#### 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];
    // conic[2] is the equation of the conic f passing through the five points
    conic[2];
    // conic[3] is a list containing the equations of the tangents
    //          through the five points
    conic[3];
    // conic[4] is an ideal representing the tropicalisation of the conic f
    conic[4];
    // conic[5] is a list containing the tropicalisation
    //          of the five tangents in conic[3]
    conic[5];
    // conic[6] is a list containing the vertices of the tropical conic
    conic[6];
    // conic[7] is a list containing the vertices of the five tangents
    conic[7];
    // 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 &");
    // 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

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