/*
 * Copyright (C) 2010 Mark H. Weaver <mhw@netris.org>
 *
 * drawdf: Direction field plotting package for Maxima
 * 
 * This program is free software; you can redistribute it and/or
 * modify it under the terms of the GNU General Public License
 * as published by the Free Software Foundation; either version 2
 * of the License, or (at your option) any later version.
 * 
 * This program is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 * GNU General Public License for more details.
 * 
 * You should have received a copy of the GNU General Public License
 * along with this program; if not, write to the Free Software
 * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301, USA.
 */

/*
 * If you encounter stack overflows when drawing complex plots,
 * try redefining draw-transform (from draw.lisp) as follows:
 *
 * (defun draw-transform (expr transform)
 *   (reduce #'append
 * 	     (mapcar #'(lambda (x) (draw-transform-one x transform))
 * 		     expr)
 * 	     :from-end t))
 */

if not get('draw,'version) then load("draw") $

default_aspect_ratio: 4/3 $
default_horiz_grid_steps: 30 $
default_horiz_arrow_spots: 30 $

vec_mag(v) := sqrt(v.v)$
normalize_vec(v) := v/sqrt(v.v)$
if drawdf_translate='drawdf_translate then drawdf_translate:false;
if drawdf_translate then translate(normalize_vec);

make_f_rk_4(dxdt_expr, dydt_expr, x_var, y_var, x_min, x_max, y_min, y_max, partition_expr) := (
  define(funmake('f_rk_4,vars),
    buildq(
      [ dxdt_expr, dydt_expr, dxdt_var:?gensym(), dydt_var:?gensym(),
        x_var, y_var, x_min, x_max, y_min, y_max, partition_expr ],
      block([dxdt_var, dydt_var],
        if x_var<x_min or x_var>x_max or y_var<y_min or y_var>y_max then error(),
        if partition_expr#current_partition then (
          if current_partition='init then current_partition:partition_expr
          else error()),
        dxdt_var:dxdt_expr,
        dydt_var:dydt_expr,
        if imagpart(dxdt_var)^2 > 0 or imagpart(dydt_var)^2 > 0 then error(),
        [dxdt_var,dydt_var]))),
  if drawdf_translate then translate(f_rk_4)) $

/* Derived from rk in dynamics.mac by Jaime E. Villate <villate@fe.up.pt> */
calc_soln_curve(derivs, vars, init, mins, maxs, max_step, max_tstep, min_tstep, duration, max_nsteps, partition_expr) :=
block (
  [ xxx,ddd,k1,k2,k3,k4,t0,t1,r,dxxx,nxxx,nk1,tstep,
    current_partition:'init,
    numer:true,display2d:false ],

  make_f_rk_4(first(derivs), second(derivs), first(vars), second(vars),
    first(mins), first(maxs), second(mins), second(maxs), partition_expr),
  t0:0,
  xxx: init,
  r: errcatch ( k1: apply('f_rk_4,xxx) ),
  if length(r)=0 then return([]),
  ddd: [xxx],
  tstep: max_tstep,
  while max_nsteps > 0 and t0 < duration do (
    max_nsteps: max_nsteps-1,
    r: errcatch (
      t1: t0+tstep,
      k2: apply('f_rk_4,xxx+k1*tstep/2),
      k3: apply('f_rk_4,xxx+k2*tstep/2),
      k4: apply('f_rk_4,xxx+k3*tstep),
      dxxx: tstep*(k1+2*k2+2*k3+k4)/6,
      nxxx: xxx+dxxx,
      nk1: apply('f_rk_4,nxxx)),
    if length(r)=0 then (
      tstep: tstep/2,
      if tstep < min_tstep then return())
    else if (abs(first(dxxx)) > first(max_step) or
             abs(second(dxxx)) > second(max_step)) and
           tstep/2 >= min_tstep then (
      tstep: tstep/2)
    else (
      /* print("t:", t1, "dt:", tstep, "x:", nxxx, "dx:", dxxx), */
      tstep: max_tstep,
      k1: nk1,
      t0: t1,
      xxx: nxxx,
      ddd: cons(xxx, ddd))),
  reverse(ddd))$
if drawdf_translate then translate(calc_soln_curve);

/*
 * Note: if dir='reverse, this will return the curve in
 *   reverse time order (starting from the initial point),
 *   in order to help solution_arrows() draw the arrows
 *   around saddle points more nicely.  In general, you
 *   should not rely on the direction of the returned curve.
 */
calc_dir_soln_curve(derivs, vars, init, mins, maxs, max_step, dir, max_tstep, min_tstep, duration, max_nsteps, partition_expr) := (
  if elementp(dir, {'forward,'right}) then (
    calc_soln_curve(derivs, vars, init, mins, maxs, max_step, max_tstep, min_tstep, duration, max_nsteps, partition_expr))
  elseif elementp(dir, {'reverse,'left}) then (
    calc_soln_curve(-derivs, vars, init, mins, maxs, max_step, max_tstep, min_tstep, duration, max_nsteps, partition_expr))
  elseif dir = 'both then block(
    [ rev:calc_soln_curve(-derivs, vars, init, mins, maxs, max_step, max_tstep, min_tstep, duration, max_nsteps, partition_expr),
      fwd:calc_soln_curve(derivs, vars, init, mins, maxs, max_step, max_tstep, min_tstep, duration, max_nsteps, partition_expr) ],
    if fwd = [] then reverse(rev)
    else if rev = [] then fwd
    else append(reverse(rev), rest(fwd)))
  else error("Invalid direction:",dir))$

disallow_arrows_in(x_lo, y_lo, x_hi, y_hi) := block(
  [ x_step, y_step, a_spot_x, a_spot_y, b_spot_x, b_spot_y,
    a_spot_x_hi, a_spot_y_hi, b_spot_x_hi, b_spot_y_hi ],

  x_step: (x_max-x_min) / horiz_arrow_spots,
  y_step: (y_max-y_min) / vert_arrow_spots,

  calc_point_vars([x_hi,y_hi]),
  a_spot_x_hi:a_spot_x, a_spot_y_hi:a_spot_y,
  b_spot_x_hi:b_spot_x, b_spot_y_hi:b_spot_y,
  calc_point_vars([x_lo,y_lo]),

  for y:a_spot_y thru a_spot_y_hi do (
    for x:a_spot_x thru a_spot_x_hi do (
      soln_arrow_a_spots[x,y]:2 )),

  for y:b_spot_y thru b_spot_y_hi do (
    for x:b_spot_x thru b_spot_x_hi do (
      soln_arrow_b_spots[x,y]:2 )))$

solution_arrows(curve) := block(
  [ arrows:[], margin:0.5,
    spacing:12, start_space:6, end_space:4, edge_space:4,
    possible_arrow_dist:0, possible_arrow_pt:false, 
    last_arrow_dist:0, new_dist, last_pt:first(curve),
    gr_x, gr_y, a_spot_x, a_spot_y, b_spot_x, b_spot_y,
    x_step, y_step, grid_step_vec, dv,
    numer:true ],

  x_step: (x_max-x_min) / horiz_arrow_spots,
  y_step: (y_max-y_min) / vert_arrow_spots,
  grid_step_vec: [x_step,y_step] * 0.3,

  define(funmake('f_calc_dv,[x_var,y_var]),
    float(grid_step_vec * normalize_vec([dxdt,dydt] / grid_step_vec))),
  if drawdf_translate then translate(f_calc_dv),

  last_arrow_dist:spacing - start_space,
  for pt in curve do (
    calc_point_vars(pt),
    if (gr_x < margin or gr_x > horiz_arrow_spots - margin or
        gr_y < margin or gr_y > vert_arrow_spots - margin or
        soln_arrow_a_spots[a_spot_x,a_spot_y] = 2 or
        soln_arrow_b_spots[b_spot_x,b_spot_y] = 2) then (
      if (possible_arrow_pt # false and possible_arrow_dist >= edge_space) then (
        confirm_solution_arrow(possible_arrow_pt) ),
      last_pt:pt, possible_arrow_pt:false, last_arrow_dist:spacing - edge_space )
    else (
      new_dist:vec_mag((pt - last_pt) / grid_step_vec),
      last_pt:pt,
      possible_arrow_dist:possible_arrow_dist + new_dist,
      last_arrow_dist:last_arrow_dist + new_dist,
      if (soln_arrow_a_spots[a_spot_x,a_spot_y] = 1 or
          soln_arrow_b_spots[b_spot_x,b_spot_y] = 1) then (
        if (possible_arrow_pt # false and possible_arrow_dist < spacing) then (
          possible_arrow_pt:false ),
        last_arrow_dist:0 )
      elseif (possible_arrow_pt = false and last_arrow_dist >= spacing) then (
        possible_arrow_pt:pt, possible_arrow_dist:0 )
      elseif (possible_arrow_pt # false and possible_arrow_dist >= spacing) then (
        confirm_solution_arrow(possible_arrow_pt),
        possible_arrow_pt:pt, possible_arrow_dist:0 ))),
  if (possible_arrow_pt # false and possible_arrow_dist >= end_space) then (
    confirm_solution_arrow(possible_arrow_pt) ),
  arrows )$

calc_point_vars(pt) := (
  gr_x: (first(pt)-x_min)/x_step,
  gr_y: (second(pt)-y_min)/y_step,
  a_spot_x: fix(gr_x + 1),
  a_spot_y: fix(gr_y + 1),
  b_spot_x: fix(gr_x + 0.5),
  b_spot_y: fix(gr_y + 0.5) )$

confirm_solution_arrow(pt) := block (
  [ gr_x, gr_y, a_spot_x, a_spot_y, b_spot_x, b_spot_y ],
  calc_point_vars(pt),
  last_arrow_dist:possible_arrow_dist,
  soln_arrow_a_spots[a_spot_x,a_spot_y]:1,
  soln_arrow_b_spots[b_spot_x,b_spot_y]:1,
  dv:apply('f_calc_dv, pt),
  arrows:cons(funmake('vector, [pt-dv/2, dv]), arrows) )$

solution_plot(pt) := block( [ curve, duration, numer:true ],
  duration: (
    if soln_duration#false then soln_duration
    else if soln_nsteps#false then soln_nsteps*soln_tstep
    else 10.0),
  curve: calc_dir_soln_curve(
    [dxdt,dydt], [x_var,y_var], float(pt),
    [x_min-(x_max-x_min), y_min-(y_max-y_min)],
    [x_max+(x_max-x_min), y_max+(y_max-y_min)],
    [x_max-x_min, y_max-y_min]/100,
    soln_direction, soln_tstep, soln_tstep/300, duration, 100000, partition_expr),
  if curve=[] then curve:[pt],
  [ 'point_type='none, funmake('soln_curve, [curve, 'soln_arrows = soln_arrows]) ] )$

/*
 * Ideally, this should probably detect and remove overlaps in the curve.
 */
equipotential_plot(pt) := block(
  [ dxdt:dydt, dydt:-dxdt, soln_arrows:false, soln_direction:'both ],
  solution_plot(pt) )$

linearize_at(pt, derivs, vars) := block(
  [ subs: map("=", vars, pt) ],
  subst(subs, apply('matrix, outermap('diff, derivs, vars))))$

/*
 * WARNING: this currently assumes you are running a new enough version
 * of Maxima to have the new improved eigenvectors function!
 */
saddle_plot(pt) := block(
  [ eigen_info, eigvals, eigval, eigvects, the_graphics:[], soln_direction ],

  eigen_info: block( [numer:false, keepfloat:false],
    float(eigenvectors(linearize_at(pt, [dxdt,dydt], [x_var,y_var])))),

  eigvals: inpart(eigen_info,1,1),
  for i:1 thru length(eigvals) do (
    eigval: inpart(eigvals,i),
    if (imagpart(eigval) = 0 and eigval # 0) then (
      soln_direction: (if eigval > 0 then 'forward else 'reverse),
      eigvects: inpart(eigen_info,2,i),
      for eigvect in eigvects do (
        the_graphics: append(
          solution_plot(pt + 0.001 * normalize_vec(eigvect)),
          solution_plot(pt - 0.001 * normalize_vec(eigvect)),
          the_graphics)))),
  the_graphics)$

valid_var_lo_hi_spec(spec) := (
  is(listp(spec) and length(spec)=3 and symbolp(first(spec)) and
    numberp(float(second(spec))) and numberp(float(third(spec)))))$

list_of_points_from(params) := block([],
  params: args(params),
  if length(params) # 1 then return(params),
  params: first(params),
  if (length(params) # 2) or listp(first(params)) then return(params),
  return([params]))$

df_graphics(derivs_spec, [params]) := block(
  [ dxdt:1, dydt, x_var:'x, y_var:'y, x_min:-10, x_max:10, y_min:-10, y_max:10,
    aspect_ratio:'auto, horiz_grid_steps:'auto, vert_grid_steps:'auto,
    horiz_arrow_spots:'auto, vert_arrow_spots:'auto,
    soln_tinitial:0, soln_tstep:0.1, soln_nsteps:false, soln_duration:false,
    field_tstep:0.1, field_nsteps:false, field_duration:false,
    soln_direction:'both, soln_arrows:false,
    show_field:true, field_arrows:'auto, field_degree:1,
    field_horiz_offset:0, field_vert_offset:0, partition_expr:false,
    field_color:'auto, init_soln_color:'auto,
    field_head_size:0.5, field_head_type:'filled_circle,
    temp_params:[], param, pvar, pvar_table,
    x_step, y_step, grid_step_vec, the_graphics:[], pre_graphics:[],
    soln_arrow_a_spots, soln_arrow_b_spots,
    numer:true, keepfloat:true ],
  local(f_calc_dv),

  if listp(derivs_spec) then (
    if length(derivs_spec) = 1 then [dydt]: derivs_spec
    elseif length(derivs_spec) = 2 then [dxdt,dydt]: derivs_spec
    else error("Invalid derivs_spec", derivs_spec))
  else dydt: derivs_spec,

  if (length(params) >= 1 and listp(first(params)) and
      length(first(params)) = 2 and every('symbolp, first(params))) then (
    [x_var,y_var]: first(params),
    params: rest(params))
  elseif (length(params) >= 2 and valid_var_lo_hi_spec(first(params)) and
          valid_var_lo_hi_spec(second(params))) then (
    [x_var,x_min,x_max]: float(first(params)),
    [y_var,y_min,y_max]: float(second(params)),
    params: rest(params,2)),

  while params#[] do (
    param: first(params),
    params: rest(params),
    if (operatorp(param, "=") and elementp(lhs(param), {'aspect_ratio,
        'horiz_grid_steps,'vert_grid_steps,'horiz_arrow_spots,'vert_arrow_spots}))
    then lhs(param) :: float(rhs(param))
    else if (operatorp(param, "=") and lhs(param)='field_grid and
             listp(rhs(param)) and length(rhs(param))=2)
    then [horiz_grid_steps,vert_grid_steps]: float(rhs(param))
    else if (listp(param) and first(param) = x_var and length(param) = 3)
    then [x_min,x_max]: float(rest(param))
    else if (listp(param) and first(param) = y_var and length(param) = 3)
    then [y_min,y_max]: float(rest(param))
    else if (listp(param) and not atom(first(param)))
    then params: append(param, params)
    else temp_params: cons(param, temp_params) ),
  params: reverse(temp_params),

  if aspect_ratio = 'auto then aspect_ratio:(
    if (horiz_grid_steps # 'auto and vert_grid_steps # 'auto)
    then horiz_grid_steps / vert_grid_steps
    elseif (horiz_arrow_spots # 'auto and vert_arrow_spots # 'auto)
    then horiz_arrow_spots / vert_arrow_spots
    else default_aspect_ratio ),

  if horiz_grid_steps = 'auto then horiz_grid_steps:(
    if vert_grid_steps # 'auto
    then fix(vert_grid_steps * aspect_ratio)
    else default_horiz_grid_steps ),
  if vert_grid_steps = 'auto
  then vert_grid_steps:fix(horiz_grid_steps / aspect_ratio),

  if horiz_arrow_spots = 'auto then horiz_arrow_spots:(
    if vert_arrow_spots # 'auto
    then fix(vert_arrow_spots * aspect_ratio)
    else default_horiz_arrow_spots ),
  if vert_arrow_spots = 'auto
  then vert_arrow_spots:fix(horiz_arrow_spots / aspect_ratio),

  soln_arrow_a_spots: make_array(fixnum, horiz_arrow_spots+2, vert_arrow_spots+2),
  soln_arrow_b_spots: make_array(fixnum, horiz_arrow_spots+2, vert_arrow_spots+2),

  pvar_table: '[ direction        = soln_direction,
                 dir              = soln_direction,
                 tinitial         = soln_tinitial,
                 tstep            = soln_tstep,
                 nsteps           = soln_nsteps,
                 duration         = soln_duration,
                 field_tstep      = field_tstep,
                 field_nsteps     = field_nsteps,
                 field_duration   = field_duration,
                 soln_arrows      = soln_arrows,
                 show_field       = show_field,
                 partition_expr   = partition_expr,
                 field_horiz_offset = field_horiz_offset,
                 field_vert_offset = field_vert_offset,
                 field_color      = field_color,
                 field_arrows     = field_arrows,
                 field_head_size  = field_head_size,
                 field_head_type  = field_head_type,
                 field_degree     = field_degree ],
                 
  for param in params do (
    if listp(param) then (
      if (pvar:assoc(first(param),pvar_table)) # false then pvar :: second(param)
      elseif (first(param) = 'trajectory_at and length(param) >= 3)
      then pre_graphics: append(reverse(solution_plot(rest(param))), pre_graphics)
      else error("Invalid parameter:", param))
    else if operatorp(param, "=") then (
      if (pvar:assoc(lhs(param),pvar_table)) # false then pvar :: rhs(param)
      else pre_graphics: cons(param, pre_graphics))
    else if operatorp(param, 'point_at)
    then pre_graphics: append(reverse([ 'point_type = 'filled_circle, funmake('points, [[args(param)]]) ]), pre_graphics)
    else if operatorp(param, 'points_at)
    then for pt in list_of_points_from(param) do (
      pre_graphics: append(reverse([ 'point_type = 'filled_circle, funmake('points, [[pt]]) ]), pre_graphics))
    else if operatorp(param, 'soln_at)
    then pre_graphics: append(reverse(solution_plot(args(param))), pre_graphics)
    else if operatorp(param, 'solns_at)
    then for pt in list_of_points_from(param) do pre_graphics: append(reverse(solution_plot(pt)), pre_graphics)
    else if operatorp(param, 'equipot_at)
    then pre_graphics: append(reverse(equipotential_plot(args(param))), pre_graphics)
    else if operatorp(param, 'equipots_at)
    then for pt in list_of_points_from(param) do pre_graphics: append(reverse(equipotential_plot(pt)), pre_graphics)
    else if operatorp(param, 'saddle_at)
    then pre_graphics: append(reverse(saddle_plot(args(param))), pre_graphics)
    else if operatorp(param, 'saddles_at)
    then for pt in list_of_points_from(param) do pre_graphics: append(reverse(saddle_plot(pt)), pre_graphics)
    else if operatorp(param, 'no_arrows_in)
    then apply('disallow_arrows_in, args(param))
    else pre_graphics: cons(param, pre_graphics)),

  for element in pre_graphics do (
    if operatorp(element, 'soln_curve) then (
      the_graphics: cons(funmake('points, [first(element)]), the_graphics),
      if assoc('soln_arrows, rest(args(element)), false) then (
        the_graphics: append(solution_arrows(first(element)), the_graphics)))
    else the_graphics: cons(element, the_graphics)),

  if soln_arrows or not show_field then (
    if field_arrows = 'auto then field_arrows: false,
    if field_color = 'auto then field_color: red,
    if init_soln_color = 'auto then init_soln_color: black )
  else (
    if field_arrows = 'auto then field_arrows: true,
    if field_color = 'auto then field_color: black,
    if init_soln_color = 'auto then init_soln_color: red ),

  x_step: (x_max-x_min) / horiz_grid_steps,
  y_step: (y_max-y_min) / vert_grid_steps,
  grid_step_vec: [x_step,y_step] * (if field_degree#1 then 0.7 else if field_arrows then 0.8 else 0.5),

  the_graphics: append(
    [ 'point_type   = 'none,
      'point_size   = 1,
      'color        = init_soln_color,
      'head_length  = 0.6 * x_step,
      'head_angle   = 30 ],
    the_graphics ),

  if show_field then block ( [numer:true],
    if field_degree=1 then (
      define(funmake('f_calc_dv,[x_var,y_var]),
        float(grid_step_vec * normalize_vec([dxdt,dydt] / grid_step_vec))),
      if drawdf_translate then translate(f_calc_dv),
      for x: (x_min + x_step/2 + field_horiz_offset) step x_step thru x_max do (
        for y: (y_min + y_step/2 + field_vert_offset) step y_step thru y_max do (
          the_graphics: append(
            errcatch(
              block([v:[x,y], dv:apply('f_calc_dv,[x,y]), rdv, idv],
                rdv: realpart(dv), idv: imagpart(dv),
                if idv.idv > ratepsilon then error()
                else if field_arrows then funmake('vector, [v-rdv/2, rdv])
                else funmake('points, [[v-rdv/2, v+rdv/2]]))),
            the_graphics))))
    else if field_degree=2 then block (
      [ dxdt2: diff(dxdt,x_var)*dxdt + diff(dxdt,y_var)*dydt,
        dydt2: diff(dydt,x_var)*dxdt + diff(dydt,y_var)*dydt,
        t_var:?gensym() ],
      define(funmake('f_calc_dv,[x_var,y_var]),
        float([dxdt,dydt])),
      define(funmake('f_calc_dv2,[x_var,y_var]),
        float([dxdt2,dydt2])),
      if drawdf_translate then translate(f_calc_dv),
      if drawdf_translate then translate(f_calc_dv2),
      for x: (x_min + x_step/2 + field_horiz_offset) step x_step thru x_max do (
        for y: (y_min + y_step/2 + field_vert_offset) step y_step thru y_max do (
          the_graphics: append(
            apply('append,
              errcatch(
                block(
                  [ v:[x,y], dv:apply('f_calc_dv,[x,y]), rdv, idv,
                    dv2:apply('f_calc_dv2,[x,y]), rdv2, idv2,
                    curve, t_min:false, t_max:false, programmode:true ],
                  rdv: realpart(dv), idv: imagpart(dv),
                  rdv2: realpart(dv2), idv2: imagpart(dv2),
                  if idv.idv > ratepsilon or idv2.idv2 > ratepsilon then error(),
                  curve: rdv2*t_var^2/2 + rdv*t_var,
                  for unit_vec in [[1,0],[-1,0],[0,1],[0,-1]] do (
                    for root in map('float,map('rhs,realroots(((unit_vec.(curve/grid_step_vec))-0.5)))) do (
                      if root>0 then (
                        if t_max=false or t_max>root then t_max:root
                      ) else if root<0 then (
                        if t_min=false or t_min<root then t_min:root
                      ))),
                  append(
                    [ funmake('parametric, append(curve + v, [t_var,t_min,t_max])) ],
                    if field_arrows then [ point_type = field_head_type, funmake('points, [[at(curve+v, [t_var=t_max])]]) ] else [])))),
            the_graphics))))
    else if field_degree='solns then block (
      [ curve, duration ],
      duration: (
        if field_duration#false then field_duration
        else if field_nsteps#false then field_nsteps*field_tstep
        else 10.0),
      for x: (x_min + x_step/2 + field_horiz_offset) step x_step thru x_max do (
        for y: (y_min + y_step/2 + field_vert_offset) step y_step thru y_max do (
          the_graphics: append(
            apply('append,
              errcatch(
                for fudge in [[0,0],[1,1],[-1,1],[-1,-1],[1,-1],'done] do (
                  if fudge='done then error(),
                  curve:reverse(
                    calc_dir_soln_curve([dxdt,dydt], [x_var,y_var], [x,y]+fudge*grid_step_vec*1e-8,
                      [x,y]-grid_step_vec/2, [x,y]+grid_step_vec/2,
                      grid_step_vec/5, 'both,
                      field_tstep, field_tstep/300, duration, 100, partition_expr)),
                  if curve#[] then return(curve)),
                append(
                  [ point_type = 'none, funmake('points, [curve]) ],
                  if field_arrows then [ point_type = field_head_type, funmake('points, [[first(curve)]]) ] else []))),
            the_graphics))))
    else error("field_degree must be either 1, 2, or 'solns")),

  the_graphics: append(
    [ 'head_length   = 0.3 * x_step,
      'head_angle    = 20,
      'point_type    = 'none,
      'point_size    = field_head_size,
      'points_joined = true,
      'xrange        = [x_min, x_max],
      'yrange        = [y_min, y_max],
      'color         = field_color ],
    the_graphics),

  apply('gr2d, the_graphics))$

drawdf([params]) := (draw(apply('df_graphics, params)), 0)$
wxdrawdf([params]) := (wxdraw(apply('df_graphics, params)), 0)$

/*

/* Examples */

drawdf(x^2+y^2, [x,-5,5], [y,-5,5], solns_at([1,1], [2,1], [3,4]))$

drawdf([y,-9*sin(x)-y/5], [x,1,5], [y,-2,2])$
drawdf([y,-9*sin(x)-y/5], [x,1,5], [y,-2,2], field_degree=2)$
drawdf([y,-9*sin(x)-y/5], [x,1,5], [y,-2,2], tstep=0.01, soln_at(3.14,0))$

drawdf(y, [x,-2,2], [y,-10,10], tstep=0.05, duration=3, solns_at([0,1], [0,-1], [0,0]))$
drawdf(y, [x,-2,2], [y,-10,10], tstep=0.05, duration=3, soln_arrows=true, solns_at([0,1], [0,-1], [0,0]))$

drawdf([x-x*y/2, (x*y - 3*y)/4], [x,2.5,3.5], [y,1.5,2.5], solns_at([3,1.8],[3,1.85],[3,1.9],[3,1.95])) $

drawdf(exp(-t)+y, [t,y], [t,-5,5], [y,-20,20], soln_at(0,-0.5))$

drawdf(3*sin(t)+1+y, [t,y], solns_at([0,-2.6],[0,-2.4]), color=blue, soln_at(0,-2.5))$
drawdf(3*sin(t)+1+y, [t,y], soln_arrows=true, solns_at([0,-2.6],[0,-2.4],[0,-2.5]))$

/* competing species */
drawdf([x*(1-x-y), y*(3/4-y-x/2)], [x,0,1.1], [y,0,1], duration=40, soln_arrows=true, point_at(1/2,1/2), solns_at([0.1,0.2], [0.2,0.1], [1,0.8], [0.8,1], [0.1,0.1], [0.6,0.05], [0.05,0.4], [1,0.01], [0.01,0.75])) $

/* damped pendulum phase portraits */
drawdf([y,-9*sin(x)-y/5], tstep=0.05, soln_arrows=true, saddles_at([%pi,0], [-%pi,0]))$
drawdf([y,-9*sin(x)-y/5], tstep=0.05, show_field=false, soln_arrows=true, saddles_at([3*%pi,0], [-3*%pi,0], [%pi,0], [-%pi,0]))$

/* direction fields sampled quadratically and numerically */
drawdf([x*(1-x-y), y*(3/4-y-x/2)], [x,0,1.1], [y,0,1], duration=40, soln_arrows=true, field_degree=2, point_at(1/2,1/2), solns_at([0.1,0.2], [0.2,0.1], [1,0.8], [0.8,1], [0.1,0.1], [0.6,0.05], [0.05,0.4], [1,0.01], [0.01,0.75])) $
drawdf([x*(1-x-y), y*(3/4-y-x/2)], [x,0,1.1], [y,0,1], duration=40, soln_arrows=true, field_degree='solns, point_at(1/2,1/2), solns_at([0.1,0.2], [0.2,0.1], [1,0.8], [0.8,1], [0.1,0.1], [0.6,0.05], [0.05,0.4], [1,0.01], [0.01,0.75])) $

drawdf([y,-9*sin(x)-y/5], tstep=0.05, soln_arrows=true, field_degree=2, saddles_at([%pi,0], [-%pi,0]))$
drawdf([y,-9*sin(x)-y/5], tstep=0.05, soln_arrows=true, field_degree='solns, saddles_at([%pi,0], [-%pi,0]))$

*/
