Cgi-script "Graph" provides very simple way to draw the graphs
online without installing any software. Just put the formula or
points into the address string of the browser after the question mark of our
script and press "Enter". As a result the graph will appear in the browser
window as in the sample below.
You can also save this graph to computer as PNG-image with
transparent background and insert it into documents. It you enter into the
address string not the formula of function, but the mathematical
expression as, for example,
http://physics-animations.com/cgi-bin/gra.pl?exp(3)*sin(5)+ln(4),
then instead of graph you will receive the result of calculations:
-17.8742145641675.
There are many different parameters, which allows to calculate the roots of
equation, construct the animation, draw the best approximation lines,
work with 3D coordinates, trigonometric and complex values. All these
possibilities are explained below. If it will be not convenient to enter the
formula, data and parameters directly into the address string of the browser,
then you can use a special form
http://physics-animations.com/cgi-bin/graph.pl Type every parameter
(function, point, etc.) on the new string without comma-point (;).
This package lets you create and manipulate complex numbers. Complex numbers
can be written in cartesian form as a+bi or in polar form as r·exp(it). A
complex number z can be transformed to cartesian form by operator z->{d}
and to polar form by operator z->{p}. Absolute value of the complex number z can
be found as abs(z), angle in radians is arg(z). Re(z), Im(z) are consequently
the real and the complex parts of z. You can enter the complex number simply as
3+4i (or cplx(a,b)). In polar coordinates enter z=cplxe(r,t), where r is radius
and t is angle. Result of calculations in polar coordinated are displayed as [r,t].
And finally if zn = [r,t], then n-th root of [r,t] can be calculated
ambiguously. For example the cube root of [8,pi] could
be [2,pi/3], or [2,pi], or [2,5pi/3]. The k-th
solution of n-th root for z = [r,t] is given by (root(z,n))[k].
In the opposite cbrt(z) can find only one solution of
the cube root.
You can add, subtract, divide complex numbers act to them
by different functions. Some examples are given below.
Expressions of functions,
constants and additional functions are written in the separate lines or
separated by symbol comma-point (;)
You can write for example gр, g(t)=sin(t)
and then function will be g, and
argument is t. Comments are separated by symbol %%.
By default f is the function and x
is the argument (variable). Both
function and its argument can be real or complex. If function is a complex
value, then vertical axis shows complex pert of function and horizontal axis
shows its real value. In this case complex number fmin define left bottom point
of the graph and complex fmax define right top point. Argument is changed
linearly from xmin to xmax.
Both xmin and xmax
can be complex too. If fmin
or fmax are not defined, then they are calculated
automatically. There are 2 regimes of scaling of
complex function: ratio=0 or 1.
If ratio=0, then the graph of
function is stretched vertically and horizontally to fmin, fmax.
If ratio=1, then scale of both axis are the same and fmin and fmax
are chosen automatically to show all the function. If
you have two function of one parameter and you wish to draw these functions in
axes XY, then output them as f1(t)+i*f2(t).
Minimal value of
argument. In case of complex function xmin
may be complex too.
xmax
xmax=2*pi
Maximal value of
argument. In case of complex function xmax
may be complex too.
fmin
-
Minimal value of
the function.
If fmin is absent, then it is calculated automatically. If function
is a complex value, then fmin must be
complex too. In this case fmin
defines the coordinate of the left bottom corner
of the graph.
fmax
-
Maximal value of
the function.
If fmax is absent, then it is calculated automatically. If function
is a complex value, then fmax must be complex too. In
this case fmax
defines the coordinate of the right top corner of the graph.
points
points=100
Number of the
points are used to draw the graph. The
greater the points
number,
the smother the curve, but the bigger time
requires for calculations. If parameter
points is too big, then calculation time can exceed the server limit
and error 500 will appear. In such a case diminish the points number.
fbox
fbox=1
Parameter fbox
may be 0 or 1.
If fbox=1,
then the frame box, grid and captions are drawn;
if fbox=0, then
not.
spline
spline=0
Parameter fbox
may be 0 or 1.
If spline=0, then the
graph is drawn from point to point by lines. If spline=1,
then the graph is drawn by parabolas/splines. In
some cases splines can be used to make the curve more smooth, while
this increase the time of calculations and may distort the curve
(for rapidly varying function).
ratio
ratio=0
If the graph of
complex function is drawn, then the curve
is automatically stretched to the border of the rectangle (ratio=0).
In some cases this can be not convenient because
distort the proportions between real and complex value of function.
For example, if we need to draw the graph of function exp(i*x),
then we expect to see a circle, but not an
ellipse. In such a case please use
spline=1. Scale in both axes will be
identical and graph will be stretched
automatically in one direction taking into account proportions of
graph and rectangle sizes.
nmax
nmax=1
If we need to
draw several functions which differ by a parameter, then nmax sets
the maximal number of functions, while the variable n will be
changed from
0 to nmax-1.
n
-
If nmax
is set,
then n varies from
0 to nmax-1.
roots
roots=1
Parameter roots
may be set to 0 or 1.
If roots=0,
then the roots of equation f(x)=0
are not calculated. If
roots=1, then the roots of equation are
calculated. By defolt the roots are
calculated. If there are several
functions, then the roots are calculated for the main function
(without indexesЕсли
имеется несколько функций, то корни вычисляются у главной (без
цифровых индексов).
width
width=300
The width of
graph in pixels.
height
height=200
The height of
graph in pixels.
animated(time,transparent)
animated(0,0)
Add
operator "animated" if you need to construct GIF-animation. Every frame differs by value of n.
For this reason for animation you can set the total number of frames
nmax. Parameter time sets the time delay
between the frames of animation in milliseconds. time=0
sets the maximal possible frame-per-second speed. transparent=1
sets transparent background.
In this case curves will overlap each other.
Curve by points:
begin
-
Begin of points
array.
end
-
End of points
array.
polygon
-
Drawing of the filled area by points.
Next parameter after begin.
polyline
On
Drawing of the
broken line by points.
Next parameter after begin. By default.
stat
-
Drawing an
approximation line y=a+bx by points (linear regression).
Next parameter after begin.
points
On
If operator
"points" follows operator "begin", then
besides of broken line or filled area the points will be displayed
in the form of arrows.
ymin
Calculated
Minimal value by Y
ymax
Calculated
Maximal value by Y
inv
-
If there is operator "inv"
in any place of code, then points with z<0 will not be displayed.
Sample 1:
xmin=0 %% Minimal value of argument (by
default xmin=0)
xmax=5 %% Maximal value of argument (by
default xmax=2*pi)
fmin=-2 %% Minimal value of function (if
empty, then calculated)
fmax=10 %% Maximal value of function (if
empty, then calculated)
points=50 %% Number of points at the curve (by
default - 100)
width=400 %% Width of the graph in pixels (by
default - 300)
height=200 %% Height of the graph
in pixels (by default - 200)
fbox=1 %% Presence of frame box (by
default - 1, i.e. yes)
a=x^4 %% Additional function
b=10*x^3 %% Additional function
c=35*x^2 %% Additional function
d=50*x %% Additional function
e=24 %% Additional functionя
a-b+c-d+e %% Main function
Program allows also to draw the complex functions of a complex value. In the
next sample value x changes from i to 1. Let us draw the function x^n for
several different n. Result is shown below.
Verification for f=x^2: x=t+i*(1-t),
where t=0..1
f=x^2=(2t-1)+i*2t(1-t)
Re(f)=2t-1; Im(f)=2t(1-t)
We can see that Im(f)=0.5(1-Re(f)^2) That is if we draw this dependence
in coordinates XY, then it will be y=0.5(1-x^2), x=-1..1. (see in the
right side). We can see that this is the same as for the complex form of
the function.
For animation of the graphs the function animated(time,transparent) is used.
In this case the plotted image is stored as Animated GIF file. Parameter time
set the time delay between the frames in milliseconds. Parameter transparent
can be -1 or
0. If parameter "transparent" is set to 1, then the transparent background is
used and all drawn graphs are seen. If parameter "transparent" is 0,
the the background is white and frames consequently substitute each other. Graphs in different frames of animation differ by parameter
n which is changed from 0 to nmax-1.
The points and the curves drawn with these points can also be transformed in GIF animation with varying parameter
n.
In a sample below the best line is drawn by points (least-squares method).
Function "rand" adds to coordinates of all the points the random value
simulating an error of the measurement.
begin stat points
nmax=50; animated(10,0)
0 0; 1 1; 2 2; 3 3; 4 4;
5 5; 6 6; 7 7; 8 8; 9 9
x=x+rand(1)
y=y+rand(1)
end
The main function "f" must be without the number. Other functions must begin
from the same letter "f" followed by the arbitrary integer numbers. If the main
function is redefined as, for example, g(t)=..., then other functions must be
g1=.., g2=.., g101=.. etc. In this case all additional functions will also have
t as an argument. Roots are calculated only for the main function.
Color of every curve is set as color0=.., color1=...
followed by RGB components of color in hexadecimal
form (as in HTML tegs).
color0 defines the color of the main function.
If the color is omitted, then the default value is darck-blue.
Curves and filled geometrical figures
can be drawn by points. Array of points is given by pairs of digits "x y" in a
column. This array begins by heading begin and finished by operator
end. You can draw several curves in one field. Filled figure is called polygon.
Colors: black, white, red, green, blue и darkblue (by default). xmin, xmax, fmin
and fmax define the external frame. In case of the curves - polyline -
you can use smoothing tool: spline=1. If parameter spline is set to
1, then the curve is drawn by parabolas instead of the lines (see sample below).
In a sample below 3 figures are drawn: two filled areas (polygon) and one not
filled curve (polyline).
In the next sample the curve is drawn by 3 points using parabolas. This
takes more processor time, but makes an inquiry string very compact and
smoothes the curve well in the case of insufficiency of the points.
begin
spline=1
0 5
5 0
0 -5
xmin=0, xmax=5,
fmin=-5, fmax=5
Coordinates may be transferred using the rules as follows:
x = f(x); y= g(y); z=h(z)
Например: x=x**2; y=log10(y); z=2*z;
The sense of the color scheme is as follows. We draw the graph of function f(x,y).
Red is the area (x,y) where f>0 and blue is the area where f<0. Color is more
saturated near zero. The border between red and blue shows the solution of the
equation f(x,y)=0. Other regions are filled in green color (which can be removed
by bitmap=3). There are other color schemes which can be set by parameter
bitmap=1...9
