Cfrgtkky

I should draw a Brownian motion starting into a disk until the first exit time from this disk in my LaTeX work (and after i should do similar things with SLE_k), how can i do?

EDIT:
I draw some planar Brownian motion using the following code, the only thing i don't know how to do is to make a circle and find an intersection

documentclass[parskip]{scrartcl}

usepackage[margin=15mm]{geometry}

usepackage{tikz}



begin{document}



newcommand{Emmett}[5]{% points, advance, rand factor, options, end label

draw[#4] (0,0)

foreach x in {1,...,#1}

{   -- ++(rand*#2,rand*#3)

}

node[right] {#5};

}



begin{tikzpicture}

Emmett{750}{0.1}{0.1}{red}{first one}

Emmett{750}{0.1}{0.1}{green}{second one}

Emmett{750}{0.1}{0.1}{blue}{third one}

end{tikzpicture}



end{document}

This adds circles around the Brownian motions.

documentclass[parskip]{scrartcl}

usepackage[margin=15mm]{geometry}

usepackage{tikz}

usetikzlibrary{fit}

begin{document}



newcommand{Emmett}[5]{% points, advance, rand factor, options, end label

begin{scope}[local bounding box=Emmett]

draw[#4] (0,0)

foreach x in {1,...,#1}

{   -- ++(rand*#2,rand*#3)

} coordinate(Emmett-last);

end{scope}

node[anchor=west,#4] at (Emmett-last) {#5};

node[circle,draw,#4,fit=(Emmett.south west) (Emmett.north east),inner sep=0.5pt]{};

}



begin{tikzpicture}

Emmett{750}{0.1}{0.1}{red}{first one}

Emmett{750}{0.1}{0.1}{green}{second one}

Emmett{750}{0.1}{0.1}{blue}{third one}

end{tikzpicture}

I do, however, not quite understand what you mean by intersections. Something like this perhaps?

documentclass[parskip]{scrartcl}

usepackage[margin=15mm]{geometry}

usepackage{tikz}

usetikzlibrary{fit,intersections}

begin{document}



newcommand{Emmett}[6]{% points, advance, rand factor, options, end label

begin{scope}[local bounding box=Emmett]

draw[#4] (0,0)

foreach x in {1,...,#1}

{   -- ++(rand*#2,rand*#3)

} coordinate(Emmett-last);

end{scope}

node[anchor=west,#4] at (Emmett-last) {#5};

node[name path=#6,circle,draw,#4,fit=(Emmett.south west) (Emmett.north east),inner sep=0.5pt]{};

}



begin{tikzpicture}

Emmett{750}{0.1}{0.1}{red}{first one}{first}

Emmett{750}{0.1}{0.1}{green}{second one}{second}

Emmett{750}{0.1}{0.1}{blue}{third one}{third}

fill[name intersections={of=first and second}] (intersection-1) circle(1pt)

(intersection-2) circle(1pt);

end{tikzpicture}

end{document}

I've put together a command using the basic layer of pgf (the front end tikz didn't provide quite enough flexibility for your problem). When called from a pgfpicture environment, it starts the Brownian motion from the centre of the disc and continues until an intersection occurs. The coordinates of the intersection point are then printed. A number of walks can be run on the same disc (in different colours).

I hadn't done any work with the pgf basic layer before so it is quite likely that improvements can be made to the code (open to suggestions here).
I've tried to comment the code but will admit the pgf layer syntax can appear quite alien if you are only familiar with the front end. Here I will summarise the salient points:

• The tikz intersections library is used to calculate the intersection of two paths (the disc and a walk). See p.1058 of the tikz manual (V. 3.1).


• The pgf command pgfgetlastxy provides access to the last coordinates of a path, and is the key player in the foreach loop used to construct the walk


• At each step, the distance of the walk from the origin can be calculated, allowing us to terminate the walk if it has gone outside of the disc.

The code:

documentclass{article}

usepackage{geometry}

usepackage{tikz} % Includes the basic layer (pgf) and mathematical engine

% Library to determine points of intersection of two paths

usetikzlibrary{intersections} % See manual p.1058

% Command to be used inside a pgfpicture. Requires tikz, tikz.intersections

newcommanddiscmotion[4]

{ % #1: Walk number (currently only used to make sure intersection points are printed in different places)

  % #2: Number of steps in walk

  % #3: If 1, Terminate after first intersection. 0 (or any other integer): Always complete walk.

  % #4: Path color (intersections always circled in red)

pgfintersectionsortbyfirstpath % If multiple intersections, order according to first path (the random walk)

pgfintersectionofpaths % Record intersections between the following two paths pgfintersectionofpaths{PATH 1}{PATH 2}

{

    pgfpathmoveto{pgfpointorigin} % Start path at origin

    foreach x in {1,...,#2} % Walk loop

    {

        pgfgetlastxy{macrox}{macroy} % Get coordinates from last step

        pgfmathsetlengthmacro{dist}{veclen(macrox,macroy)} % Calculate distance from origin (pt)

        ifnum#3=1relax % Only check if outside disc if #3 has a value of 1

            ifdim  dist < 50pt % Check whether path has gone outside disc

                pgfpathlineto{pgfpointadd{pgfpoint{macrox}{macroy}}{pgfpoint{rand*10}{rand*10}}} % Add random vector onto path

            fi

        else % Otherwise always continue path

                pgfpathlineto{pgfpointadd{pgfpoint{macrox}{macroy}}{pgfpoint{rand*10}{rand*10}}}

        fi

    }

    % Now actually draw the path

    color{#4}  % Change color temporarily

    pgfgetpathtemppath

    pgfusepath{stroke}

    pgfsetpathtemppath

    color{black} % Restore color (for circle)

}

{

    % Path for disc

    pgfpathcircle{pgfpointorigin}{50pt}

    pgfgetpathtemppath

    pgfusepath{stroke}

    pgfsetpathtemppath

}

% Lengths to store intersections points (origin = centre of disc), in pt.

newdimenxintersect

newdimenyintersect

pgfintersectionsortbyfirstpath % Repeat of earlier command to be sure :)

foreach s in {1,...,pgfintersectionsolutions} % Iterate through intersection object (list of coordinates)

{

    pgfpathcircle{pgfpointintersectionsolution{s}}{2pt} % Circle the intersection point

    pgfextractx{xintersect}{pgfpointintersectionsolution{s}} % Get x coordinate of current intersection

    pgfextractx{yintersect}{pgfpointintersectionsolution{s}} % Get y coordinate of current intersection

    % Add a text object to right of disc, using pgfmathparse to change the coordinates in pt to cm (precision 4)

    pgftext[at=pgfpointscale{0.5}{% Just some scaling

    pgfpointadd{pgfpointadd{pgfpoint{0cm}{6cm}}{pgfpoint{0cm}{-#1 cm}}}{pgfpoint{12cm}{-s cm}}}] % {test}

        {small#4: $x=$pgfmathparse{xintersect/28.45274}pgfmathprintnumber[fixed,precision=4]{pgfmathresult} $y=$pgfmathparse{yintersect/28.45274}pgfmathprintnumber[fixed,precision=4]{pgfmathresult}}

}

color{red}

% pgfusepath{fill} % Fill intersection circles (instead of stroke)

pgfusepath{stroke} % Draw intersection paths  pgfusepath{fill} could be used here

color{black} % Restore color

}

begin{document}

begin{pgfpicture}

discmotion{1}{250}{1}{black}

discmotion{2}{250}{1}{blue}

discmotion{3}{250}{1}{orange}

end{pgfpicture}

end{document}

Example output with above code:

Note that the step sizes are quite large (rand*10); you may want to reduce this. You may also want to add additional arguments, specifying the radius of the disc, for example (currently fixed at 50pt). Lastly, there is also the option to not stop at the firsrt intersection, but always complete the prescribed number of steps (recording all intersections). Change the first argument from 1 to use this. For example,

begin{pgfpicture}

discmotion{1}{200}{0}{black} % Don't stop until 200!

end{pgfpicture}

just produced