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## 8.27 graph

This package implements two-dimensional linear and logarithmic graphs, including automatic scale and tick selection (with the ability to override manually). A graph is a guide (that can be drawn with the draw command, with an optional legend) constructed with one of the following routines:

• guide graph(picture pic=currentpicture, real f(real), real a, real b,
int n=ngraph, real T(real)=identity,
interpolate join=operator --);
guide[] graph(picture pic=currentpicture, real f(real), real a, real b,
int n=ngraph, real T(real)=identity, bool3 cond(real),
interpolate join=operator --);

Returns a graph using the scaling information for picture pic (see automatic scaling) of the function f on the interval [T(a),T(b)], sampling at n points evenly spaced in [a,b], optionally restricted by the bool3 function cond on [a,b]. If cond is:
• true, the point is added to the existing guide;
• default, the point is added to a new guide;
• false, the point is omitted and a new guide is begun.

The points are connected using the interpolation specified by join:

• operator -- (linear interpolation; the abbreviation Straight is also accepted);
• operator .. (piecewise Bezier cubic spline interpolation; the abbreviation Spline is also accepted);
• Hermite (standard cubic spline interpolation using boundary condition notaknot, natural, periodic, clamped(real slopea, real slopeb)), or monotonic. The abbreviation Hermite is equivalent to Hermite(notaknot) for nonperiodic data and Hermite(periodic) for periodic data).
• guide graph(picture pic=currentpicture, real x(real), real y(real),
real a, real b, int n=ngraph, real T(real)=identity,
interpolate join=operator --);
guide[] graph(picture pic=currentpicture, real x(real), real y(real),
real a, real b, int n=ngraph, real T(real)=identity,
bool3 cond(real), interpolate join=operator --);

Returns a graph using the scaling information for picture pic of the parametrized function (x(t),y(t)) for t in the interval [T(a),T(b)], sampling at n points evenly spaced in [a,b], optionally restricted by the bool3 function cond on [a,b], using the given interpolation type.
• guide graph(picture pic=currentpicture, pair z(real), real a, real b,
int n=ngraph, real T(real)=identity,
interpolate join=operator --);
guide[] graph(picture pic=currentpicture, pair z(real), real a, real b,
int n=ngraph, real T(real)=identity, bool3 cond(real),
interpolate join=operator --);

Returns a graph using the scaling information for picture pic of the parametrized function z(t) for t in the interval [T(a),T(b)], sampling at n points evenly spaced in [a,b], optionally restricted by the bool3 function cond on [a,b], using the given interpolation type.
• guide graph(picture pic=currentpicture, pair[] z,
interpolate join=operator --);
guide[] graph(picture pic=currentpicture, pair[] z, bool3[] cond,
interpolate join=operator --);

Returns a graph using the scaling information for picture pic of the elements of the array z, optionally restricted to those indices for which the elements of the boolean array cond are true, using the given interpolation type.
• guide graph(picture pic=currentpicture, real[] x, real[] y,
interpolate join=operator --);
guide[] graph(picture pic=currentpicture, real[] x, real[] y,
bool3[] cond, interpolate join=operator --);

Returns a graph using the scaling information for picture pic of the elements of the arrays (x,y), optionally restricted to those indices for which the elements of the boolean array cond are true, using the given interpolation type.
• guide polargraph(picture pic=currentpicture, real f(real), real a,
real b, int n=ngraph, interpolate join=operator --);

Returns a polar-coordinate graph using the scaling information for picture pic of the function f on the interval [a,b], sampling at n evenly spaced points, with the given interpolation type.
• guide polargraph(picture pic=currentpicture, real[] r, real[] theta,
interpolate join=operator--);

Returns a polar-coordinate graph using the scaling information for picture pic of the elements of the arrays (r,theta), using the given interpolation type.

An axis can be drawn on a picture with one of the following commands:

• void xaxis(picture pic=currentpicture, Label L="", axis axis=YZero,
real xmin=-infinity, real xmax=infinity, pen p=currentpen,
ticks ticks=NoTicks, arrowbar arrow=None, bool above=false);

Draw an x axis on picture pic from x=xmin to x=xmax using pen p, optionally labelling it with Label L. The relative label location along the axis (a real number from [0,1]) defaults to 1 (see Label), so that the label is drawn at the end of the axis. An infinite value of xmin or xmax specifies that the corresponding axis limit will be automatically determined from the picture limits. The optional arrow argument takes the same values as in the draw command (see arrows). The axis is drawn before any existing objects in pic unless above=true. The axis placement is determined by one of the following axis types:
YZero(bool extend=true)

Request an x axis at y=0 (or y=1 on a logarithmic axis) extending to the full dimensions of the picture, unless extend=false.

YEquals(real Y, bool extend=true)

Request an x axis at y=Y extending to the full dimensions of the picture, unless extend=false.

Bottom(bool extend=false)

Request a bottom axis.

Top(bool extend=false)

Request a top axis.

BottomTop(bool extend=false)

Request a bottom and top axis.

Custom axis types can be created by following the examples in graph.asy. One can easily override the default values for the standard axis types:

import graph;

YZero=new axis(bool extend=true) {
return new void(picture pic, axisT axis) {
real y=pic.scale.x.scale.logarithmic ? 1 : 0;
axis.value=I*pic.scale.y.T(y);
axis.position=1;
axis.side=right;
axis.align=2.5E;
axis.value2=Infinity;
axis.extend=extend;
};
};
YZero=YZero();



The default tick option is NoTicks. The options LeftTicks, RightTicks, or Ticks can be used to draw ticks on the left, right, or both sides of the path, relative to the direction in which the path is drawn. These tick routines accept a number of optional arguments:

ticks LeftTicks(Label format="", ticklabel ticklabel=null,
bool beginlabel=true, bool endlabel=true,
int N=0, int n=0, real Step=0, real step=0,
bool begin=true, bool end=true, tickmodifier modify=None,
real Size=0, real size=0, bool extend=false,
pen pTick=nullpen, pen ptick=nullpen);


If any of these parameters are omitted, reasonable defaults will be chosen:

Label format

override the default tick label format (defaultformat, initially "$%.4g$"), rotation, pen, and alignment (for example, LeftSide, Center, or RightSide) relative to the axis. To enable LaTeX math mode fonts, the format string should begin and end with  see format. If the format string is trailingzero, trailing zeros will be added to the tick labels; if the format string is "%", the tick label will be suppressed; ticklabel is a function string(real x) returning the label (by default, format(format.s,x)) for each major tick value x; bool beginlabel include the first label; bool endlabel include the last label; int N when automatic scaling is enabled (the default; see automatic scaling), divide a linear axis evenly into this many intervals, separated by major ticks; for a logarithmic axis, this is the number of decades between labelled ticks; int n divide each interval into this many subintervals, separated by minor ticks; real Step the tick value spacing between major ticks (if N=0); real step the tick value spacing between minor ticks (if n=0); bool begin include the first major tick; bool end include the last major tick; tickmodifier modify; an optional function that takes and returns a tickvalue structure having real[] members major and minor consisting of the tick values (to allow modification of the automatically generated tick values); real Size the size of the major ticks (in PostScript coordinates); real size the size of the minor ticks (in PostScript coordinates); bool extend; extend the ticks between two axes (useful for drawing a grid on the graph); pen pTick an optional pen used to draw the major ticks; pen ptick an optional pen used to draw the minor ticks. For convenience, the predefined tickmodifiers OmitTick(... real[] x), OmitTickInterval(real a, real b), and OmitTickIntervals(real[] a, real[] b) can be used to remove specific auto-generated ticks and their labels. The OmitFormat(string s=defaultformat ... real[] x) ticklabel can be used to remove specific tick labels but not the corresponding ticks. The tickmodifier NoZero is an abbreviation for OmitTick(0) and the ticklabel NoZeroFormat is an abbrevation for OmitFormat(0). It is also possible to specify custom tick locations with LeftTicks, RightTicks, and Ticks by passing explicit real arrays Ticks and (optionally) ticks containing the locations of the major and minor ticks, respectively: ticks LeftTicks(Label format="", ticklabel ticklabel=null, bool beginlabel=true, bool endlabel=true, real[] Ticks, real[] ticks=new real[], real Size=0, real size=0, bool extend=false, pen pTick=nullpen, pen ptick=nullpen)  • void yaxis(picture pic=currentpicture, Label L="", axis axis=XZero, real ymin=-infinity, real ymax=infinity, pen p=currentpen, ticks ticks=NoTicks, arrowbar arrow=None, bool above=false, bool autorotate=true);  Draw a y axis on picture pic from y=ymin to y=ymax using pen p, optionally labelling it with a Label L that is autorotated unless autorotate=false. The relative location of the label (a real number from [0,1]) defaults to 1 (see Label). An infinite value of ymin or ymax specifies that the corresponding axis limit will be automatically determined from the picture limits. The optional arrow argument takes the same values as in the draw command (see arrows). The axis is drawn before any existing objects in pic unless above=true. The tick type is specified by ticks and the axis placement is determined by one of the following axis types: XZero(bool extend=true) Request a y axis at x=0 (or x=1 on a logarithmic axis) extending to the full dimensions of the picture, unless extend=false. XEquals(real X, bool extend=true) Request a y axis at x=X extending to the full dimensions of the picture, unless extend=false. Left(bool extend=false) Request a left axis. Right(bool extend=false) Request a right axis. LeftRight(bool extend=false) Request a left and right axis. • For convenience, the functions void xequals(picture pic=currentpicture, Label L="", real x, bool extend=false, real ymin=-infinity, real ymax=infinity, pen p=currentpen, ticks ticks=NoTicks, bool above=true, arrowbar arrow=None);  and void yequals(picture pic=currentpicture, Label L="", real y, bool extend=false, real xmin=-infinity, real xmax=infinity, pen p=currentpen, ticks ticks=NoTicks, bool above=true, arrowbar arrow=None);  can be respectively used to call yaxis and xaxis with the appropriate axis types XEquals(x,extend) and YEquals(y,extend). This is the recommended way of drawing vertical or horizontal lines and axes at arbitrary locations. • void axes(picture pic=currentpicture, Label xlabel="", Label ylabel="", bool extend=true, pair min=(-infinity,-infinity), pair max=(infinity,infinity), pen p=currentpen, arrowbar arrow=None, bool above=false);  This convenience routine draws both x and y axes on picture pic from min to max, with optional labels xlabel and ylabel and any arrows specified by arrow. The axes are drawn on top of existing objects in pic only if above=true. • void axis(picture pic=currentpicture, Label L="", path g, pen p=currentpen, ticks ticks, ticklocate locate, arrowbar arrow=None, int[] divisor=new int[], bool above=false, bool opposite=false);  This routine can be used to draw on picture pic a general axis based on an arbitrary path g, using pen p. One can optionally label the axis with Label L and add an arrow arrow. The tick type is given by ticks. The optional integer array divisor specifies what tick divisors to try in the attempt to produce uncrowded tick labels. A true value for the flag opposite identifies an unlabelled secondary axis (typically drawn opposite a primary axis). The axis is drawn before any existing objects in pic unless above=true. The tick locator ticklocate is constructed by the routine ticklocate ticklocate(real a, real b, autoscaleT S=defaultS, real tickmin=-infinity, real tickmax=infinity, real time(real)=null, pair dir(real)=zero);  where a and b specify the respective tick values at point(g,0) and point(g,length(g)), S specifies the autoscaling transformation, the function real time(real v) returns the time corresponding to the value v, and pair dir(real t) returns the absolute tick direction as a function of t (zero means draw the tick perpendicular to the axis). • These routines are useful for manually putting ticks and labels on axes (if the variable Label is given as the Label argument, the format argument will be used to format a string based on the tick location): void xtick(picture pic=currentpicture, Label L="", explicit pair z, pair dir=N, string format="", real size=Ticksize, pen p=currentpen); void xtick(picture pic=currentpicture, Label L="", real x, pair dir=N, string format="", real size=Ticksize, pen p=currentpen); void ytick(picture pic=currentpicture, Label L="", explicit pair z, pair dir=E, string format="", real size=Ticksize, pen p=currentpen); void ytick(picture pic=currentpicture, Label L="", real y, pair dir=E, string format="", real size=Ticksize, pen p=currentpen); void tick(picture pic=currentpicture, pair z, pair dir, real size=Ticksize, pen p=currentpen); void labelx(picture pic=currentpicture, Label L="", explicit pair z, align align=S, string format="", pen p=currentpen); void labelx(picture pic=currentpicture, Label L="", real x, align align=S, string format="", pen p=currentpen); void labelx(picture pic=currentpicture, Label L, string format="", explicit pen p=currentpen); void labely(picture pic=currentpicture, Label L="", explicit pair z, align align=W, string format="", pen p=currentpen); void labely(picture pic=currentpicture, Label L="", real y, align align=W, string format="", pen p=currentpen); void labely(picture pic=currentpicture, Label L, string format="", explicit pen p=currentpen);  Here are some simple examples of two-dimensional graphs: 1. This example draws a textbook-style graph of y= exp(x), with the y axis starting at y=0: import graph; size(150,0); real f(real x) {return exp(x);} pair F(real x) {return (x,f(x));} xaxis("x$"); yaxis("$y$",0); draw(graph(f,-4,2,operator ..),red); labely(1,E); label("$e^x$",F(1),SE);  2. The next example draws a scientific-style graph with a legend. The position of the legend can be adjusted either explicitly or by using the graphical user interface xasy (see section Graphical User Interface). If an UnFill(real xmargin=0, real ymargin=xmargin) or Fill(pen) option is specified to add, the legend will obscure any underlying objects. Here we illustrate how to clip the portion of the picture covered by a label: import graph; size(400,200,IgnoreAspect); real Sin(real t) {return sin(2pi*t);} real Cos(real t) {return cos(2pi*t);} draw(graph(Sin,0,1),red,"$\sin(2\pi x)$"); draw(graph(Cos,0,1),blue,"$\cos(2\pi x)$"); xaxis("$x$",BottomTop,LeftTicks); yaxis("$y$",LeftRight,RightTicks(trailingzero)); label("LABEL",point(0),UnFill(1mm)); add(legend(),point(E),20E,UnFill);  To specify a fixed size for the graph proper, use attach: import graph; size(250,200,IgnoreAspect); real Sin(real t) {return sin(2pi*t);} real Cos(real t) {return cos(2pi*t);} draw(graph(Sin,0,1),red,"$\sin(2\pi x)$"); draw(graph(Cos,0,1),blue,"$\cos(2\pi x)$"); xaxis("$x$",BottomTop,LeftTicks); yaxis("$y$",LeftRight,RightTicks(trailingzero)); label("LABEL",point(0),UnFill(1mm)); attach(legend(),truepoint(E),20E,UnFill);  A legend can have multiple entries per line: import graph; size(8cm,6cm,IgnoreAspect); typedef real realfcn(real); realfcn F(real p) { return new real(real x) {return sin(p*x);}; }; for(int i=1; i < 5; ++i) draw(graph(F(i*pi),0,1),Pen(i), "$\sin("+(i == 1 ? "" : (string) i)+"\pi x)$"); xaxis("$x$",BottomTop,LeftTicks); yaxis("$y$",LeftRight,RightTicks(trailingzero)); attach(legend(2),(point(S).x,truepoint(S).y),10S,UnFill);  3. This example draws a graph of one array versus another (both of the same size) using custom tick locations and a smaller font size for the tick labels on the y axis. import graph; size(200,150,IgnoreAspect); real[] x={0,1,2,3}; real[] y=x^2; draw(graph(x,y),red); xaxis("$x$",BottomTop,LeftTicks); yaxis("$y$",LeftRight, RightTicks(Label(fontsize(8pt)),new real[]{0,4,9}));  4. This example shows how to graph columns of data read from a file. import graph; size(200,150,IgnoreAspect); file in=input("filegraph.dat").line(); real[][] a=in.dimension(0,0); a=transpose(a); real[] x=a[0]; real[] y=a[1]; draw(graph(x,y),red); xaxis("$x$",BottomTop,LeftTicks); yaxis("$y",LeftRight,RightTicks);  5. The next example draws two graphs of an array of coordinate pairs, using frame alignment and data markers. In the left-hand graph, the markers, constructed with marker marker(path g, markroutine markroutine=marknodes, pen p=currentpen, filltype filltype=NoFill, bool above=true);  using the path unitcircle (see filltype), are drawn below each node. Any frame can be converted to a marker, using marker marker(frame f, markroutine markroutine=marknodes, bool above=true);  In the right-hand graph, the unit n-sided regular polygon polygon(int n) and the unit n-point cyclic cross cross(int n, bool round=true, real r=0) (where r is an optional “inner” radius) are used to build a custom marker frame. Here markuniform(bool centered=false, int n, bool rotated=false) adds this frame at n uniformly spaced points along the arclength of the path, optionally rotated by the angle of the local tangent to the path (if centered is true, the frames will be centered within n evenly spaced arclength intervals). Alternatively, one can use markroutine marknodes to request that the marks be placed at each Bezier node of the path, or markroutine markuniform(pair z(real t), real a, real b, int n) to place marks at points z(t) for n evenly spaced values of t in [a,b]. These markers are predefined: marker[] Mark={ marker(scale(circlescale)*unitcircle), marker(polygon(3)),marker(polygon(4)), marker(polygon(5)),marker(invert*polygon(3)), marker(cross(4)),marker(cross(6)) }; marker[] MarkFill={ marker(scale(circlescale)*unitcircle,Fill),marker(polygon(3),Fill), marker(polygon(4),Fill),marker(polygon(5),Fill), marker(invert*polygon(3),Fill) };  The example also illustrates the errorbar routines: void errorbars(picture pic=currentpicture, pair[] z, pair[] dp, pair[] dm={}, bool[] cond={}, pen p=currentpen, real size=0); void errorbars(picture pic=currentpicture, real[] x, real[] y, real[] dpx, real[] dpy, real[] dmx={}, real[] dmy={}, bool[] cond={}, pen p=currentpen, real size=0);  Here, the positive and negative extents of the error are given by the absolute values of the elements of the pair array dp and the optional pair array dm. If dm is not specified, the positive and negative extents of the error are assumed to be equal. import graph; picture pic; real xsize=200, ysize=140; size(pic,xsize,ysize,IgnoreAspect); pair[] f={(5,5),(50,20),(90,90)}; pair[] df={(0,0),(5,7),(0,5)}; errorbars(pic,f,df,red); draw(pic,graph(pic,f),"legend", marker(scale(0.8mm)*unitcircle,red,FillDraw(blue),above=false)); scale(pic,true); xaxis(pic,"x$",BottomTop,LeftTicks); yaxis(pic,"$y$",LeftRight,RightTicks); add(pic,legend(pic),point(pic,NW),20SE,UnFill); picture pic2; size(pic2,xsize,ysize,IgnoreAspect); frame mark; filldraw(mark,scale(0.8mm)*polygon(6),green,green); draw(mark,scale(0.8mm)*cross(6),blue); draw(pic2,graph(pic2,f),marker(mark,markuniform(5))); scale(pic2,true); xaxis(pic2,"$x$",BottomTop,LeftTicks); yaxis(pic2,"$y$",LeftRight,RightTicks); yequals(pic2,55.0,red+Dotted); xequals(pic2,70.0,red+Dotted); // Fit pic to W of origin: add(pic.fit(),(0,0),W); // Fit pic2 to E of (5mm,0): add(pic2.fit(),(5mm,0),E);  6. A custom mark routine can be also be specified: import graph; size(200,100,IgnoreAspect); markroutine marks() { return new void(picture pic=currentpicture, frame f, path g) { path p=scale(1mm)*unitcircle; for(int i=0; i <= length(g); ++i) { pair z=point(g,i); frame f; if(i % 4 == 0) { fill(f,p); add(pic,f,z); } else { if(z.y > 50) { pic.add(new void(frame F, transform t) { path q=shift(t*z)*p; unfill(F,q); draw(F,q); }); } else { draw(f,p); add(pic,f,z); } } } }; } pair[] f={(5,5),(40,20),(55,51),(90,30)}; draw(graph(f),marker(marks())); scale(true); xaxis("$x$",BottomTop,LeftTicks); yaxis("$y$",LeftRight,RightTicks);  7. This example shows how to label an axis with arbitrary strings. import graph; size(400,150,IgnoreAspect); real[] x=sequence(12); real[] y=sin(2pi*x/12); scale(false); string[] month={"Jan","Feb","Mar","Apr","May","Jun", "Jul","Aug","Sep","Oct","Nov","Dec"}; draw(graph(x,y),red,MarkFill[0]); xaxis(BottomTop,LeftTicks(new string(real x) { return month[round(x % 12)];})); yaxis("$y$",LeftRight,RightTicks(4));  8. The next example draws a graph of a parametrized curve. The calls to xlimits(picture pic=currentpicture, real min=-infinity, real max=infinity, bool crop=NoCrop);  and the analogous function ylimits can be uncommented to set the respective axes limits for picture pic to the specified min and max values. Alternatively, the function void limits(picture pic=currentpicture, pair min, pair max, bool crop=NoCrop);  can be used to limit the axes to the box having opposite vertices at the given pairs). Existing objects in picture pic will be cropped to lie within the given limits if crop=Crop. The function crop(picture pic) can be used to crop a graph to the current graph limits. import graph; size(0,200); real x(real t) {return cos(2pi*t);} real y(real t) {return sin(2pi*t);} draw(graph(x,y,0,1)); //limits((0,-1),(1,0),Crop); xaxis("$x$",BottomTop,LeftTicks); yaxis("$y$",LeftRight,RightTicks(trailingzero));  The next example illustrates how one can extract a common axis scaling factor. import graph; axiscoverage=0.9; size(200,IgnoreAspect); real[] x={-1e-11,1e-11}; real[] y={0,1e6}; real xscale=round(log10(max(x))); real yscale=round(log10(max(y)))-1; draw(graph(x*10^(-xscale),y*10^(-yscale)),red); xaxis("$x/10^{"+(string) xscale+"}$",BottomTop,LeftTicks); yaxis("$y/10^{"+(string) yscale+"}",LeftRight,RightTicks(trailingzero));  Axis scaling can be requested and/or automatic selection of the axis limits can be inhibited with one of these scale routines: void scale(picture pic=currentpicture, scaleT x, scaleT y); void scale(picture pic=currentpicture, bool xautoscale=true, bool yautoscale=xautoscale, bool zautoscale=yautoscale);  This sets the scalings for picture pic. The graph routines accept an optional picture argument for determining the appropriate scalings to use; if none is given, it uses those set for currentpicture. Two frequently used scaling routines Linear and Log are predefined in graph. All picture coordinates (including those in paths and those given to the label and limits functions) are always treated as linear (post-scaled) coordinates. Use pair Scale(picture pic=currentpicture, pair z);  to convert a graph coordinate into a scaled picture coordinate. The x and y components can be individually scaled using the analogous routines real ScaleX(picture pic=currentpicture, real x); real ScaleY(picture pic=currentpicture, real y);  The predefined scaling routines can be given two optional boolean arguments: automin=false and automax=automin. These default to false but can be respectively set to true to enable automatic selection of "nice" axis minimum and maximum values. The Linear scaling can also take as optional final arguments a multiplicative scaling factor and intercept (e.g. for a depth axis, Linear(-1) requests axis reversal). For example, to draw a log/log graph of a function, use scale(Log,Log): import graph; size(200,200,IgnoreAspect); real f(real t) {return 1/t;} scale(Log,Log); draw(graph(f,0.1,10)); //limits((1,0.1),(10,0.5),Crop); dot(Label("(3,5)",align=S),Scale((3,5))); xaxis("x$",BottomTop,LeftTicks); yaxis("$y$",LeftRight,RightTicks);  By extending the ticks, one can easily produce a logarithmic grid: import graph; size(200,200,IgnoreAspect); real f(real t) {return 1/t;} scale(Log,Log); draw(graph(f,0.1,10),red); pen thin=linewidth(0.5*linewidth()); xaxis("$x$",BottomTop,LeftTicks(begin=false,end=false,extend=true, ptick=thin)); yaxis("$y$",LeftRight,RightTicks(begin=false,end=false,extend=true, ptick=thin));  One can also specify custom tick locations and formats for logarithmic axes: import graph; size(300,175,IgnoreAspect); scale(Log,Log); draw(graph(identity,5,20)); xlimits(5,20); ylimits(1,100); xaxis("$M/M_\odot$",BottomTop,LeftTicks(DefaultFormat, new real[] {6,10,12,14,16,18})); yaxis("$\nu_{\rm upp}$[Hz]",LeftRight,RightTicks(DefaultFormat));  It is easy to draw logarithmic graphs with respect to other bases: import graph; size(200,IgnoreAspect); // Base-2 logarithmic scale on y-axis: real log2(real x) {static real log2=log(2); return log(x)/log2;} real pow2(real x) {return 2^x;} scaleT yscale=scaleT(log2,pow2,logarithmic=true); scale(Linear,yscale); real f(real x) {return 1+x^2;} draw(graph(f,-4,4)); yaxis("$y$",ymin=1,ymax=f(5),RightTicks(Label(Fill(white))),EndArrow); xaxis("$x$",xmin=-5,xmax=5,LeftTicks,EndArrow);  Here is an example of "broken" linear x and logarithmic y axes that omit the segments [3,8] and [100,1000], respectively. In the case of a logarithmic axis, the break endpoints are automatically rounded to the nearest integral power of the base. import graph; size(200,150,IgnoreAspect); // Break the x axis at 3; restart at 8: real a=3, b=8; // Break the y axis at 100; restart at 1000: real c=100, d=1000; scale(Broken(a,b),BrokenLog(c,d)); real[] x={1,2,4,6,10}; real[] y=x^4; draw(graph(x,y),red,MarkFill[0]); xaxis("$x$",BottomTop,LeftTicks(Break(a,b))); yaxis("$y$",LeftRight,RightTicks(Break(c,d))); label(rotate(90)*Break,(a,point(S).y)); label(rotate(90)*Break,(a,point(N).y)); label(Break,(point(W).x,ScaleY(c))); label(Break,(point(E).x,ScaleY(c)));  9. Asymptote can draw secondary axes with the routines picture secondaryX(picture primary=currentpicture, void f(picture)); picture secondaryY(picture primary=currentpicture, void f(picture));  In this example, secondaryY is used to draw a secondary linear y axis against a primary logarithmic y axis: import graph; texpreamble("\def\Arg{\mathop {\rm Arg}\nolimits}"); size(10cm,5cm,IgnoreAspect); real ampl(real x) {return 2.5/(1+x^2);} real phas(real x) {return -atan(x)/pi;} scale(Log,Log); draw(graph(ampl,0.01,10)); ylimits(0.001,100); xaxis("$\omega\tau_0$",BottomTop,LeftTicks); yaxis("$|G(\omega\tau_0)|$",Left,RightTicks); picture q=secondaryY(new void(picture pic) { scale(pic,Log,Linear); draw(pic,graph(pic,phas,0.01,10),red); ylimits(pic,-1.0,1.5); yaxis(pic,"$\Arg G/\pi$",Right,red, LeftTicks("$% #.1f$", begin=false,end=false)); yequals(pic,1,Dotted); }); label(q,"(1,0)",Scale(q,(1,0)),red); add(q);  A secondary logarithmic y axis can be drawn like this: import graph; size(9cm,6cm,IgnoreAspect); string data="secondaryaxis.csv"; file in=input(data).line().csv(); string[] titlelabel=in; string[] columnlabel=in; real[][] a=in.dimension(0,0); a=transpose(a); real[] t=a[0], susceptible=a[1], infectious=a[2], dead=a[3], larvae=a[4]; real[] susceptibleM=a[5], exposed=a[6],infectiousM=a[7]; scale(true); draw(graph(t,susceptible,t >= 10 & t <= 15)); draw(graph(t,dead,t >= 10 & t <= 15),dashed); xaxis("Time ($\tau$)",BottomTop,LeftTicks); yaxis(Left,RightTicks); picture secondary=secondaryY(new void(picture pic) { scale(pic,Linear(true),Log(true)); draw(pic,graph(pic,t,infectious,t >= 10 & t <= 15),red); yaxis(pic,Right,red,LeftTicks(begin=false,end=false)); }); add(secondary); label(shift(5mm*N)*"Proportion of crows",point(NW),E);  10. Here is a histogram example, which uses the stats module. import graph; import stats; size(400,200,IgnoreAspect); int n=10000; real[] a=new real[n]; for(int i=0; i < n; ++i) a[i]=Gaussrand(); draw(graph(Gaussian,min(a),max(a)),blue); // Optionally calculate "optimal" number of bins a la Shimazaki and Shinomoto. int N=bins(a); histogram(a,min(a),max(a),N,normalize=true,low=0,lightred,black,bars=false); xaxis("$x$",BottomTop,LeftTicks); yaxis("$dP/dx$",LeftRight,RightTicks(trailingzero));  11. Here is an example of reading column data in from a file and a least-squares fit, using the stats module. size(400,200,IgnoreAspect); import graph; import stats; file fin=input("leastsquares.dat").line(); real[][] a=fin.dimension(0,0); a=transpose(a); real[] t=a[0], rho=a[1]; // Read in parameters from the keyboard: //real first=getreal("first"); //real step=getreal("step"); //real last=getreal("last"); real first=100; real step=50; real last=700; // Remove negative or zero values of rho: t=rho > 0 ? t : null; rho=rho > 0 ? rho : null; scale(Log(true),Linear(true)); int n=step > 0 ? ceil((last-first)/step) : 0; real[] T,xi,dxi; for(int i=0; i <= n; ++i) { real first=first+i*step; real[] logrho=(t >= first & t <= last) ? log(rho) : null; real[] logt=(t >= first & t <= last) ? -log(t) : null; if(logt.length < 2) break; // Fit to the line logt=L.m*logrho+L.b: linefit L=leastsquares(logt,logrho); T.push(first); xi.push(L.m); dxi.push(L.dm); } draw(graph(T,xi),blue); errorbars(T,xi,dxi,red); crop(); ylimits(0); xaxis("$T$",BottomTop,LeftTicks); yaxis("$\xi",LeftRight,RightTicks);  12. Here is an example that illustrates the general axis routine. import graph; size(0,100); path g=ellipse((0,0),1,2); scale(true); axis(Label("C",align=10W),g,LeftTicks(endlabel=false,8,end=false), ticklocate(0,360,new real(real v) { path h=(0,0)--max(abs(max(g)),abs(min(g)))*dir(v); return intersect(g,h)[0];}));  13. To draw a vector field of n arrows evenly spaced along the arclength of a path, use the routine picture vectorfield(path vector(real), path g, int n, bool truesize=false, pen p=currentpen, arrowbar arrow=Arrow);  as illustrated in this simple example of a flow field: import graph; defaultpen(1.0); size(0,150,IgnoreAspect); real arrowsize=4mm; real arrowlength=2arrowsize; typedef path vector(real); // Return a vector interpolated linearly between a and b. vector vector(pair a, pair b) { return new path(real x) { return (0,0)--arrowlength*interp(a,b,x); }; } real f(real x) {return 1/x;} real epsilon=0.5; path g=graph(f,epsilon,1/epsilon); int n=3; draw(g); xaxis("x$"); yaxis("$y\$");



14. To draw a vector field of nx\timesny arrows in box(a,b), use the routine
picture vectorfield(path vector(pair), pair a, pair b,
int nx=nmesh, int ny=nx, bool truesize=false,
real maxlength=truesize ? 0 : maxlength(a,b,nx,ny),
bool cond(pair z)=null, pen p=currentpen,
arrowbar arrow=Arrow, margin margin=PenMargin)


as illustrated in this example:

import graph;
size(100);

pair a=(0,0);
pair b=(2pi,2pi);

path vector(pair z) {return (0,0)--(sin(z.x),cos(z.y));}


15. The following scientific graphs, which illustrate many features of Asymptote’s graphics routines, were generated from the examples diatom.asy and westnile.asy, using the comma-separated data in diatom.csv and westnile.csv.
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