How to plot a 2D curve
For plotting a 2D curve it's required 3 steps:
-
define the parametric expressions
of the curve for coordinates X and Y by filling the appropriate fields
X(t),
Y(t)
For example 
For example 
-
press the Plot-it! button
The 2D curve can then be modified,
and printed
, moreover the plotting
settings and the visual
settings can be modified to adjust the curve rendering.
How to plot the curvature and the evolute
of 2D curve
The curvature and evolute are characteristic function in the differential
geometry of a differential curve. For viewing their graphic it's necessary
first of all defining the parametric expressions
of the curve for coordinates X and Y in the respectly fields, the following
steps are:
For example
-
press the Curvature button for plotting curvature
function
-
press the Evolute button for plotting the evolute
function
How to plot
the osculating circle and the normal and tangent vectors
to a 2D curve
The osculating circle and the vector and tangent vectors are plotted
together to the 2D curve, whit a different color, for a better graphic
visualization. So, it's needed first of all filling the fields X(t) and
Y(t) for the expression and the field range = for defining the interval
in which is defined the 2D curve, then:
-
define a value for the parameter t, in its definition interval, in which
is evaluateded the osculating circle or the system of normal-tangent vectors
For example: 
-
press the C.Osculatore button
for plotting osculating circle in the fixed point
For example: 
-
press the Tang-Norm button
for plotting normal and tangent vectors in the fixed point
For example: 
Modifying a 2D curve
A 2D curve can be modified by changing its definitions in the X(t),
Y(t) fields and/or the range
fields and plot-it again.
The modifications will no be visibible until the Plot-it! button is
pressed and the curve is plotted again.
Plotting
In general a 2D curve is painted by rendering in 2D form a finite set of
points (sample points) calculated by using the definitions for X(t)
and Y(t) and connecting the points with lines.
Undefined Values/ Infinite Values
If the intervals for parameters contains values in which one of the parametric
function X(t)Y(t) is undefined or it has infinite values,
this can or cannot be detected by Plot-it2D depending on the values of
parameters intervals.
In the case that the set of sample points will include values for which
functions X(t)Y(t) are undefined or infinite value:
-
a function whose value is infinite or undefined will produce
the conventional value 0 (zero) and the event (infinite value/undefined
value) is signaled in the message bar.
Visual Settings
Settings visual parameters will affect the current plotting and any further
plotting.
The effect of a change in a visual parameters is visible after any
action is made on the plotted image (i.e. clicking on the image or zooming
the image), it is not required to plot the image again after a change on
visual settings.
-
Adapt (default not checked). If Adapt is not checked the
curve will appear in real proportion, that is to say that the figure is
downsized with the same proportional factor on the two axes to fit the
screen. In the case in which the range of min/max values has great differences
between two axis, the displayed figure will eventually reduce to a point.
In this case it prefereable to use the Adapt option. If Adapt is checked
the curve will be drawn reducing the proportion of each axis indipentently
in order to fit the screen, and to allow the maximum available space to
each axis. In case of great difference between the ranges of the axes this
option can result in a better visualization of the figure. It must be noted
that since the reduction factor is independent on each axes this can result
in deforming figures such as circles or perpendicular lines. So, for plotting
the osculating circle and the normal and tangent vectors, it's needed that
Adapt is checked.
Printing a 2D curve
Currently only the browser default printing facilities is available, in
order to print the 2D curve with the web page containing the parameters:
choose
Print command on the File menu.
Since plot-it! will adapt to a variety of browser the result of printing
depends on the available version of the browser.
How evaluate the coefficients of Frenet Trihedon of
2D curve in a fixed point
After have defined the parametric expressions
of the curve for coordinates X and Y and defined the range
for parameters
t it's possibile evaluating the coefficients of Frenet's apparato of
2D curve :
-
define a value, in the definition interval, for parameter
t , in the field named t =
For example 
-
press the Ap.Frenet button for evaluating the factors
-
the fields named T'=, N'= and K= show the values of coifficients
of Frenet's apparato and respectively curvature of 2D curve in the fixed
point;
Expressions
A valid Plot-it! expression is any expression built by parameter
t , numerical constants,
predefined
constant symbols, operators
and functions.
Invalid expressions can cause no effects on the plotting or unpredictable
results.
Parameter
parameter t can be specified inside any function expression,
values will be assigned to the parameter for evaluating functions,
depending on the specificied ranges
and settings
Ranges
Parameters ranges, represent the interval of values which are used
to evaluate the target functions.
min, max values are specified inserting the appropriate fields
constant expressions, i.e. expressions cointaining either numerical
constants o special
constant symbols, but no parameters.
A range is valid if the values are not empty and min <= max.
If a value is omitted and/or the interval is invalid the result is
unpredictable.
An invalid range is signaled in the message bar.
Example of valid [min,max] ranges specifications are:
[-1,1] [0,1000] [0, 2*pi] [log(10),1-pi/2,]
Constants
Two type of constants are allowed in Plot-it expressions: numerical
constants and special
constants.
Numerical Constant
Plot-it accepts in input numerical constants in standard notation:
-
standard notation with/without sign and decimal point
Example: 2345 0 –3.1002 150000
-
scientific notation is used for output of very large/small
numbers:
Example: -9.03342e+021 1.3653e-012
Special Constants
Predefined special constants are available in order to enhance the numerical
precision of computations and improving clarity of expressions:
-
e = 2.7182… constant basis for exponential and logarithm functions
-
pi = 3.1415… trigonometric constant
Operators
Conventional arithmetical operators and additional math
functions can be used in Plot-it expressions:
-
– unary minus sign –<expr> Es. –(–234) is equivalent to 234
-
+ sum <expr>+<expr> Es. 3 + 12
-
– minus <expr>–<expr> Es. 3 – 12
-
* multiplication <expr>*<expr> Es. 3 * 12
-
/ division <expr>/<expr> Es. 3 / 12
-
^ power operator <expr1>^<expr2> , <expr1> is the basis and
<expr2> is the exponent Es. 3^2 that is 9.
-
( ) parentheses, alterate standard precedence rules Es. (3 – 12)* 4
/ (2 / 3)
Functions
Several mathematical functions are available in Plot-it!.
Functions calls in expressions are specified in usual prefix notation
fun(arg1,…,argn)
where fun is the function name and arg1,…,argn are the function
arguments separated by comma.
Function arguments, can recursively contains expressions and functions
as in sqrt(2-sin(s*e))
Basic Functions
-
abs(<arg>) absolute value of <arg>
-
sqrt(<arg>) square root of <arg>
-
exp(<arg>) exponential function,
constant e raised to the power of <arg>
-
log(<arg>) logarithm of <arg>
to the base e
Trigonometric functions
-
sin(<arg>) sine of <arg>
-
cos(<arg>) cosine of <arg>
-
asin(<arg>) inverse sine of <arg>
-
acos(<arg>) inverse cosine of <arg>
-
tan(<arg>) tangent of <arg>
-
cot(<arg>) cotangent of <arg>
-
atan(<arg>) inverse tangent of <arg>
-
acot(<arg>) inverse cotangent of
<arg>
-
sinh(<arg>) hyperbolic sine of <arg>
-
cosh(<arg>) hyperbolic cosine of
<arg>
-
tanh(<arg>) hyperbolic tangent of
<arg>
-
asinh(<arg>) inverse hyperbolic
sine of <arg>
-
acosh(<arg>) inverse hyperbolic
cosine of <arg>
-
atanh(<arg>) inverse hyperbolic
tangent of <arg>
Il presente software plot-it!, plot-it!3d, plot-it!2d,
i sorgenti, le relative classi java, e pagine html sono di esclusiva
proprietà degli autori,
qualsiasi utilizzo per fini commerciali è escluso.