Snippets
Lilian Besson (Naereen) Petits programmes en OCaml pour tester l'algorithme de Bresenham pour des segments et des cercles. Illustre https://fr.wikipedia.org/wiki/Algorithme_de_tracé_de_segment_de_Bresenham
Created by
Lilian Besson (Naereen)
last modified
Lilian Besson
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 | (** <h1>Représentation des segments en infographie.</h1>
<h1>Prépa. agreg 2013-2014 : Texte de Modélisation.</h1>
<b>Date:</b> Mardi 29 Avril 2014.
@see <http://besson.qc.to/a/m/29-04/public2012-D5.pdf> Sujet.
@see <http://besson.qc.to/a/m/29-04/> Documentation.
@author Lilian BESSON
*)
(** <h2>Préliminaires.</h2> *)
(** <h3>Type [point] pour représenter un point [A = (a1, a2)].</h3> *)
type point = int * int;;
(** Deux points, venant de l'exemple de la <b>Figure 1</b>. *)
let a : point = (0, 0)
and b : point = (18, 7) ;;
(** <h3>Type [segment] pour représenter un segment, sous forme de liste de points.</h3> *)
type segment = point array;;
(** L'exemple de la <b>Figure 1</b>. *)
let d1 : segment = [|
(0, 0);
(1, 0);
(2, 1);
(3, 1);
(4, 1);
(5, 2);
(6, 2);
(7, 3);
(8, 3);
(9, 3);
(10, 4);
(11, 4);
(12, 4);
(13, 5);
(14, 5);
(15, 5);
(16, 6);
(17, 6);
(18, 7);
|];;
(** <h3>Calculer la pente.</h3> *)
let pente ((a1, a2) : point) ((b1, b2) : point) =
(float_of_int (b2 - a2) ) /. (float_of_int (b1 - a1) );;
(** Choisit la partie inférieure ou supérieure, selon laquelle est la plus proche. *)
let ceil_or_floor myfloat =
let n1 = ceil myfloat and n2 = floor myfloat in
(** [n1 <= myfloat <= n2] *)
if (myfloat -. n2 ) > 0.5 then n1 else n2;;
(** <h3>Une première heuristique pour trouver les ordonnées du segment.</h3> *)
(** Le calcul dépend d'une fonction [mytrunc] qui décide comment arrondir les ordonnées [y_i] en valeurs entières. *)
let calc_segment mytrunc ((a1, a2) : point) ((b1, b2) : point) =
let n = b1 - a1 in
assert(n > 0);
let alpha = pente (a1, a2) (b1, b2) in
assert(alpha < 1.);
assert(alpha > 0.);
(** Le segment [s] dont on calcule directement les ordonnées [y_i] petit à petit. *)
let s = Array.init (n+1)
(fun i -> (
a1 + i,
truncate (mytrunc ( (float_of_int a2) +. ( alpha *. (float_of_int i) ) ) )
) ) in
(s:segment)
;;
(** <h4>On a trois exemples, avec [floor], [ceil] et [ceil_or_floor].</h4> *)
(** Cette fonction là est donc l'autre approche proposée en partie [3.] ! *)
let lower = calc_segment floor
and upper = calc_segment ceil
and best = calc_segment ceil_or_floor
;;
(** <h2>L'algorithme de Bresenham, tel qu'il était demandé.</h2> *)
let bresenham ((a1, a2) : point) ((b1, b2) : point) =
let n = b1 - a1 in
let e = ref (b1 - a1) in
let dx = 2 * n in
let dy = 2 * (b2 - a2) in
let alpha = pente (a1, a2) (b1, b2) in
assert(alpha < 1.);
assert(alpha > 0.);
(** Le segment [s] dont on va modifier les abscisses [x_i] et les ordonnées [y_i] petit à petit. *)
let s = Array.create (n + 1) (a1,a2) in
(** Début ! Adapté depuis wikipédia, comme le pseudo code de l'énoncé était "faux". *)
let x1 = ref a1 and x2 = ref b1 in
let y1 = ref a2 in
let i = ref 0 in
while (!x1 <= !x2) do
s.(!i) <- (!x1, !y1);
incr x1;
e := !e - dy;
if (!e <= 0) then begin
incr y1;
e:= !e + dx;
end;
incr i;
done;
(s:segment)
;;
(** <h3>Un exemple pour chaque fonction.</h3> *)
let d2 = lower a b;;
let d3 = upper a b;;
let d4 = best a b;;
let d5 = bresenham a b;;
(** <h2>Illustration graphique de l'algorithme de Bresenham.</h2> *)
open Graphics;;
(** Alterner les couleurs des carrés quand on en ajoute un, c'est <i>(plus lisible !)</i>. *)
(** Quelques couleurs supplémentaires *)
let marron : color = rgb 84 47 0
and purple : color = rgb 165 0 255
;;
let colors = [| black; red; green; blue; marron; purple |];;
let colors_name = [| "black"; "red"; "green"; "blue"; "marron"; "purple" |];;
let indice_color = ref 0;;
let initcolor () = indice_color := 0;;
let nextcolor () =
indice_color := (!indice_color + 1) mod (Array.length colors);
set_color colors.(!indice_color)
;;
(** <h3>Afficher une droite discrète sous forme de petits carrés.</h3> *)
let plot_segment ?(taille = 40) ?(nommethode = "par défaut") (monseg : segment) () =
let n = Array.length monseg in
(** Adapter la taille ici, selon le nombre de point. *)
let origine = 30 in (** Adapter la position de l'origine ici, selon le nombre de point. *)
open_graph " 600x500";
set_color black;
set_window_title (Printf.sprintf ".: Segment de taille %i avec la methode %s :." n nommethode);
(* resize_window (2*origine + taille*n + 10) (2*origine + taille*n + 10); *)
initcolor ();
set_line_width 2;
let x0, y0 = monseg.(0) in
(** On recentre en [origine, origine] *)
let ms = Array.map (fun (x, y) -> (x - x0, y - y0)) monseg in
(** La droite *)
draw_segments [| origine, origine, origine + taille*(1 + (fst ms.(n-1) ) ), origine + taille*(snd ms.(n-1) ) |];
(** On dessine chaque carré, de taille [taille] sur [taille]. *)
Array.iter (fun (x, y) ->
nextcolor ();
draw_rect (origine + x*taille) (origine + y*taille) taille taille;
) ms;
;;
(** <h3>Un exemple pour chaque méthode.</h3> *)
(** Cf. test dans le terminal. *)
let test5 (a:point) (b:point) () =
let d2 = lower a b in
Printf.printf "\nExemple par la méthode de la sélection inférieure : (appuyer sur [Enter]).";
plot_segment ~nommethode:"sélection inférieure (lower)" d2 ();
read_line ();
let d3 = upper a b in
Printf.printf "\nExemple par la méthode de la sélection supérieure : (appuyer sur [Enter]).";
plot_segment ~nommethode:"sélection supérieure (upper)" d3 ();
read_line ();
let d4 = best a b in
Printf.printf "\nExemple par la méthode de la sélection optimale : (appuyer sur [Enter]).";
plot_segment ~nommethode:"sélection optimale (best)" d4 ();
read_line ();
let d5 = bresenham a b in
Printf.printf "\nExemple par la méthode de la sélection de Bresenham : (appuyer sur [Enter]).";
plot_segment ~nommethode:"de Bresenham" d5 ();
read_line ();
;;
(*
Printf.printf "\nExemple en recopiant le segment de l'énoncé : (appuyer sur [Enter]).";;
plot_segment ~nommethode:"énoncé" d1 ();;
read_line ();;
*)
(* test5 a b ();; *)
let test_segment taille =
Printf.printf "Taille = %i" taille;
let a : point = (0, 0) and alpha = 2. /. 3. in
let b : point = (200/taille, int_of_float (alpha *. float_of_int(200/taille)) ) in
Printf.printf "Avec a = (%i, %i) et b = (%i, %i)" (fst a) (snd a) (fst b) (snd b);
let s : segment = bresenham a b in
plot_segment ~nommethode:"de Bresenham" ~taille:taille s ();
Printf.printf "Avec taille = %i. Appuyer sur [Enter] pour fermer la fenetre ..." taille;
ignore (read_line ());
clear_graph ();
;;
let tests () =
for i = 6 downto 2 do
for j = 1 downto 0 do
test_segment (10 * i + 5 * j);
done;
done;
for i = 20 downto 1 do
test_segment i;
done;
;;
tests();;
(** <h2>Fin. Des questions ?</h2> *)
(** {%html: <hr> %} *)
(** {%html: <center><small>par Lilian BESSON, avril 2014.</small></center> %} *)
|
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 | (** <h1>Représentation des segments en infographie.</h1>
<h1>Prépa. agreg 2013-2014 : Texte de Modélisation.</h1>
<b>Date:</b> Mardi 29 Avril 2014.
@see <http://besson.qc.to/a/m/29-04/public2012-D5.pdf> Sujet.
@see <http://besson.qc.to/a/m/29-04/> Documentation.
@author Lilian BESSON
*)
(** <h2>Préliminaires.</h2> *)
(** <h3>Type [point] pour représenter un point [A = (a1, a2)].</h3> *)
type point = int * int;;
(** <h3>Type [courbe] pour représenter une courbe, sous forme de liste de points.</h3> *)
type courbe = point array;;
(** <h3>Type [cercle] pour représenter un cercle, avec son centre et son rayon.</h3> *)
type cercle = { centre: point; rayon: int};;
(** Un exemple *)
let c1 : cercle = { centre = (0, 0); rayon = 7 };;
let c2 : cercle = { centre = (0, 3); rayon = 12 };;
(** <h2>Adaptation de cet algorithme pour des cercles</h2> *)
let get_x ({centre; _}: cercle) = fst centre
and get_y ({centre; _}: cercle) = snd centre
and get_r ({rayon; _}: cercle) = rayon;;
(** Trois types de deplacements *)
let deltax = (1, 0)
and deltay = (0, 1)
and deltaxy = (1, 1);;
(** Qu'on peut appliquer avec cette fonction. *)
let transl (xy: point) delta =
let x = fst xy and y = snd xy in
let dx = fst delta and dy = snd delta in
(x + dx, y + dy);;
(** Et celle la qui modifie les deux references comme il faut. *)
let transl_ref (x: int ref) (y: int ref) delta =
let dx = fst delta and dy = snd delta in
x := !x + dx;
y := !y + dy;
;;
(** Mise au carre d'un entier, puisque [x ** 2] ne marche que pour les flottants... *)
let carre x = x * x;;
(** Une certaine distance entre la courbe cercle et le point xy. *)
let distance_au_cercle (c: cercle) (xy: point) =
let r = get_r c and x0 = get_x c and y0 = get_y c in
let x = fst xy and y = snd xy in
abs( (carre r) - ((carre (x - x0)) + (carre (y - y0))) )
;;
(** Depuis le point xy, trouve lequel des deplacement elementaries dx, dy ou dx+dy permet de rester le plus pret du cercle. *)
let plus_petit_delta (c: cercle) (xy: point) =
let dx = distance_au_cercle c (transl xy deltax)
and dy = distance_au_cercle c (transl xy deltay)
and dxy = distance_au_cercle c (transl xy deltaxy) in
if dx < min dy dxy then
deltax
else begin
if dy < min dx dxy then
deltay
else
deltaxy
end
;;
(** On va recuperer la liste des pixels de l'arc (pi, pi/2) en sens anti trigonometrique *)
let pixels_de_cercle (c: cercle): point array =
let r = get_r c in
(** On part a gauche, theta = pi *)
let x0 = (get_x c) - r and y0 = get_y c in
(** On va en haut, theta = pi/2 *)
let xfin = get_x c and yfin = (get_y c) + r in
(** Liste des pixels a afficher, construite progressivement *)
let pixels = ref [(x0, y0)] in
let xi = ref x0 and yi = ref y0 in
(** On est en (xi, yi) *)
(* while ((!xi < xfin) && (!yi <= yfin) ) || ((!xi <= xfin) && (!yi < yfin) ) *)
while (!xi, !yi) < (xfin, yfin) do (** Ordre lexicographique *)
let delta = plus_petit_delta c (!xi, !yi) in
transl_ref xi yi delta;
pixels := (!xi, !yi) :: !pixels;
done;
Array.of_list (List.rev !pixels)
;;
(** Un exemple *)
let p1 = pixels_de_cercle c1;;
let p2 = pixels_de_cercle c2;;
(** --------------------------------------- *)
(** <h2>Illustration graphique de l'algorithme de Bresenham.</h2> *)
open Graphics;;
(** Alterner les couleurs des carrés quand on en ajoute un, c'est <i>(plus lisible !)</i>. *)
let colors = [| black; red; green; blue; cyan; magenta |];;
let colors_name = [| "black"; "red"; "green"; "blue"; "cyan"; "magenta" |];;
let indice_color = ref 0;;
let initcolor () = indice_color := 0;;
let nextcolor () =
indice_color := (!indice_color + 1) mod (Array.length colors);
set_color colors.(!indice_color)
;;
(** <h3>Afficher une droite discrète sous forme de petits carrés.</h3> *)
let plot_cercle_courbe ?(taille=40) (c: cercle) (monseg: courbe) () =
let r = get_r c and centrex = get_x c and centrey = get_y c in
let n = Array.length monseg in
let taille = taille in (** Adapter la taille ici, selon le nombre de point. *)
let origine = 30 in (** Adapter la position de l'origine ici, selon le nombre de point. *)
open_graph " 600x500";
set_color black;
set_window_title (Printf.sprintf ".: Quart de cercle de rayon %i et taille %i :." r taille);
(* resize_window (2*origine + taille*n + 10) (2*origine + taille*n + 10); *)
initcolor ();
set_line_width 2;
let x0, y0 = monseg.(0) in
(** On recentre en [origine, origine] *)
let ms = Array.map (fun (x, y) -> (x - x0, y - y0)) monseg in
(** On dessine le cercle *)
draw_circle (origine + taille * (centrex - x0)) (origine + taille * (centrey - y0)) (r * taille);
(** On dessine chaque carré, de taille [taille] sur [taille]. *)
Array.iter (fun (x, y) ->
nextcolor ();
draw_rect (origine + x*taille) (origine + y*taille) taille taille;
) ms;
;;
(** Exemples *)
plot_cercle_courbe c1 p1 ();;
ignore (read_line ());;
clear_graph ();;
plot_cercle_courbe ~taille:20 c2 p2 ();;
ignore (read_line ());;
clear_graph ();;
let test_cercle taille =
let c = { centre = (0, 0); rayon = 200/taille } in
let p = pixels_de_cercle c in
plot_cercle_courbe ~taille:taille c p ();
Printf.printf "Avec taille = %i. Appuyer sur [Enter] pour fermer la fenetre ..." taille;
ignore (read_line ());
clear_graph ();
;;
let tests () =
for i = 7 downto 2 do
for j = 1 downto 0 do
test_cercle (10 * i + 5 * j);
done;
done;
for i = 20 downto 1 do
test_cercle i;
done;
;;
tests();;
(** <h2>Fin. Des questions ?</h2> *)
(** {%html: <hr> %} *)
(** {%html: <center><small>par Lilian BESSON, avril 2014.</small></center> %} *)
|
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