6e98cd67-3922-400b-b4c9-c7b0a5c668fc 9 6 9 decimal

&[DATE] &[TIME] - &[FILENAME]

This example is based on http://twt.mpei.ac.ru/ochkov/Lace/Lace_eng.htm

x y f 2 x * sin y cos * 0.4 - :

t0 1 time :

Specify what region of the input variables you want to plot (generically, we'll call these variables x and y):

xmin 3 - :

xmax 3 :

ymin 3 - :

ymax 3 :

Specify the fineness of display resolution of the plot:

xstep 0.0625 :

ystep 0.0625 :

Plot size graph

xsize xmax xmin - (xstep / 1 + :

xsize 97

The graph region is this many points wide

ysize ymax ymin - (ystep / 1 + :

ysize 97

The graph region is this many points tall

xsize ysize * 9409

Total number of points potentially graphed

The implicit-function graphing routine:

It works by computing the value of the function at and near each one of the xsize*ysize points, scanning across the specified region (of the input variables) in grid fashion. The value ("Pvalue") at each point P is multiplied by the value ("Qvalue") at a point Q that is close to point P (distance "xstep" away from P). - If P or Q is a root of the equation, then Pvalue or Qvalue is 0, so Pvalue * Qvalue = 0. - If a root is between P and Q, then, typically, one of Pvalue or Qvalue is positive and the other is negative, so Pvalue * Qvalue < 0. This requires that: (a) xstep is small enough so that no other roots are between P and Q, and (b) there does not happen to be a root between P and Q that is also a local maximum or minimum [example: if f(x)=x^2 and P is at x=-1 and Q is at x=+1, then Pvalue and Qvalue are both positive, even though there is a root x=0 between P and Q]. So if Pvalue * Qvalue <= 0, the coordinates (x,y) of P are added (concatenated) to a list of points to plot. The same procedure that has just been described for points P and Q is also done for points P and R, where R is close in the y-direction (by distance ystep) to point P. If only P and Q were considered, then nearly horizontal parts of the graph might not have as many points, which would make them harder to see. The names P, Q, R are used only in this text, not in the graphing routine itself. P is (x, y) Q is (x-xstep, y) R is (x, y-ystep)

Points 0012 mat :

Initialize the list of points to plot; otherwise, when the first point is to be added, the software won't be able to detect the dimensions of the list and will indicate an error.

y ymin ymax ymin ystep + range x xmin xmax xmin xstep + range x xstep - y f x y f * 0 ≤ (x y f x y ystep - f * 0 ≤ (| (Points Points x y 12 mat stack : out range if for for

Points Points 2 Points rows 12 submatrix :

Remove the first "point" that was added when the list of points was initialized. This will NOT erase the point (0, 0), if it happened to be found as a root of the equation or found to be near a root - any (0, 0) that was found will still appear in the list.

Points rows 383

The number of points that were found to be roots, or to have roots near them.

data spec pts i 1 data rows range p i el a b c spec : for v p 1 el : Unest 'p' j 2 p rows range v v p j el stack : for data v augment 51 line :

Remember the meaning of "implicit function"

x y f 2 x * sin y cos * : 0 ≡

box 3 - 3 - 3 - 33333 - 3 - 3 - 52 mat :

An unspecified "implicit function", plots the zero contour level. Define your contour level like this => for the 0.4 contour level.

x y f 2 x * sin y cos * 0.4 - :

a b c spec . 5 red 13 mat : pts Points a b c spec pts : 21 line

#A0A0A0#

1 time t0 - 18 sec *

At the opening, the canvas is rectangular. Make it square exactly [calippers]. The mesh we have chosen will produce 97 pixels values in both directions. Click in the plot, wheel mouse is in steps of "one pica [8 pixels]". At 96 pixels wheel mouse, the plot is OK wrt the Smath canvas that is in "one" pixel fineness. This is the best contour plot Smath can render. It renders in points only. Whereas the high number of points to plot, pts(data,spec) renders the contour plot pretty fast. It will plot in all web colors as you wish specify or in base colors.

box pts 2 1 sys