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Les interpolations avec Lazarus (3/3)

Les courbes de Bézier cubiques

L'auteur

Gilles Vasseur

L'article

Publié le 30 juin 2018

Version PDF Version hors-ligne

ePub, Azw et Mobi

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I. Introduction▲

Les programmes de test sont présents dans le répertoire exemples accompagnant le tutoriel.

II. Les courbes de Bézier cubiques (cubic bezier)▲

II-A. Un exemple interactif▲

II-B. La production de listes de coordonnées▲

II-C. Les courbes de Bézier en action▲

III. Des classes pour les interpolations, fonctions d'easing et les courbes de Bézier cubiques▲

IV. Une unité pour les nostalgiques▲

[Exemple Interpolations 08]

Comme souvent en Pascal, il est possible de travailler avec des fonctions hors des classes. Nous fournissons donc une unité baptisée interpolationfunctions composée des fonctions étudiées, mais accessibles directement.

En voici l'interface :

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unit interpolationfunctions;

{$mode objfpc}{$H+}

interface

uses
  Classes, SysUtils;

type
  TInterpolationFunction = function(AStart, AEnd, ADuration, AStep: Single):
   Single;

function InPower(AStart, AChange, ADuration, AStep: Single; APower: Integer): Single;
function OutPower(AStart, AChange, ADuration, AStep: Single; APower: Integer): Single;
function InOutPower(AStart, AChange, ADuration, AStep: Single; APower: Integer): Single;
function EaseNone(AStart, AEnd, ADuration, AStep: Single): Single;
function EaseInQuad(AStart, AEnd, ADuration, AStep: Single): Single;
function EaseOutQuad(AStart, AEnd, ADuration, AStep: Single): Single;
function EaseInOutQuad(AStart, AEnd, ADuration, AStep: Single): Single;
function EaseOutInQuad(AStart, AEnd, ADuration, AStep: Single): Single;
function EaseInCubic(AStart, AEnd, ADuration, AStep: Single): Single;
function EaseOutCubic(AStart, AEnd, ADuration, AStep: Single): Single;
function EaseInOutCubic(AStart, AEnd, ADuration, AStep: Single): Single;
function EaseOutInCubic(AStart, AEnd, ADuration, AStep: Single): Single;
function EaseInQuart(AStart, AEnd, ADuration, AStep: Single): Single;
function EaseOutQuart(AStart, AEnd, ADuration, AStep: Single): Single;
function EaseInOutQuart(AStart, AEnd, ADuration, AStep: Single): Single;
function EaseOutInQuart(AStart, AEnd, ADuration, AStep: Single): Single;
function EaseInQuint(AStart, AEnd, ADuration, AStep: Single): Single;
function EaseOutQuint(AStart, AEnd, ADuration, AStep: Single): Single;
function EaseInOutQuint(AStart, AEnd, ADuration, AStep: Single): Single;
function EaseOutInQuint(AStart, AEnd, ADuration, AStep: Single): Single;
function EaseInSine(AStart, AEnd, ADuration, AStep: Single): Single;
function EaseOutSine(AStart, AEnd, ADuration, AStep: Single): Single;
function EaseInOutSine(AStart, AEnd, ADuration, AStep: Single): Single;
function EaseOutInSine(AStart, AEnd, ADuration, AStep: Single): Single;
function EaseInExpo(AStart, AEnd, ADuration, AStep: Single): Single;
function EaseOutExpo(AStart, AEnd, ADuration, AStep: Single): Single;
function EaseInOutExpo(AStart, AEnd, ADuration, AStep: Single): Single;
function EaseOutInExpo(AStart, AEnd, ADuration, AStep: Single): Single;
function EaseInCirc(AStart, AEnd, ADuration, AStep: Single): Single;
function EaseOutCirc(AStart, AEnd, ADuration, AStep: Single): Single;
function EaseInOutCirc(AStart, AEnd, ADuration, AStep: Single): Single;
function EaseOutInCirc(AStart, AEnd, ADuration, AStep: Single): Single;
function EaseInElastic(AStart, AEnd, ADuration, AStep: Single): Single;
function EaseOutElastic(AStart, AEnd, ADuration, AStep: Single): Single;
function EaseInOutElastic(AStart, AEnd, ADuration, AStep: Single): Single;
function EaseOutInElastic(AStart, AEnd, ADuration, AStep: Single): Single;
function EaseInBack(AStart, AEnd, ADuration, AStep: Single): Single;
function EaseOutBack(AStart, AEnd, ADuration, AStep: Single): Single;
function EaseInOutBack(AStart, AEnd, ADuration, AStep: Single): Single;
function EaseOutInBack(AStart, AEnd, ADuration, AStep: Single): Single;
function EaseInBounce(AStart, AEnd, ADuration, AStep: Single): Single;
function EaseOutBounce(AStart, AEnd, ADuration, AStep: Single): Single;
function EaseInOutBounce(AStart, AEnd, ADuration, AStep: Single): Single;
function EaseOutInBounce(AStart, AEnd, ADuration, AStep: Single): Single;

Le type TInterpolationfunction permet d'utiliser ces fonctions comme paramètres de procédures ou d'autres fonctions.

Ainsi, l'entête de ComputeInterpolation pourra devenir :

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function TMainForm.ComputeInterpolation(AInter: TInterpolationFunction; AStart,
  AEnd, AStep: Single; ABack: Boolean): Single;

L'appel de cette fonction se fera en précisant qu'il ne s'agit pas d'exécuter la fonction fournie comme premier paramètre, mais d'accéder à son adresse. Nous utiliserons alors un @ devant son identificateur :

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btnEaseInQuad.Left := ComputeInterpolationInt(@EaseInQuad, 0, LWidth, Li, LBack);

Pour le reste, la modification essentielle tient au fait que la durée n'est plus accessible directement puisque nous nous situons hors de la classe : il faut par conséquent prévoir un paramètre supplémentaire à toutes nos fonctions (Aduration).

Si nous reprenons l'exemple 03 donné dans le premier tutoriel pour afficher les fonctions d'easing, quelques modifications sont à apporter à sa réalisation. Voici le code source de l'unité principale modifiée :

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unit main;

{$mode objfpc}{$H+}

interface

uses
  Classes, SysUtils, FileUtil, Forms, Controls, Graphics, Dialogs, ExtCtrls,
  StdCtrls, ComCtrls,
  interpolationfunctions;

const
 C_DefaultDuration = 100;
 C_End = 200;

type

  { TMainForm }

  TMainForm = class(TForm)
    btnEaseOutInQuart: TButton;
    btnLinear2: TButton;
    btnEaseInQuad: TButton;
    btnEaseOutCubic: TButton;
    btnEaseInCubic: TButton;
    btnEaseOutQuad: TButton;
    btnEaseInOutCubic: TButton;
    btnEaseInOutQuart: TButton;
    btnEaseOutInCubic: TButton;
    btnEaseInQuart: TButton;
    btnEaseOutQuart: TButton;
    btnEaseInQuint: TButton;
    btnEaseOutQuint: TButton;
    btnEaseOutInQuint: TButton;
    btnEaseInOutQuint: TButton;
    btnEaseInSine: TButton;
    btnEaseOutSine: TButton;
    btnEaseInOutSine: TButton;
    btnEaseOutInSine: TButton;
    btnEaseInOutQuad: TButton;
    btnEaseOutInQuad: TButton;
    btnEaseInExpo: TButton;
    btnEaseInCirc: TButton;
    btnEaseInElastic: TButton;
    btnEaseInBack: TButton;
    btnEaseInBounce: TButton;
    btnEaseOutCirc: TButton;
    btnEaseOutElastic: TButton;
    btnEaseOutBack: TButton;
    btnEaseOutBounce: TButton;
    btnEaseOutExpo: TButton;
    btnLinear: TButton;
    btnEaseOutInCirc: TButton;
    btnEaseOutInElastic: TButton;
    btnEaseOutInBack: TButton;
    btnEaseOutInBounce: TButton;
    btnEaseInOutExpo: TButton;
    btnEaseInOutCirc: TButton;
    btnEaseInOutElastic: TButton;
    btnEaseInOutBack: TButton;
    btnEaseOutInExpo: TButton;
    btnEaseInOutBounce: TButton;
    pnlRight: TPanel;
    pnlLeft: TPanel;
    pnlBottom: TPanel;
    pnlMain: TPanel;
    Splitter1: TSplitter;
    tbarWait: TTrackBar;
    procedure btnEaseInOutQuadClick(Sender: TObject);
    procedure btnEaseInQuadClick(Sender: TObject);
    procedure btnEaseOutInQuadClick(Sender: TObject);
    procedure btnEaseOutQuadClick(Sender: TObject);
    procedure btnLinear2Click(Sender: TObject);
    procedure FormCreate(Sender: TObject);
    procedure tbarWaitChange(Sender: TObject);
  private
    fDuration: Cardinal;
    procedure SetDuration(AValue: Cardinal);
  public
    property Duration: Cardinal read fDuration write SetDuration default C_DefaultDuration;
    function ComputeInterpolation(AInter: TInterpolationFunction;
      AStart, AEnd, AStep: Single; ABack: Boolean = False):
      Single;
    function ComputeInterpolationInt(AInter: TInterpolationFunction; AStart, AEnd,
      AStep: Integer; ABack: Boolean = False): Integer;
  end;

var
  MainForm: TMainForm;

implementation

{$R *.lfm}

{ TMainForm }

procedure TMainForm.btnLinear2Click(Sender: TObject);
// *** lancement des animations ***
var
  Li, LWidth, LWidth2: Integer;
  LBack: Boolean;
begin
  tbarWait.Enabled := False;
  LWidth := pnlLeft.Width - btnLinear2.Width;
  LWidth2 := pnlRight.Width - btnLinear.Width;
  for LBack := False to True do
    for Li := 1 to Duration do
    begin
      btnLinear.Left := ComputeInterpolationInt(@EaseNone, 0, LWidth, Li, LBack);
      btnLinear2.Left := ComputeInterpolationInt(@EaseNone, 0, LWidth2, Li, LBack);

      btnEaseInQuad.Left := ComputeInterpolationInt(@EaseInQuad, 0, LWidth, Li, LBack);
      btnEaseOutQuad.Left := ComputeInterpolationInt(@EaseOutQuad, 0, LWidth, Li, LBack);
      btnEaseInOutQuad.Left := ComputeInterpolationInt(@EaseInOutQuad, 0, LWidth2, Li, LBack);
      btnEaseOutInQuad.Left := ComputeInterpolationInt(@EaseOutInQuad, 0, LWidth2, Li, LBack);

      btnEaseInCubic.Left := ComputeInterpolationInt(@EaseInCubic, 0, LWidth, Li, LBack);
      btnEaseOutCubic.Left := ComputeInterpolationInt(@EaseOutCubic, 0, LWidth, Li, LBack);
      btnEaseInOutCubic.Left := ComputeInterpolationInt(@EaseInOutCubic, 0, LWidth2, Li, LBack);
      btnEaseOutInCubic.Left := ComputeInterpolationInt(@EaseOutInCubic, 0, LWidth2, Li, LBack);

      btnEaseInQuart.Left := ComputeInterpolationInt(@EaseInQuart, 0, LWidth, Li, LBack);
      btnEaseOutQuart.Left := ComputeInterpolationInt(@EaseOutQuart, 0, LWidth, Li, LBack);
      btnEaseInOutQuart.Left := ComputeInterpolationInt(@EaseInOutQuart, 0, LWidth2, Li, LBack);
      btnEaseOutInQuart.Left := ComputeInterpolationInt(@EaseOutInQuart, 0, LWidth2, Li, LBack);

      btnEaseInQuint.Left := ComputeInterpolationInt(@EaseInQuint, 0, LWidth, Li, LBack);
      btnEaseOutQuint.Left := ComputeInterpolationInt(@EaseOutQuint, 0, LWidth, Li, LBack);
      btnEaseInOutQuint.Left := ComputeInterpolationInt(@EaseInOutQuint, 0, LWidth2, Li, LBack);
      btnEaseOutInQuint.Left := ComputeInterpolationInt(@EaseOutInQuint, 0, LWidth2, Li, LBack);

      btnEaseInSine.Left := ComputeInterpolationInt(@EaseInSine, 0, LWidth, Li, LBack);
      btnEaseOutSine.Left := ComputeInterpolationInt(@EaseOutSine, 0, LWidth, Li, LBack);
      btnEaseInOutSine.Left := ComputeInterpolationInt(@EaseInOutSine, 0, LWidth2, Li, LBack);
      btnEaseOutInSine.Left := ComputeInterpolationInt(@EaseOutInSine, 0, LWidth2, Li, LBack);

      btnEaseInExpo.Left := ComputeInterpolationInt(@EaseInExpo, 0, LWidth, Li, LBack);
      btnEaseOutExpo.Left := ComputeInterpolationInt(@EaseOutExpo, 0, LWidth, Li, LBack);
      btnEaseInOutExpo.Left := ComputeInterpolationInt(@EaseInOutExpo, 0, LWidth2, Li, LBack);
      btnEaseOutInExpo.Left := ComputeInterpolationInt(@EaseOutInExpo, 0, LWidth2, Li, LBack);

      btnEaseInCirc.Left := ComputeInterpolationInt(@EaseInCirc, 0, LWidth, Li, LBack);
      btnEaseOutCirc.Left := ComputeInterpolationInt(@EaseOutCirc, 0, LWidth, Li, LBack);
      btnEaseInOutCirc.Left := ComputeInterpolationInt(@EaseInOutCirc, 0, LWidth2, Li, LBack);
      btnEaseOutInCirc.Left := ComputeInterpolationInt(@EaseOutInCirc, 0, LWidth2, Li, LBack);

      btnEaseInElastic.Left := ComputeInterpolationInt(@EaseInElastic, 0, LWidth, Li, LBack);
      btnEaseOutElastic.Left := ComputeInterpolationInt(@EaseOutElastic, 0, LWidth, Li, LBack);
      btnEaseInOutElastic.Left := ComputeInterpolationInt(@EaseInOutElastic, 0, LWidth2, Li, LBack);
      btnEaseOutInElastic.Left := ComputeInterpolationInt(@EaseOutInElastic, 0, LWidth2, Li, LBack);

      btnEaseInBack.Left := ComputeInterpolationInt(@EaseInBack, 0, LWidth, Li, LBack);
      btnEaseOutBack.Left := ComputeInterpolationInt(@EaseOutBack, 0, LWidth, Li, LBack);
      btnEaseInOutBack.Left := ComputeInterpolationInt(@EaseInOutBack, 0, LWidth2, Li, LBack);
      btnEaseOutInBack.Left := ComputeInterpolationInt(@EaseOutInBack, 0, LWidth2, Li, LBack);

      btnEaseInBounce.Left := ComputeInterpolationInt(@EaseInBounce, 0, LWidth, Li, LBack);
      btnEaseOutBounce.Left := ComputeInterpolationInt(@EaseOutBounce, 0, LWidth, Li, LBack);
      btnEaseInOutBounce.Left := ComputeInterpolationInt(@EaseInOutBounce, 0, LWidth2, Li, LBack);
      btnEaseOutInBounce.Left := ComputeInterpolationInt(@EaseOutInBounce, 0, LWidth2, Li, LBack);

      sleep(10);
      Repaint;
      Application.ProcessMessages;
    end;
  tbarWait.Enabled := True;
end;

procedure TMainForm.btnEaseInQuadClick(Sender: TObject);
// *** fonctions IN ***
var
  Li, LWidth: Integer;
  LBack: Boolean;
begin
  tbarWait.Enabled := False;
  LWidth := pnlLeft.Width - btnLinear2.Width;
  for LBack := False to True do
    for Li := 1 to fDuration do
    begin
      btnLinear.Left := ComputeInterpolationInt(@EaseNone, 0, LWidth, Li, LBack);
      btnEaseInQuad.Left := ComputeInterpolationInt(@EaseInQuad, 0, LWidth, Li, LBack);
      btnEaseInCubic.Left := ComputeInterpolationInt(@EaseInCubic, 0, LWidth, Li, LBack);
      btnEaseInQuart.Left := ComputeInterpolationInt(@EaseInQuart, 0, LWidth, Li, LBack);
      btnEaseInQuint.Left := ComputeInterpolationInt(@EaseInQuint, 0, LWidth, Li, LBack);
      btnEaseInSine.Left := ComputeInterpolationInt(@EaseInSine, 0, LWidth, Li, LBack);
      btnEaseInExpo.Left := ComputeInterpolationInt(@EaseInExpo, 0, LWidth, Li, LBack);
      btnEaseInCirc.Left := ComputeInterpolationInt(@EaseInCirc, 0, LWidth, Li, LBack);
      btnEaseInElastic.Left := ComputeInterpolationInt(@EaseInElastic, 0, LWidth, Li, LBack);
      btnEaseInBack.Left := ComputeInterpolationInt(@EaseInBack, 0, LWidth, Li, LBack);
      btnEaseInBounce.Left := ComputeInterpolationInt(@EaseInBounce, 0, LWidth, Li, LBack);
      sleep(10);
      pnlLeft.Repaint;
      Application.ProcessMessages;
    end;
  tbarWait.Enabled := True;
end;

procedure TMainForm.btnEaseInOutQuadClick(Sender: TObject);
// *** fonctions INOUT ***
var
  Li, LWidth2: Integer;
  LBack: Boolean;
begin
  tbarWait.Enabled := False;
  LWidth2 := pnlRight.Width - btnLinear.Width;
  for LBack := False to True do
    for Li := 1 to fDuration do
    begin
      btnLinear2.Left := ComputeInterpolationInt(@EaseNone, 0, LWidth2, Li, LBack);
      btnEaseInOutQuad.Left := ComputeInterpolationInt(@EaseInOutQuad, 0, LWidth2, Li, LBack);
      btnEaseInOutCubic.Left := ComputeInterpolationInt(@EaseInOutCubic, 0, LWidth2, Li, LBack);
      btnEaseInOutQuart.Left := ComputeInterpolationInt(@EaseInOutQuart, 0, LWidth2, Li, LBack);
      btnEaseInOutQuint.Left := ComputeInterpolationInt(@EaseInOutQuint, 0, LWidth2, Li, LBack);
      btnEaseInOutSine.Left := ComputeInterpolationInt(@EaseInOutSine, 0, LWidth2, Li, LBack);
      btnEaseInOutExpo.Left := ComputeInterpolationInt(@EaseInOutExpo, 0, LWidth2, Li, LBack);
      btnEaseInOutCirc.Left := ComputeInterpolationInt(@EaseInOutCirc, 0, LWidth2, Li, LBack);
      btnEaseInOutElastic.Left := ComputeInterpolationInt(@EaseInOutElastic, 0, LWidth2, Li, LBack);
      btnEaseInOutBack.Left := ComputeInterpolationInt(@EaseInOutBack, 0, LWidth2, Li, LBack);
      btnEaseInOutBounce.Left := ComputeInterpolationInt(@EaseInOutBounce, 0, LWidth2, Li, LBack);
      sleep(10);
      pnlRight.Repaint;
      Application.ProcessMessages;
    end;
  tbarWait.Enabled := True;
end;

procedure TMainForm.btnEaseOutInQuadClick(Sender: TObject);
// *** fonctions OUTIN ***
var
  Li, LWidth2: Integer;
  LBack: Boolean;
begin
  tbarWait.Enabled := False;
  LWidth2 := pnlRight.Width - btnLinear.Width;
  for LBack := False to True do
    for Li := 1 to fDuration do
    begin
      btnLinear2.Left := ComputeInterpolationInt(@EaseNone, 0, LWidth2, Li, LBack);
      btnEaseOutInQuad.Left := ComputeInterpolationInt(@EaseOutInQuad, 0, LWidth2, Li, LBack);
      btnEaseOutInCubic.Left := ComputeInterpolationInt(@EaseOutInCubic, 0, LWidth2, Li, LBack);
      btnEaseOutInQuart.Left := ComputeInterpolationInt(@EaseOutInQuart, 0, LWidth2, Li, LBack);
      btnEaseOutInQuint.Left := ComputeInterpolationInt(@EaseOutInQuint, 0, LWidth2, Li, LBack);
      btnEaseOutInSine.Left := ComputeInterpolationInt(@EaseOutInSine, 0, LWidth2, Li, LBack);
      btnEaseOutInExpo.Left := ComputeInterpolationInt(@EaseOutInExpo, 0, LWidth2, Li, LBack);
      btnEaseOutInCirc.Left := ComputeInterpolationInt(@EaseOutInCirc, 0, LWidth2, Li, LBack);
      btnEaseOutInElastic.Left := ComputeInterpolationInt(@EaseOutInElastic, 0, LWidth2, Li, LBack);
      btnEaseOutInBack.Left := ComputeInterpolationInt(@EaseOutInBack, 0, LWidth2, Li, LBack);
      btnEaseOutInBounce.Left := ComputeInterpolationInt(@EaseOutInBounce, 0, LWidth2, Li, LBack);
      sleep(10);
      pnlRight.Repaint;
      Application.ProcessMessages;
    end;
  tbarWait.Enabled := True;
end;

procedure TMainForm.btnEaseOutQuadClick(Sender: TObject);
// *** fonctions OUT ***
var
  Li, LWidth: Integer;
  LBack: Boolean;
begin
  tbarWait.Enabled := False;
  LWidth := pnlLeft.Width - btnLinear2.Width;
  for LBack := False to True do
    for Li := 1 to fDuration do
    begin
      btnLinear.Left := ComputeInterpolationInt(@EaseNone, 0, LWidth, Li, LBack);

      btnEaseOutQuad.Left := ComputeInterpolationInt(@EaseOutQuad, 0, LWidth, Li, LBack);
      btnEaseOutCubic.Left := ComputeInterpolationInt(@EaseOutCubic, 0, LWidth, Li, LBack);
      btnEaseOutQuart.Left := ComputeInterpolationInt(@EaseOutQuart, 0, LWidth, Li, LBack);
      btnEaseOutQuint.Left := ComputeInterpolationInt(@EaseOutQuint, 0, LWidth, Li, LBack);
      btnEaseOutSine.Left := ComputeInterpolationInt(@EaseOutSine, 0, LWidth, Li, LBack);
      btnEaseOutExpo.Left := ComputeInterpolationInt(@EaseOutExpo, 0, LWidth, Li, LBack);
      btnEaseOutCirc.Left := ComputeInterpolationInt(@EaseOutCirc, 0, LWidth, Li, LBack);
      btnEaseOutElastic.Left := ComputeInterpolationInt(@EaseOutElastic, 0, LWidth, Li, LBack);
      btnEaseOutBack.Left := ComputeInterpolationInt(@EaseOutBack, 0, LWidth, Li, LBack);
      btnEaseOutBounce.Left := ComputeInterpolationInt(@EaseOutBounce, 0, LWidth, Li, LBack);
      sleep(10);
      pnlLeft.Repaint;
      Application.ProcessMessages;
    end;
  tbarWait.Enabled := True;
end;

procedure TMainForm.FormCreate(Sender: TObject);
// *** création de la fiche ***
begin
  Duration := C_DefaultDuration;
  tbarWait.Max := C_End;
  tbarWait.Position := fDuration;
end;

procedure TMainForm.tbarWaitChange(Sender: TObject);
// *** changement de la vitesse ***
begin
  Duration := tbarWait.Position;
end;

procedure TMainForm.SetDuration(AValue: Cardinal);
// *** détermination de la durée d'interpolation ***
begin
  if fDuration = AValue then
    Exit;
  fDuration := AValue;
end;

function TMainForm.ComputeInterpolation(AInter: TInterpolationFunction; AStart,
  AEnd, AStep: Single; ABack: Boolean): Single;
// *** calcul des interpolations ***
begin
  if AStep < 0 then
    Result := AStart
  else
  if AStep > Duration then
    Result := AEnd
  else
    Result := AInter(AStart, AEnd, fDuration, AStep);
  if ABack then
    Result := AEnd - Result;
end;

function TMainForm.ComputeInterpolationInt(AInter: TInterpolationFunction;
  AStart, AEnd, AStep: Integer; ABack: Boolean): Integer;
// *** interpolation entière ***
begin
  Result := Round(ComputeInterpolation(AInter, AStart, AEnd, AStep, ABack));
end;

end.

À l'exécution, le résultat est exactement le même que celui obtenu avec une inclusion dans la classe de la fiche principale du projet. Comme notre objectif était de créer des classes autonomes, nous n'irons pas plus loin dans l'exploration de cette voie qui privilégie la programmation structurée sans passer par des classes et des objets.

Si le titre de ce chapitre pouvait sembler ironique, la souplesse de Pascal qui accepte aussi bien une programmation séquentielle qu'une approche POO (Programmation Orientée Objet) permet à chacun de s'y retrouver et de ne pas forcément mettre en œuvre des outils lourds pour des résultats médiocres !

V. Conclusion▲

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