Contour plot: Difference between revisions

From Citizendium
Jump to navigation Jump to search
imported>Dmitrii Kouznetsov
(I see no need to copy anything from Wikipedia for this article.)
 
mNo edit summary
 
(7 intermediate revisions by 4 users not shown)
Line 1: Line 1:
[[Image:ContourPlotExample.jpg|right|400px|thumb| Fig.1. Function <math>
{{subpages}}
{{Image|ContourPlotExample.jpg|right|400px| Fig.1. Function <math>
f(x,y)=
f(x,y)=
\sin\left(\frac{\pi}{4}x\right)
\sin\left(\frac{\pi}{4}x\right)
Line 5: Line 6:
<math>-10\le x\le 10~</math>,
<math>-10\le x\le 10~</math>,
<math>-10\le y\le 10~</math>.
<math>-10\le y\le 10~</math>.
Levels <math>f(x,y)=-0.1</math> are shown wiht red lines.
Levels <math>f(x,y)=-0.1</math> are shown with red lines.
Levels <math>f(x,y)= 0.1</math> are shown wiht blue lines.]]
Levels <math>f(x,y)= 0.1</math> are shown with blue lines.}}
'''Contour plot''' is kind of [[graphical image]] that shows some function <math>f(x,y)</math> with lines  
'''Contour plot''' is kind of [[graphical image]] that shows some function <math>f(x,y)</math> with lines  
  (1) <math>f(x,y)=L</math>,
  (1) <math>f(x,y)=L</math>,
Line 18: Line 19:
<math>-10\le x\le 10~</math>,
<math>-10\le x\le 10~</math>,
<math>-10\le y\le 10~</math>.
<math>-10\le y\le 10~</math>.
Levels <math>f(x,y)=-0.1</math> are shown wiht red lines.
Levels <math>f(x,y)=-0.1</math> are shown with red lines.
Levels <math>f(x,y)= 0.1</math> are shown wiht blue lines.
Levels <math>f(x,y)= 0.1</math> are shown with blue lines.


An isoline actually is graphic of solution of [[equation]] (1). One of variables (for example, <math> x </math>) can be considered as [[independent variable]], then, another variable can be treated as ''solution''. Graphic of one isoline is called [[implicit plot]]. Contour plot can be considered as set of [[implicit plot]]s, corresponding some sequence of levels.
An isoline actually is graphic of solution of [[equation]] (1). One of variables (for example, <math> x </math>) can be considered as [[independent variable]], then, another variable can be treated as ''solution''. Graphic of one isoline is called [[implicit plot]]. Contour plot can be considered as set of [[implicit plot]]s, corresponding some sequence of levels.
Line 25: Line 26:
Countour plots are used to map some surface: [[relief]] of the Earth surface at a [[geographic map]], real and imaginary part of a [[function of complex variable]] - any bidimensional distribution.
Countour plots are used to map some surface: [[relief]] of the Earth surface at a [[geographic map]], real and imaginary part of a [[function of complex variable]] - any bidimensional distribution.


[[Programming language]]s of high level ([[Matlab]], [[Mathematica]], [[Maple]]) offer the special [[procedure]]s (operators, "functions") for ''contour plot'' and ''implicit plot''. These procedures allow quick programming of contour plots, but may be slow in execution and sometimes do not provide a good compromise between quality of the image and size of the file. Customised codes (like that used to plot Figure 1) may be more efficient, but require some effort for programming.
==Implementation of contour plot in high level languages==
[[Image:FixedPointsLoge00.png|200px|thumb|Fig.2. Contour plot of function <math>f=|z-\ln(z)|</math> in the complex
<math>z</math>-plane, made with [[Mathematica]].]]
[[Programming language]]s of high level ([[Matlab]], [[Mathematica]], [[Maple]]) offer the special [[procedure]]s (operators, "functions") for ''contour plot'' and ''implicit plot''. These procedures allow quick programming of contour plots of simple functions (see, for ex., Fig.2 and its source), but may be slow in execution and sometimes do not provide a good compromise between quality of the image and size of the file. Customised codes (see examples below) may be more efficient, but require some effort for programming.


More examples of contour plots and speculations around can be found at Wikipedia http://en.wikipedia.org/wiki/Contour_plot
==Use of Contour plot==
The contour plot can be used to represent a [[function of complex variable]]. The two sequences of [[implicit plot]]s,
for example, for the real and for the imaginary parts of the function can be plotted versus real and imaginary parts of the argument.
For a [[holomorphic function]], the contour lines for the real part are orthogonal to those for the imaginary part, forming the curvilinear rectangular mesh. The modulus and phase of a complex function can be used instead of the real and imaginary part.
Some deviations from the holomorphism may be revealed with the contour plot.
 
The following figures were generated using the contour plot:
 
[[Image:Logez02.jpg|70px]]
[[Image:TaylorExampleZ.jpg|140px]]
<!-- Factorial !-->
[[Image:LogFactorialZ.jpg‎|100px]]
[[Image:Factorialz.jpg|100px]]
[[Image:OneOverFactorial.jpg|100px]]
<!-- tetration !-->
[[Image:Sqrt(exp)(z).jpg|120px]][[Image:ZpluxSinZ.jpg|100px]]
[[Image:TetrationModified.jpg|140px]]
[[Image:TetrationModifiedZoom.jpg|120px]]
[[Image:AnalyticTetrationBaseSqrt2u00.png|80px]]
[[Image:AnalyticTetrationBaseEv00.gif|140px]]
[[Image:Analytic4thAckermannFunction00.jpg|160px]]
[[Image:SlogFitFixedPoint04.jpg|70px]]
[[Image:SLOGtailor16.jpg|70px]]
[[Image:SLOGappro50.jpg|70px]]
[[Image:SLOGtest50.jpg|200px]][[Category:Suggestion Bot Tag]]

Latest revision as of 17:00, 1 August 2024

This article is developing and not approved.
Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
Code [?]
 
This editable Main Article is under development and subject to a disclaimer.
Fig.1. Function is plotted in the range , . Levels are shown with red lines. Levels are shown with blue lines.

Contour plot is kind of graphical image that shows some function with lines

(1) ,

where is constant, called level. Such lines are called isolines. If many isolines are shown, their density indicate the slop of the function.

One example of the contour plot is shown in Figure 1. Function is plotted in the range , . Levels are shown with red lines. Levels are shown with blue lines.

An isoline actually is graphic of solution of equation (1). One of variables (for example, ) can be considered as independent variable, then, another variable can be treated as solution. Graphic of one isoline is called implicit plot. Contour plot can be considered as set of implicit plots, corresponding some sequence of levels.

Countour plots are used to map some surface: relief of the Earth surface at a geographic map, real and imaginary part of a function of complex variable - any bidimensional distribution.

Implementation of contour plot in high level languages

Fig.2. Contour plot of function in the complex -plane, made with Mathematica.

Programming languages of high level (Matlab, Mathematica, Maple) offer the special procedures (operators, "functions") for contour plot and implicit plot. These procedures allow quick programming of contour plots of simple functions (see, for ex., Fig.2 and its source), but may be slow in execution and sometimes do not provide a good compromise between quality of the image and size of the file. Customised codes (see examples below) may be more efficient, but require some effort for programming.

Use of Contour plot

The contour plot can be used to represent a function of complex variable. The two sequences of implicit plots, for example, for the real and for the imaginary parts of the function can be plotted versus real and imaginary parts of the argument. For a holomorphic function, the contour lines for the real part are orthogonal to those for the imaginary part, forming the curvilinear rectangular mesh. The modulus and phase of a complex function can be used instead of the real and imaginary part. Some deviations from the holomorphism may be revealed with the contour plot.

The following figures were generated using the contour plot:

Logez02.jpg TaylorExampleZ.jpg LogFactorialZ.jpg Factorialz.jpg OneOverFactorial.jpg Sqrt(exp)(z).jpgZpluxSinZ.jpg TetrationModified.jpg TetrationModifiedZoom.jpg AnalyticTetrationBaseSqrt2u00.png AnalyticTetrationBaseEv00.gif Analytic4thAckermannFunction00.jpg SlogFitFixedPoint04.jpg SLOGtailor16.jpg SLOGappro50.jpg SLOGtest50.jpg