| #### D.13.6.10 conicWithTangents Procedure from library `tropical.lib` (see tropical_lib). Usage: conicWithTangents(points[,#]); points list, # optional list Assume: points is a list of five points in the plane over K(t) Return: list, l[1] = the list points of the five given points l[2] = the conic f passing through the five points l[3] = list of equations of tangents to f in the given points l[4] = ideal, tropicalisation of f (i.e. list of linear forms) l[5] = a list of the tropicalisation of the tangents l[6] = a list containing the vertices of the tropical conic f l[7] = a list containing lists with vertices of the tangents l[8] = a string which contains the latex-code to draw the tropical conic and its tropicalised tangents l[9] = if # is non-empty, this is the same data for the dual conic and the points dual to the computed tangents Note: the points must be generic, i.e. no three on a line Example:
    LIB "tropical.lib";
    ring r=(0,t),(x,y),dp;
    // the input consists of a list of five points in the plane over Q(t)
    list points=list(1/t2,t),list(1/t,t2),list(1,1),list(t,1/t2),list(t2,1/t);
    list conic=conicWithTangents(points);
    // conic[1] is the list of the given five points
    conic[1];
    ==> [1]:
    ==>    [1]:
    ==> 1/(t2)
    ==>    [2]:
    ==> (t)
    ==> [2]:
    ==>    [1]:
    ==> 1/(t)
    ==>    [2]:
    ==> (t2)
    ==> [3]:
    ==>    [1]:
    ==>       1
    ==>    [2]:
    ==>       1
    ==> [4]:
    ==>    [1]:
    ==> (t)
    ==>    [2]:
    ==> 1/(t2)
    ==> [5]:
    ==>    [1]:
    ==> (t2)
    ==>    [2]:
    ==> 1/(t)
    // conic[2] is the equation of the conic f passing through the five points
    conic[2];
    ==> (2t3)*x2+(-t6+t5+2t4+t3+2t2+t-1)*xy+(2t3)*y2+(-t5-2t4-3t3-2t2-t)*x+(-t5-2\
       t4-3t3-2t2-t)*y+(t6+t5+2t4+t3+2t2+t+1)
    // conic[3] is a list containing the equations of the tangents
    //          through the five points
    conic[3];
    ==> [1]:
    ==>    (-t7+t6+t5-t4-t3-t2+2t)*x+(-t7+t6-2t5+2t2+t-1)/(t2)*y+(t7+t5-t4+t3-t2-\
       1)/(t)
    ==> [2]:
    ==>    (-t8+t7+2t6-2t3+t2-t)*x+(2t6-t5-t4-t3+t2+t-1)/(t)*y+(-t7-t5+t4-t3+t2+1\
       )
    ==> [3]:
    ==>    (-t6+2t3-1)*x+(-t6+2t3-1)*y+(2t6-4t3+2)
    ==> [4]:
    ==>    (-t7+t6-2t5+2t2+t-1)/(t2)*x+(-t7+t6+t5-t4-t3-t2+2t)*y+(t7+t5-t4+t3-t2-\
       1)/(t)
    ==> [5]:
    ==>    (2t6-t5-t4-t3+t2+t-1)/(t)*x+(-t8+t7+2t6-2t3+t2-t)*y+(-t7-t5+t4-t3+t2+1\
       )
    // conic[4] is an ideal representing the tropicalisation of the conic f
    conic[4];
    ==> [1]:
    ==>    2*x+3
    ==> [2]:
    ==>    x+y
    ==> [3]:
    ==>    2*y+3
    ==> [4]:
    ==>    x+1
    ==> [5]:
    ==>    y+1
    ==> [6]:
    ==>    0
    // conic[5] is a list containing the tropicalisation
    //          of the five tangents in conic[3]
    conic[5];
    ==> [1]:
    ==>    [1]:
    ==>       x+1
    ==>    [2]:
    ==>       y-2
    ==>    [3]:
    ==>       -1
    ==> [2]:
    ==>    [1]:
    ==>       x+1
    ==>    [2]:
    ==>       y-1
    ==>    [3]:
    ==>       0
    ==> [3]:
    ==>    [1]:
    ==>       x
    ==>    [2]:
    ==>       y
    ==>    [3]:
    ==>       0
    ==> [4]:
    ==>    [1]:
    ==>       x-2
    ==>    [2]:
    ==>       y+1
    ==>    [3]:
    ==>       -1
    ==> [5]:
    ==>    [1]:
    ==>       x-1
    ==>    [2]:
    ==>       y+1
    ==>    [3]:
    ==>       0
    // conic[6] is a list containing the vertices of the tropical conic
    conic[6];
    ==> [1]:
    ==>    [1]:
    ==>       -2
    ==>    [2]:
    ==>       1
    ==> [2]:
    ==>    [1]:
    ==>       1
    ==>    [2]:
    ==>       -2
    ==> [3]:
    ==>    [1]:
    ==>       -1
    ==>    [2]:
    ==>       1
    ==> [4]:
    ==>    [1]:
    ==>       1
    ==>    [2]:
    ==>       -1
    // conic[7] is a list containing the vertices of the five tangents
    conic[7];
    ==> [1]:
    ==>    [1]:
    ==>       [1]:
    ==>          -2
    ==>       [2]:
    ==>          1
    ==> [2]:
    ==>    [1]:
    ==>       [1]:
    ==>          -1
    ==>       [2]:
    ==>          1
    ==> [3]:
    ==>    [1]:
    ==>       [1]:
    ==>          0
    ==>       [2]:
    ==>          0
    ==> [4]:
    ==>    [1]:
    ==>       [1]:
    ==>          1
    ==>       [2]:
    ==>          -2
    ==> [5]:
    ==>    [1]:
    ==>       [1]:
    ==>          1
    ==>       [2]:
    ==>          -1
    // conic[8] contains the latex code to draw the tropical conic and
    //          its tropicalised tangents; it can written in a file, processed and
    //          displayed via xdg-open
    write(":w /tmp/conic.tex",conic[8]);
    system("sh","cd /tmp; latex /tmp/conic.tex; dvips /tmp/conic.dvi -o;
    xdg-open conic.ps &");
    ==> This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017) (preloaded\
        format=latex)
    ==>  restricted \write18 enabled.
    ==> entering extended mode
    ==> (/tmp/conic.tex
    ==> LaTeX2e <2017/01/01> patch level 3
    ==> Babel <3.10> and hyphenation patterns for 3 language(s) loaded.
    ==> (/usr/share/texmf-dist/tex/latex/amscls/amsart.cls
    ==> Document Class: amsart 2015/03/04 v2.20.2
    ==> (/usr/share/texmf-dist/tex/latex/amsmath/amsmath.sty
    ==> For additional information on amsmath, use the `?' option.
    ==> (/usr/share/texmf-dist/tex/latex/amsmath/amstext.sty
    ==> (/usr/share/texmf-dist/tex/latex/amsmath/amsgen.sty))
    ==> (/usr/share/texmf-dist/tex/latex/amsmath/amsbsy.sty)
    ==> (/usr/share/texmf-dist/tex/latex/amsmath/amsopn.sty))
    ==> (/usr/share/texmf-dist/tex/latex/amsfonts/umsa.fd)
    ==> (/usr/share/texmf-dist/tex/latex/amsfonts/amsfonts.sty))
    ==>
    ==> ! LaTeX Error: File `texdraw.sty' not found.
    ==>
    ==> Type X to quit or <RETURN> to proceed,
    ==> or enter new name. (Default extension: sty)
    ==>
    ==> Enter file name: warning: kpathsea: $.sty: Unrecognized variable construc\
       t `$.'.
    ==>
    ==> ! LaTeX Error: File `$.sty' not found.
    ==>
    ==> Type X to quit or <RETURN> to proceed,
    ==> or enter new name. (Default extension: sty)
    ==>
    ==> Enter file name:
    ==> ! Emergency stop.
    ==> <read *>
    ==>
    ==> l.3 \begin
    ==>           {document}^^M
    ==> No pages of output.
    ==> Transcript written on conic.log.
    ==> This is dvips(k) 5.997 Copyright 2017 Radical Eye Software (www.radicaley\
       e.com)
    ==> dvips: DVI file can't be opened: /tmp/conic.dvi: No such file or director\
       y
    ==> sh: Zeile 1: xdg-open: Kommando nicht gefunden.
    ==> 0
    // with an optional argument the same information for the dual conic is computed
    //         and saved in conic[9]
    conic=conicWithTangents(points,1);
    conic[9][2]; // the equation of the dual conic
    ==> (t6-6t5+11t4-6t3+t2)*x2+(-4t8+8t7-2t6-4t5+6t4-4t3-2t2+8t-4)*xy+(t6-6t5+11\
       t4-6t3+t2)*y2+(-2t7+2t6+4t5-6t4+4t3+2t2-2t)*x+(-2t7+2t6+4t5-6t4+4t3+2t2-2\
       t)*y+(t8-4t7+2t6+8t5-13t4+8t3+2t2-4t+1)

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