Summer 1992
Work by Cindy Ruan
(report edited by RJF)
Simple plotting
1. We wrote a common-lisp plotting procedure, using xgraph for output,
that takes a lisp function of one variable and produces a plot. Most
of the options that are available to the direct user of xgraph are
also available to the lisp user. In order to avoid the phenomenon of
``beats'' that can make the plot of a periodic function look like it
has a different period (or is a constant), the sample points are
``jiggled'' a little. Several functions may be plotted on the same
graph.
2. We wrote a lisp plotting procedure to plot one or more
parametric functions of x and y as t varies, similar to the above.
[limits: no adaptation of sample density, no treatment of outlying
points]
Interval plot:
1. We wrote a lisp plotting procedure, using xgraph for output, that
takes a lisp function of one variable and produces an interval-based
plot, as described in [fateman:honest plotting]. The lisp function
being plotted must be evaluable in interval arithmetic.
So far, only these operations are included in the interval arithmetic:
+ - * / sin cos tan cot csc sec abs exp log.
2. Those y values that are too far away from the majority of the
points are removed from the plot by default. The way we decide is to
first remove the points that are outside three standard deviations
from the median. We next recalculate sigma = standard deviation of the
remaining points. Only those points that are between [median - 3
sigma, median + 3 sigma] are displayed. This technique is an
attempt to preserve the appearance of the curve in the presence of
extreme variations like those present in a plot of tan(x).)
(This technique should also be added in the normal plotting program.)
3. If the left and right edges of the plot are too close together
for the scales to be useful, a rescaling is computed. For
example, plotting between x=1.00001 and x=1.00002 is going to have
x-axis ticks all labelled 1.00.
The plot is rescaled to f(x + min) from 0 to (max - min).
Rescaling is done if max and min are
too close: say abs(max / (max - min)) > 100.
Other activities
``Combinatorica'' and Graph Theory:
1. Only the primitive functions involving permutations
have been programmed in lisp, as a first step in implementing
Skiena's Combinatorica programs (in Mathematica) directly in Lisp.
See the file comb.lisp for details.
.............
Suggestions for further development of plotting
2D Plotting of Function(s) of One Variable:
We should apply some adaptive plotting techniques such as divided differences,
rescaling the x-axis when the interval of x to plot is too small, and etc. to
make the plot more accurate. We also need to take more care about errors,
floating-point infinities etc. in plotting of simple functions.
(The interval package takes some of this into account).
Interval Plot:
We should add more mathematics functions to the set that can be evaluated
in interval computation: square-root, general power, special cases for
polynomials.
We should try to shade the boxes displayed and do something to
distinquish between the finite y-intervals and the infinite ones.
Currently, only the border of the boxes are drawn. Since we might
sometimes limit the y range to be displayed, some of the boxes might
only be shown partially. This makes it difficult to distinquish from
the infinity intervals. Look at (intplot (/ 1 x) (x -5 5)) for
example.
We should pass more options through to customize the display as we
have done in the normal plotting program.
A parametric intplot could be written as well.
Combinatorics and Graph Theory:
Much of ``Combinatorica'' is compelling visually because of the
graphical components programmed in the book. It should be possible to
devise a data-representation of graphs so that xgraph (or some other
plotting programs) can duplicate the kinds of displays made by Skiena.
.........
Comments on xgraph:
Xgraph has been quite useful in providing a simple framework for
plotting output. It is not quite as flexible as we would like in at
least the following ways: the axis tick-mark labeling is not always as
pleasing as it might be; there seems to be no option for displaying
filled polygons (as would be useful for interval plots). Since the
lisp plotting facilities work by producing a file which is then read
by xgraph, there is a certain overhead in writing computed numbers as
ascii and then reading them in. Various other techniques would be
much faster, including direct memory access. Speed does not seem to
be an important limitation for 2-D graphs however.
Xgraph does not help for 3-D surface plotting, a logical extension to
this work. We have not found an equivalently easy to use high-quality
public-domain X-window and Postscript program for 3D plotting,
although there may be some contenders now (Zicvis?).
.................
PLOTTING UTILITIES
To use plot or parametric_plot, load the file plot.lisp into the lisp system.
To use intplot, load the file intmath.lisp.
The followings explains the usage for each of the above functions.
---------------------------------------------------------------------------
plot - plots function(s) of one variable
USAGE
(plot f (var min max) {options}*)
plots f as a function of var from min to max
(plot (list f1 f2 f3 ...) (var min max) {options}*)
plots more than one functions together
Use keywords options to customize the display. They are:
:points <N> plot at least N points. (default 100)
:title <String> display a title on the top of the plot. The argument String
must be quoted with double quotes.
:xlabel <String> label of the x axis. (default "X")
:ylabel <String> label of the y axis. (default "Y")
:boundbox <Bool> draws a bounding box around the data region if Bool evaluates
to true. (default false)
:markers <Bool> marks each data point with a large dot. (default false)
:ticks <Bool> draws tick marks rather than full grid lines. (defualt false)
:ymax <N> limits the maximum y value to be displayed to be N.
:ymin <N> limits the minimum y value to be displayed to be N.
:logx <Bool> specifies a logarithmic x axis. (default false)
:logy <Bool> specifies a logarithmic y axis. (default false)
EXAMPLES
(plot (sin x) (x 0 pi)) plots (sin x) as x ranges from 0 to pi.
(plot (list (sin x) (cos x)) (x -3 3)) plots (sin x) (cos x) from -3 to 3.
(plot (expt s 2) (s -10 10) :points 200
:title "s^2"
:ticks t) plots (expt s 2) from -10 to 10
with title "s^2" and tick marks.
----------------------------------------------------------------------------
parametric_plot - plot parametric function(s)
USAGE
(parametric_plot (list fx fy) (t tmin tmax) {options}*)
to make one parametric plot
(parametric_plot (list (list fx fy) (list gx gy) ...)
(t tmin tmax) {options}*)
to plot more than one curves together
where fx and fy are functions of t and tmin and tmax are numbers such that
tmin < tmax.
The keyword options are the same as in the plot program.
EXAMPLE
(parametric_plot (list (sin u) (cos u)) (u 0 (* 2 pi)))
(parametric_plot (list (list (sin u) (cos u))
(list (+ u 3) (* u 7)))
(u -5 10) :points 30)
------------------------------------------------------------------------
intplot - interval plot of a function of one variable
USAGE
(intplot f (var min max) {options}*)
EXAMPLES
(intplot (sin x) (x -3.14 3.14)) plots (sin x) as x ranges from -3.14 to 3.14
options for intplot can be similar to the list of options for plot, but
are not yet implemented.