly sought. Also welcome, are original contributions that are purely geographical or purely mathematical. These may be prefaced (by editor or author) with commentary suggesting directions that might lead toward the desired interaction. Individuals wishing to submit articles, either short or full-- length, as well as contributions for regular features, should send them, in triplicate, directly to the Editor--in--Chief. Contributed articles will be refereed by geographers and/or mathematicians. Invited articles will be screened by suitable members of the editorial board. IMaGe is open to having authors suggest, and furnish material for, new regular features. \vskip1in \noindent \vskip.2cm \noindent {\bf Send all correspondence to:} \vskip.1cm \centerline{\bf Institute of Mathematical Geography} \centerline{\bf 2790 Briarcliff} \centerline{\bf Ann Arbor, MI 48105-1429} \vskip.1cm \centerline{\bf (313) 761-1231} \centerline{\bf IMaGe@UMICHUM} \vfill\eject This document is produced using the typesetting program, {\TeX}, of Donald Knuth and the American Mathematical Society. Notation in the electronic file is in accordance with that of Knuth's {\sl The {\TeX}book}. The program is downloaded for hard copy for on The University of Michigan's Xerox 9700 laser-- printing Xerox machine, using IMaGe's commercial account with that University. Unless otherwise noted, all regular ``features" are written by the Editor--in--Chief. \smallskip {\nn Upon final acceptance, authors will work with IMaGe to get manuscripts into a format well--suited to the requirements of {\sl Solstice\/}. Typically, this would mean that authors would submit a clean ASCII file of the manuscript, as well as hard copy, figures, and so forth (in camera--ready form). Depending on the nature of the document and on the changing technology used to produce {\sl Solstice\/}, there may be other requirements as well. Currently, the text is typeset using {\TeX}; in that way, mathematical formul{\ae} can be transmitted as ASCII files and downloaded faithfully and printed out. The reader inexperienced in the use of {\TeX} should note that this is not a ``what--you--see--is--what--you--get" display; however, we hope that such readers find {\TeX} easier to learn after exposure to {\sl Solstice\/}'s e-files written using {\TeX}!} {\nn Copyright will be taken out in the name of the Institute of Mathematical Geography, and authors are required to transfer copyright to IMaGe as a condition of publication. There are no page charges; authors will be given permission to make reprints from the electronic file, or to have IMaGe make a single master reprint for a nominal fee dependent on manuscript length. Hard copy of {\sl Solstice\/} is available at a cost of \$15.95 per year (plus shipping and handling; hard copy is issued once yearly, in the Monograph series of the Institute of Mathematical Geography. Order directly from IMaGe. It is the desire of IMaGe to offer electronic copies to interested parties for free. Whether or not it will be feasible to continue distributing complimentary electronic files remains to be seen. Presently {\sl Solstice\/} is funded by IMaGe and by a generous donation of computer time from a member of the Editorial Board. Thank you for participating in this project focusing on environmentally-sensitive publishing.} \vskip.5cm Copyright, June, 1992, Institute of Mathematical Geography. All rights reserved. \vskip1cm {\bf ISBN: } {\bf ISSN: 1059-5325} \vfill\eject \centerline{\bf SUMMARY OF CONTENT} \smallskip \noindent{\bf 1. ARTICLES.} \smallskip\noindent {\bf Harry L. Stern}. \smallskip\noindent {\bf Computing Areas of Regions With Discretely Defined Boundaries}. \smallskip\noindent 1. Introduction 2. General Formulation 3. The Plane 4. The Sphere 5. Numerical Example and Remarks. Appendix--Fortran Program. \smallskip \noindent{\bf 2. NOTE } \smallskip\noindent {\bf Sandra L. Arlinghaus, John D. Nystuen, Michael J. Woldenberg}. \smallskip\noindent {\bf The Quadratic World of Kinematic Waves} \smallskip \noindent{\bf 3. SOFTWARE REVIEW} \smallskip 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}. \smallskip Reviewed by {\bf Yung-Jaan Lee}, University of Michigan. \smallskip \noindent{\bf 4. PRESS CLIPPINGS} \smallskip \noindent{\bf 5. INDEX to Volumes I (1990) and II (1991) of {\sl Solstice}.} \smallskip \noindent{\bf 6. OTHER PUBLICATIONS OF THE INSTITUTE OF MATHEMATICAL GEOGRAPHY } \vfill\eject \centerline{\bf Computing Areas of Regions With Discretely Defined Boundaries} \vskip.5cm \centerline{\bf Harry L. Stern} \centerline{\bf Polar Science Center} \centerline{\bf 1013 N.E. 40th Street} \centerline{\bf Seattle, WA 98195} \vskip.5cm \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. \vskip.5cm \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 $S$: $$ ({\bf \nabla} \times \hbox{\bf F}){\bf \cdot }{\bf \hat{\hbox{n}}} = 1. \eqno(2) $$ With $\hbox{\bf F} $ determined (though not uniquely) by equation (2), the left hand side of (1) simply reduces to the area of $S$, giving $$ A =\oint_C \hbox{\bf F}\cdot {{d\hbox{\bf R}}\over {dt}} \, dt. \eqno(3) $$ 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 $$ A =\sum_{k=1}^N \int_0^{L_k} \hbox{\bf F}(s)\cdot {{d\hbox{\bf R}_k}\over {ds}} \, ds \eqno(4) $$ 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, \vskip.2cm \noindent(1) Find the unit outward normal on the surface, ${{\bf \hat{\hbox{\bf n}}}} $; \vskip.1cm \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 $; \vskip.1cm \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 } $; \vskip.1cm \noindent(4) Form the integrand in equation (4) and do the integration; \vskip.1cm \noindent(5) Sum the contributions in (4) to get the area of the region. \vskip.2cm \noindent Some specific cases follow. \vskip.5cm \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 $$ {\partial {F_2} \over \partial x} - {\partial {F_1} \over \partial y} = 1. \eqno(5) $$ 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 $$ I_k = \int_0^{L_k} {1\over 2} \left( x {{dy} \over {ds}} - y {{dx} \over {ds}}\right)\, ds. \eqno(6) $$ 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 $$ \eqalign{ I_k &= {1 \over 2}\int_0^{L_k} \left\{\left(x_k+{{s\,\,\Delta x}\over{L_k}}\right) \left({{\Delta y}\over {L_k}}\right) -\left(y_k + {{s\,\,\Delta y}\over{L_k}}\right) \left({{\Delta x}\over {L_k}}\right)\right\}\,ds \cr &= {1\over 2}\int_0^{L_k}\left\{{{x_k\Delta y}\over {L_k}}- {{y_k\Delta x}\over {L_k}}\right\}\, ds \cr &= {1 \over 2}(x_k\Delta y - y_k \Delta x) \cr &= {1\over 2}(x_ky_{k+1}-y_kx_{k+1}). \cr }\eqno(8) $$ It follows that the area of the polygon in the plane whose vertexes are the points $ ( x_k , y_k ) $ is $$ A={1\over 2}\sum_{k=1}^N(x_ky_{k+1}-y_kx_{k+1}) \eqno(9) $$ 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 result. \vskip.5cm \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 $$ x=\hbox{cos}\,\phi\,\,\hbox{cos}\,\theta \quad y=\hbox{cos}\,\phi\,\,\hbox{sin}\,\theta \quad z=\hbox{sin}\,\phi \eqno(10) $$ 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 $$ {\bf \hat{\hbox{\bf u}}}_{\theta} = (-\hbox{sin}\theta ){\bf \hat{\hbox{\bf \i}}} +( \hbox{cos}\theta ){\bf \hat{\hbox{\bf \j}}} = {{-y}\over{\sqrt{1-z^2}}}{\bf \hat{\hbox{\bf \i}}} +{{ x}\over{\sqrt{1-z^2}}}{\bf \hat{\hbox{\bf \j}}} \eqno(11a) $$ $$ \eqalign{ {\bf \hat{\hbox{\bf u}}}_{\phi} &=(\hbox{sin}\phi\,\hbox{cos}\theta ){\bf \hat{\hbox{\bf \i}}} +(\hbox{sin}\phi\,\hbox{sin}\theta ){\bf \hat{\hbox{\bf \j}}} +(-\hbox{cos}\phi){\bf \hat{\hbox{\bf k}}}\cr &={{xz}\over{\sqrt{1-z^2}}}\,{\bf \hat{\hbox{\bf \i}}} +{{yz}\over{\sqrt{1-z^2}}}\,{\bf \hat{\hbox{\bf \j}}} -{\sqrt{1-z^2}}\,{\bf \hat{\hbox{\bf k}}}\cr } \eqno(11b) $$ $$ {\bf \hat{\hbox{\bf u}}}_r =(\hbox{cos}\phi\,\hbox{cos}\theta ){\bf \hat{\hbox{\bf \i}}} +(\hbox{cos}\phi\,\hbox{sin}\theta ){\bf \hat{\hbox{\bf \j}}} +(\hbox{sin}\phi ){\bf \hat{\hbox{\bf k}}} =x{\bf \hat{\hbox{\bf \i}}}+y{\bf \hat{\hbox{\bf \j}}} +z{\bf \hat{\hbox{\bf k}}}. \eqno(11c) $$ 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] $$ (\nabla \times \hbox{\bf F})\cdot {\bf \hat{\hbox{\bf u}}}_r = {1\over {\hbox{cos}\,\phi}} \left[{{\partial} \over {\partial \theta}}(F_{\phi}) - {{\partial} \over {\partial \phi}}(\hbox{cos}\,\phi\,\,F_{\theta}) \right] = 1. \eqno(12) $$ This is most naturally satisfied if we take $$ {{\partial} \over {\partial \phi}}(\hbox{cos}\,\phi\,\,F_{\theta}) =-\hbox{cos}\,\phi \qquad {{\partial} \over {\partial \theta}}(F_{\phi})=0 \eqno(13) $$ or $$ F_{\theta}=-\hbox{tan}\,\phi + {{g({\theta})} \over {\hbox{cos}\,\phi}} \qquad F_{\phi}=h(\phi) \eqno(14) $$ 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 $$ \hbox{\bf R}(\alpha)=A(\alpha)\hbox{\bf P}_1+B(\alpha)\hbox{\bf P}_2 \eqno(15) $$ where $ A ( \alpha ) $ and $ B ( \alpha ) $ are determined from the following two conditions: \vskip.2cm \noindent (1) ${\hbox{\bf R}} ( \alpha ) $ lies on the unit sphere: $ {\hbox{\bf R}} {\bf \cdot }\,{\hbox{\bf R}} = 1 $; \vskip.1cm \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 $$ A^2+B^2+2AB \hbox{cos} \Delta = 1 \qquad A+B\hbox{cos}\Delta=\hbox{cos}\,\alpha \eqno(16) $$ respectively. Solving for $A$ and $B$, we find $$ \hbox{\bf R}(\alpha)= {{\hbox{sin}\,(\Delta - \alpha )}\over {\hbox{sin}\,\Delta}} \hbox{\bf P}_1 + {{\hbox{sin}\,(\alpha)} \over {\hbox{sin}\,\Delta}} \hbox{\bf P}_2. \eqno(17) $$ 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 $$ {{d\hbox{\bf R}}\over{d\alpha }}=G(\alpha){\bf \hat{\hbox{\bf u}}}_{\theta} +H(\alpha){\bf \hat{\hbox{\bf u}}}_{\phi} \eqno(18) $$ 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 $$ \eqalign{ G(\alpha)&=\hbox{\bf R}'{\bf \cdot}{\bf \hat{\hbox{\bf u}}}_{\theta} \cr &=(x'{\bf \hat{\hbox{\bf \i}}}+y'{\bf \hat{\hbox{\bf \j}}} +z'{\bf \hat{\hbox{\bf k}}}) {\bf \cdot} \left[{{-y}\over{\sqrt{1-z^2}}}\,{\bf \hat{\hbox{\bf \i}}} +{{x} \over{\sqrt{1-z^2}}}\,{\bf \hat{\hbox{\bf \j}}}\right]\cr &={{xy'-yx'}\over{\sqrt{1-z^2}}} \cr }\eqno(19) $$ and $$ \eqalign{ H(\alpha )&=\hbox{\bf R}'{\bf \cdot}{\bf \hat{\hbox{\bf u}}}_{\phi} \cr &=(x'{\bf \hat{\hbox{\bf \i}}}+y'{\bf \hat{\hbox{\bf \j}}} +z'{\bf \hat{\hbox{\bf k}}}) {\bf \cdot} \left[{{xz}\over{\sqrt{1-z^2}}}\,{\bf \hat{\hbox{\bf \i}}} +{{yz}\over{\sqrt{1-z^2}}}\,{\bf \hat{\hbox{\bf \j}}} -{\sqrt{1-z^2}}\,{\bf \hat{\hbox{\bf k}}}\right] \cr &={{x'xz}\over{\sqrt{1-z^2}}} +{{y'yz}\over{\sqrt{1-z^2}}} -z'{\sqrt{1-z^2}}\cr &={{(x'x+y'y+z'z)z-z'}\over{\sqrt{1-z^2}}}\cr &={{-z'}\over{\sqrt{1-z^2}}}\cr }\eqno(20) $$ 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 $$ \hbox{\bf F}=\left[{{-z}\over {\sqrt{1-z^2}}} +{{g(\theta )}\over {\sqrt{1-z^2}}}\right] {\bf \hat{\hbox{\bf u}}}_{\theta} +h(\phi ){\bf \hat{\hbox{\bf u}}}_{\phi}. \eqno(21) $$ Using the components of $ d{\hbox{\bf R}} / d \alpha $ from equations (19) and (20), we have $$ \hbox{\bf F}{\bf \cdot}{{d\hbox{\bf R}}\over{d \alpha }} = \left[{{xy'-yx'}\over{\sqrt{1-z^2}}}\right] \left[{{-z}\over {\sqrt{1-z^2}}}+{{g(\theta )}\over {\sqrt{1-z^2}}}\right] -{{z'h(\phi )}\over {\sqrt{1-z^2}}} \eqno(22) $$ 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 $$ \hbox{\bf F}{\bf \cdot}{{d\hbox{\bf R}}\over{d \alpha }} = {{(xy'-yx')(g(\theta )-z)}\over{1-z^2}} \eqno(23) $$ 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) as $$ I_1=\int_0^{\Delta } {{xy'-yx'}\over{1+z}}\,d\alpha . \eqno(24) $$ 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, namely $$ x={{\hbox{sin}\,(\Delta - \alpha )}\over{\hbox{sin}\,\Delta}}x_1 +{{\hbox{sin}\,(\alpha )}\over{\hbox{sin}\,\Delta}}x_2 \eqno(25a) $$ $$ x'={-{\hbox{cos}\,(\Delta - \alpha )}\over{\hbox{sin}\,\Delta}}x_1 +{{\hbox{cos}\,(\alpha )}\over{\hbox{sin}\,\Delta}}x_2 \eqno(25b) $$ and similar equations for $ y, y' $ and $ z, z' $. Substituting these expressions into equation (24) and using standard trigonometric identities leads to $$ I_1=(x_1y_2-y_1x_2)\int_0^{\Delta} {{d\alpha }\over {\hbox{sin}\,\Delta +z_1\hbox{sin}\,(\Delta -\alpha ) +z_2\hbox{sin}\,\alpha}}. \eqno(26) $$ 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 $$ A=\sum_{k=1}^N(x_ky_{k+1}-y_kx_{k+1})J_k \eqno(27) $$ where the terms $ J_k $ are the integrals $$ J_k=\int_0^{{\Delta}_k} {{d\alpha }\over {\hbox{sin}\,({\Delta}_k) +z_k\hbox{sin}\,({\Delta}_k -\alpha ) +z_{k+1}\hbox{sin}\,\alpha}}. \eqno(28) $$ 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 $$ J_k=\int_1^{e^{i \Delta }} {{2\,\,dw}\over{aw^2+2bw+c}} \eqno(29) $$ where $$ a=z_{k+1}-z_ke^{-i \Delta} \quad b=i\,\hbox{sin}\,\Delta \quad c=z_ke^{i \Delta }-z_{k+1}. \eqno(30) $$ 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 $$ D=z_k^2+z_{k+1}^2-2z_kz_{k+1}\,\hbox{cos}\,\Delta - \hbox{sin}^2\Delta . \eqno(31) $$ The three cases are [Marsden, Appendix A] $$ J_k=\cases{ {{1}\over {\sqrt{D}}}\,\hbox{ln}\, \left[{{aw+b-\sqrt{D}}\over{aw+b+\sqrt{D}}}\right] &($D>0$)\cr {{2}\over {\sqrt{-D}}}\,\hbox{arctan}\, \left[{{aw+b}\over{\sqrt{-D}}}\right] &($D<0$)\cr {{-2}\over{aw+b}} &($D=0$)\cr }\eqno(32) $$ 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 $$ Q=z_k+z_{k+1}+1+\hbox{cos}\,\Delta \eqno(33) $$ in terms of which the expressions for $ J_k $ become $$ J_k=\cases{ {{1}\over {\sqrt{D}}}\,\hbox{ln}\, \left[{{Q+\sqrt{D}}\over{Q-\sqrt{D}}}\right] &($D>0$)\cr {{2}\over {\sqrt{-D}}}\,\hbox{arctan}\, \left[{{\sqrt{-D}}\over{Q}}\right] &($D<0$)\cr {{Q}\over{(1+z_k)(1+z_{k+1})(1+\hbox{cos}\,\Delta )}} &($D=0$)\cr }\eqno(34) $$ This completes the determination of the terms in the area formula (27). We will now summarize the steps and put them in an algorithmic format. \vskip.5cm Problem: \vskip.2cm \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. \vskip.5cm Solution: \vskip.2cm \noindent(1) Set the running sum to $0$ and set $k$ to $1$. \vskip.1cm \noindent(2) Compute $ \hbox{cos}\, \Delta$ $=$ ${\hbox{\bf P}}_k {\bf \cdot }{\hbox{\bf P}}_{k+1}$ 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. \vskip.1cm \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 $. \vskip.1cm \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} ) $. \vskip.1cm \noindent(5) Compute the integral contribution $ J_k $ in the area formula (27), using the appropriate form of equation (34). \vskip.1cm \noindent(6) Compute the first factor in the area formula (27), $ x_ky_{k+1} - y_k x_{k+1} $ or $ \hbox{cos} \phi_k $ $\hbox{cos} \phi_{k+1}$ $\hbox{sin}$ $( \theta_{k+1} - \theta_k ). $ \vskip.1cm \noindent(7) Multiply together the results of steps 5 and 6 to get the $k$-th term in the summation of (27), and add this to the running sum. \vskip.1cm \noindent(8) If $k$ is less than $N$ then increment $k$ and go to step 2. \vskip.2cm A computer program that implements the above algorithm is given in the appendix. \vskip.5cm \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] $$ x={{180}\over{\pi }}\theta \qquad y={{180}\over{\pi }}\hbox{ln}\,\left[\hbox{arctan}\,\left({{\phi}\over{2}} +{{\pi}\over{4}}\right)\right] \eqno(35) $$ 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. \vfill\eject \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. \vfill\eject {\tt \noindent{\bf Appendix -- Fortran Program} \vskip.5cm \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\hrulefill} \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\hfil} \line{c\qquad Variables.\hfil} \line{c\hfil} \line{\phantom{c}\qquad integer n, k \hfil} \line{\phantom{c}\qquad real sum, first, integral, cosDelta, D, Q, R \hfil} \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\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{c\hfil} \line{\phantom{c}\qquad read(5,*) n, filename \hfil} \line{c\hfil} \line{c\qquad Read the coordinates. Longitude is first. Both in degrees.\hfil} \line{c\hfil} \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\hfil} \line{c\qquad Convert to radians.\hfil} \line{c\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{c\hfil} \line{\phantom{c}\qquad phi(n+1) = phi(1) \hfil} \line{\phantom{c}\qquad theta(n+1) = theta(1) \hfil} \line{c\hfil} \line{c\qquad Initialize for the summation. \hfil} \line{c\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{c\hfil} \line{\phantom{c}\qquad do 20 k=1,n \hfil} \line{c\hfil} \line{c\qquad \quad Previous "k+1" values become new "k" values.\hfil} \line{c\hfil} \line{\phantom{c}\qquad \quad cosPhiK = cosPhiK1 \hfil} \line{\phantom{c}\qquad \quad sinPhiK = sinPhiK1 \hfil} \line{c\hfil} \line{c\qquad \quad Get new "k+1" values.\hfil} \line{c\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\hfil} \line{c\qquad \quad Compute first factor in k-th term of summation.\hfil} \line{c\hfil} \line{\phantom{c}\qquad \quad first = cosPhiK * cosPhiK1 * sin(theta(k+1)-theta(k))\hfil} \line{c\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{c\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 +2.0*sinPhiK*sinPhiK1)\hfil} \line{\phantom{c}\qquad \quad Q = sinPhiK + sinPhiK1 + 1.0 + cosDelta\hfil} \line{c\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) ) / R\hfil} \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) *(1.0+sinPhiK1)*(1.0+cosDelta))\hfil} \line{\phantom{c}\qquad \quad endif\hfil} \line{c\hfil} \line{c\qquad \quad Accumulate sum and go on to next segment.\hfil} \line{c\hfil} \line{\phantom{c}\qquad \quad sum = sum + first * integral\hfil} \line{c\hfil} \line{\phantom{c}$\,$ 20 continue\hfil} \line{c\hfil} \line{c\qquad Write results and stop.\hfil} \line{c\hfil} \line{\phantom{c}\qquad write(6,90) sum, sum/(4.0*pi), sum*Rearth*Rearth\hfil} \line{c\hfil} \line{\phantom{c}\qquad stop\hfil} \line{\phantom{c}$\,$90 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) = ', e14.6)\hfil} \line{\phantom{c}\qquad end\hfil} } \vfill\eject \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. \vfill\eject \centerline{\bf The Quadratic World of Kinematic Waves} \vskip.2cm \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, 1968]. 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 1992]. \vskip.5cm $^{\star }$ Author Woldenberg wishes to acknowledge input from M. Sonis regarding the analysis of kinematic waves---1981. \vfill\eject \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] Proc.\/}, 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. \vfill\eject \noindent{\bf REVIEW} of RangeMapper$^{\hbox{TM}}$ (version 1.4b). \vskip.1cm A utility for biological species range mapping, and similar mapping tasks in other fields. Price: \$350 \vskip.2cm Program and manual written by {\bf Kenelm W. Philip}. Tundra Vole Software 1590 North Becker Ridge Road Fairbanks, Alaska 99709 (907) 479-2689 \vskip.1cm Reviewed by {\bf Yung-Jaan Lee}, Ph.D. Candidate in Urban, Technological, and Environmental Planning, The University of Michigan, Ann Arbor, MI 48109. \vskip.5cm \noindent From the author's flyer: \vskip.1cm ``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." \vskip.5cm 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 pixels. 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: \vskip.2cm \qquad a {\bf Close} selection under the File menu; \vskip.2cm \qquad a {\bf Window} sub-menu in the pull-down menu; \vskip.2cm \qquad the filename displayed at the top of the screen (different from the title of the map); \vskip.2cm \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. \vskip.5cm {\bf Note}: Canvas is a trademark of Deneba Systems; \vskip.1cm SuperPaint is a trademark of Aldus Corporation; \vskip.1cm NISUS is a trademark of Paragon Concepts, Inc.; \vskip.1cm Apple and LaserWriter are registered trademarks of Apple Computer, Inc.; \vskip.1cm Macintosh is a trademark licensed to Apple Computer, Inc.; MacDraw is a trademark of Apple Computer, Inc. \vfill\eject \noindent{\bf FEATURES} \vskip.5cm \noindent{\bf Press Clippings} \vskip.2cm \centerline{\bf FROM SCIENCE, AAAS} \vskip.2cm {\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: VISBMS@UBVMS \vskip.2cm \centerline{\bf FROM SCIENCE NEWS} \vskip.2cm {\bf Math for all seasons} \vskip.1cm by Ivars Peterson \vskip.1cm January 25, 1992, Vol. 141, No. 4. Page 61. Reprinted with permission of {\sl Science News\/}. \vskip.2cm 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. \vskip.2cm \centerline{\bf LETTER AND RESPONSE IN SCIENCE NEWS} \vskip.2cm One from AAAS in reply to Peterson; one from IMaGe in reply to AAAS, during period from January through May, 1992. \vskip.2cm \centerline{\bf AAG NEWSLETTER} \vskip.2cm Volume 27, Number 6, June 1992. ``Online Geographical Journals," page 10. \vfill\eject \noindent{\bf INDEX to Volumes I (1990) and II (1991) of {\sl Solstice}.} \vskip.5cm \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 diffusion. 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. \smallskip \smallskip 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 ones. \smallskip \noindent 3. FEATURES \smallskip \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. \smallskip \noindent 4. SAMPLE OF HOW TO DOWNLOAD THE ELECTRONIC FILE \vfill\eject \noindent{\bf Volume I, Number 2, Winter, 1990} \smallskip \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 observations. \noindent 2. ARTICLES Sandra Lach Arlinghaus, {\sl Scale and Dimension: Their Logical Harmony\/}. Linkage between scale and dimension is made using the Fallacy of Division and the Fallacy of Composition in a fractal setting. \smallskip \smallskip 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. \smallskip \smallskip 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. \smallskip \smallskip 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 pixels. \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. \smallskip \noindent {\bf Volume II, Number 1, Summer, 1991} \smallskip \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.). \smallskip \smallskip \noindent 2. FEATURES \item{i} Construction Zone --- The logistic curve. \item{ii.} Educational feature --- Lectures on ``Spatial Theory" \smallskip \noindent {\bf Volume II, Number 2, Winter, 1991} \smallskip \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 Community. \noindent 3. FEATURES Press clipping; Word Search Puzzle; Software Briefs. \vfill\eject \noindent{\bf OTHER publications of } \smallskip \centerline{\it INSTITUTE OF MATHEMATICAL GEOGRAPHY (IMaGe)} \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} \vskip.2cm \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.} \smallskip 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\/}, 1986. \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\/}, 1986. \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\/}, 1987. \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 litt\'erature. 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. \vskip.1cm 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 unpublished 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; post---1979, 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. \vskip.2cm 8. James W. Fonseca, {\it The Urban Rank--size Hierarchy: A Mathematical Interpretation\/}, 1989. \vskip.1cm 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. \vskip.2cm 9. Sandra L. Arlinghaus, {\it An Atlas of Steiner Networks\/}, 1989. \vskip.1cm 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 trees. \vskip.2cm 10. Daniel A. Griffith, {\it Simulating $K=3$ Christaller Central Place Structures: An Algorithm Using A Constant Elasticity of Substitution Consumption Function\/}, 1989. \vskip.1cm 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 into 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 manifest 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. \vskip.2cm 11. Sandra L. Arlinghaus and John D. Nystuen, {\it Environmental Effects on Bus Durability\/}, 1990. \vskip.1cm This monograph draws on the authors' previous publications on ``Climatic" and ``Terrain" effects on bus durability. 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. \vskip.2cm 12. Daniel A. Griffith, Editor. {\sl Spatial Statistics: Past, Present, and Future\/}, 1990. \vskip.1cm Proceedings of a Symposium of the same name held at Syracuse University in Summer, 1989. Content includes a Preface by Griffith and the following papers: \vskip.1cm \line{{\sl Brian Ripley}, ``Gibbsian interaction models"; \hfil} \line{{\sl J. Keith Ord}, ``Statistical methods for point pattern data"; \hfil} \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 statistics";\hfil} \line{{\sl Sylvia Richardson}, ``Some remarks on the testing of association between spatial processes";\hfil} \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. \vskip.2cm 13. Sandra L. Arlinghaus, Editor. {\sl Solstice---I\/}, 1990. \vskip.2cm 14. Sandra L. Arlinghaus, {\sl Essays on Mathematical Geography--III\/}, 1991. \vskip.2cm 15. Sandra L. Arlinghaus, Editor, {\sl Solstice---II\/}, 1991. \vfill\eject \centerline{\it DISCUSSION PAPERS--ORIGINAL} \centerline{\it Editor, Daniel A. Griffith} \centerline{\it Professor of Geography} \centerline{\it Syracuse University} \centerline{Founder as an IMaGe series: Sandra L. Arlinghaus} \smallskip \noindent 1. {\sl Spatial Regression Analysis on the PC: Spatial Statistics Using Minitab}. 1989. \vskip.5cm \centerline{\it DISCUSSION PAPERS--REPRINTS} \centerline{\it Editor of MICMG Series, John D. Nystuen} \centerline{\it Professor of Geography and Urban Planning} \centerline{\it The University of Michigan} \smallskip \noindent 1. {\sl Reprint of the Papers of the Michigan InterUniversity Community of Mathematical Geographers.} Editor, John D. Nystuen. \smallskip Contents--original editor: John D. Nystuen. \smallskip \noindent 1. Arthur Getis, ``Temporal land use pattern analysis with the use of nearest neighbor and quadrat methods." July, 1963 \smallskip \noindent 2. Marc Anderson, ``A working bibliography of mathematical geography." September, 1963. \smallskip \noindent 3. William Bunge, ``Patterns of location." February, 1964. \smallskip \noindent 4. Michael F. Dacey, ``Imperfections in the uniform plane." June, 1964. \smallskip \noindent 5. Robert S. Yuill, A simulation study of barrier effects in spatial diffusion problems." April, 1965. \smallskip \noindent 6. William Warntz, ``A note on surfaces and paths and applications to geographical problems." May, 1965. \smallskip \noindent 7. Stig Nordbeck, ``The law of allometric growth." June, 1965. \smallskip \noindent 8. Waldo R. Tobler, ``Numerical map generalization;" and Waldo R. Tobler, ``Notes on the analysis of geographical distributions." January, 1966. \smallskip \noindent 9. Peter R. Gould, ``On mental maps." September, 1966. \smallskip \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. \smallskip \noindent 11. E. Casetti and R. K. Semple, ``A method for the stepwise separation of spatial terends." April, 1968. \smallskip \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. \vfill\eject \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} \smallskip 1. Allen K. Philbrick. {\sl This Human World}. \smallskip \vskip.5cm 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. \bye