Answer to "Points problem"

*To*: mathgroup at yoda.ncsa.uiuc.edu*Subject*: Answer to "Points problem"*From*: CAMERON at midd.cc.middlebury.edu*Date*: Thu, 4 Oct 90 00:49 EDT

[[ Are members of MathGroup "MathGroupies" ? ]] [[ Should I say "elements of the MathGroup" ? ]] [[ Is it abelian? ]] maxwebb at vitus.cse.ogi.edu (Max G. Webb) asks why some code like Show[Graphics3D[ pointlist, shortlinelist ]]; displays the points in pointlist, but Show[Graphics3D[ pointlist, longlinelist ]]; produces a display in which the points are missing. He is falling afoul of a subtle feature of Mathematica graphics: automatic scaling. Consider the function boundingbox[x_] := Block[ {y=N[x]}, positions = Position[y,{_?NumberQ,_?NumberQ,_?NumberQ}]; coordinates = Apply[ (y[[##]])&, positions, {1} ]; {Min[#],Max[#]}& /@ Transpose[coordinates] ] This function takes a Graphics3D object (or list of 3D graphics primitives) and returns a list {{xmin,xmax},{ymin,ymax},{zmin,zmax}} indicating the ranges of x, y, and z coordinates in the object. The original posting's values for pointlist, shortlinelist, and longlinelist are reproduced for reference at the end of this note. With these values, we get boundingbox[pointlist] {{-1.367512718901739, 1.3608552544435}, {0.00111807892934969, 0.002607945574256309}, {0.001639740038943013, 0.003728634284214638}} boundingbox[shortlinelist] {{-0.1820927732, 0.}, {0., 0.0004133633611}, {0., 0.0008263234388}} boundingbox[longlinelist] {{-0.1854201093010261, 0.}, {0., 0.0005506040866933036}, {0., 0.001102215067405295}} Note that the smallest y coordinate among the points is greater than the largest y coordinate among the lines. Similarly for the z coordinates. The x range for the points encloses the x range for the lines, but is about fourteen times as big. Whenever Mathematica displays Graphics or Graphics3D objects, if PlotRange->Automatic is in effect (as it is by default if no explicit PlotRange is given) then Mathematica goes through a complicated (and, as far as I know, completely undocumented) sequence of calculations to decide what range of values to display along each coordinate axis. This computation includes, essentially, an analysis of the mean and variance of the coordinates in the graphics list, with graphics features whose coordinates are "outliers" being rejected. This is why, if you ask Mathematica to do Plot[ Tan[x], {x,0,2Pi} ] you get a reasonable plot, chopped off at reasonable y-coordinates. The points "near" +/- infinity are chopped off because they are too far from the mean y-coordinate (most points on that graph have y-coordinates near 0; only points with x-coordinates very near Pi/2 or 3Pi/2 have y-coordinates far from 0). Although I don't know the details of how Mathematica decides what to keep and what to reject, this example suggests that it gives more weight to Line objects than to Point objects, because adding a couple of Line's to the graphic radically changes the default scale. At any rate, if you do the following: Show[Graphics3D[{pointlist,shortlinelist}], PlotRange->{{-1.4,1.4},{0.,.003},{0.,.004}}, BoxRatios->{1,1,1} ]; and Show[Graphics3D[{pointlist,longlinelist}], PlotRange->{{-1.4,1.4},{0.,.003},{0.,.004}}, BoxRatios->{1,1,1} ]; then you override the automatic scaling, and you will get pictures that I believe are much closer to what the original poster wanted and expected to get. --Cameron Smith Mathematica programming consultant CAMERON at MIDD.BITNET --or-- cameron at midd.cc.middlebury.edu The original posting had pointlist = {PointSize[0.05], GrayLevel[0.5], Point[{-1.367511195263215, 0.001195340754223037, 0.003139555787092553}], Point[{-1.367512718901739, 0.001195342899414338, 0.001639740038943013}], Point[{1.36085379675106, 0.002607945574256309, 0.00372862923999308}], Point[{1.3608552544435, 0.00111807892934969, 0.003728634284214638}] }; and shortlinelist = { Line[{{0., 0., 0.}, {-0.003327336062, 0.0004133633611, 0.0008263234388}}], Line[{{-0.003327336062, 0.0004133633611, 0.0008263234388}, {-0.1820927732, 0.0001372407256, 0.0002758916286}}] }; and longlinelist = { Line[{{0., 0., 0.}, {-0.003327336062, 0.0004133633611, 0.0008263234388}}], Line[{{-0.003327336062, 0.0004133633611, 0.0008263234388}, {-0.1820927732, 0.0001372407256, 0.0002758916286}}], Line[{{-0.1854201093010261, 0.0005506040866933036, 0.001102215067405295}, {0., 0., 0.}}], Line[{{-0.1854201093010261, 0.0005506040866933036, 0.001102215067405295}, {-0.1820927751, 0.0001345637681, 0.0002715492562}}] };