Rovinn kivky

 

Uivatelsk souadn soustava

 

Jak ji vme, DELPHI pracuje s tzv. svtovmi souadnicemi, kter jsou na pipojenm obrzku vyznaeny mode. Prce v takov soustav je pro uivatele znan nepjemn. Vytvome si proto tzv. uivatelskou souadnou soustavu, kter je vyznaena erven. Dlku osy  (interval ) jako i dlku osy  (interval ) budeme moci mnit. V obrzku je  resp.  dlka uivatelsk jednotky na ose   resp.  ve svtovch souadnicch. Zejm je

 

;                                                       (1)

 

Souadnice uivatelskho potku O  ve svtovch souadnicch jsou pak

 

 ;                                                      (2)

 

Je-li pak  libovoln bod na kreslic ploe,  jeho souadnice v uivatelsk soustav a  jeho souadnice ve svtov soustav, pak zejm plat

 

;                                         (3)

 

Realizujme nyn uivatelskou souadnou soustavu a vykresleme trojhelnk s danmi vrcholy. Na formul jsme opt umstili objekt Image, proceduru, kter trojhelnk vykresluje, jsme nazvali Triangle a budeme ji aktivovat OnClick. Po tchto zkladnch pravch za ns DELPHI zape do unity, kterou jsme pejmenovali na Graph2D,  nsledujc ern (event. modr) dky (viz dle). Vimnme si nejdve dku TForm1 = class(TForm). Ten se v unit objev automaticky, i kdy do formule nezaadme dnou komponentu. Deklaruje typ TForm1 jako ddice typu TForm. Znamen to, e TForm1 automaticky pebr vlastnosti typu TForm a navc meme definovat jeho dal vlastnosti. Nco za ns opt provede DELPHI. Zaadme-li do formule komponentu Image, objev se nov promnn ddice - Image1, kter je typu TImage. Spojme-li s udlost OnClick proceduru, kterou v Object Inspectoru nazveme Triangle, pak, jak ji vme, DELPHI tuto proceduru zape na konec zdrojovho textu [procedure TForm1.Triangle (Sender: TObject);] a automaticky ns na ni navede, abychom ji mohli vypsat.

 

unit Graph2D;

interface

uses

Windows, Messages, SysUtils, Classes,

Graphics, Controls, Forms, Dialogs,ExtCtrls;

type

   TPoint = array [1..2] of Double;

   TForm1 = class(TForm)

       Image1: TImage;

   procedure Triangle(Sender:TObject); 

                                          

private

          { Private declarations }

public

          { Public declarations }

procedure Scale(x1,x2,y1,y2:Double);

function   XCoor(x:Double):integer;

function    YCoor(y:Double):integer;

end;

 

Vimnte si vak, e se hlavika tto procedury automaticky zapsala tak jako nov vlastnost typu TForm. Tmto zpsobem meme i my sami obohacovat typ TForm o nov promnn, procedury a funkce, nejlpe do odstavce Public (zde budou tyto vlastnosti v ppad poteby dostupn i z jinch jednotek). Nyn se tedy dostvme k ervenmu textu, kter musme dopsat sami a kterm realizujeme uivatelskou souadnou soustavu.

 

var  Form1                    :TForm1;

     Unit_X,Unit_Y,x1,x2,y1,y2:Double;

     O                        :TPoint;

 

implementation

 

{$R *.DFM}

procedure TForm1.Scale(x1,x2,y1,y2:Double);

  begin

    Unit_X:= Image1.Width/(x2-x1);

    Unit_Y:=Image1.Height/(y2-y1);

    O[1]:= -x1*Unit_X;O[2]:=y2*Unit_Y;

  end;

 

 function  TForm1.XCoor(x:Double):integer;

    begin  XCoor:=trunc(O[1]+Unit_X*x);end;

 

  function  TForm1.YCoor(y:Double):integer;

     begin YCoor:=trunc(O[2]-Unit_Y*y);end;

 

procedure TForm1.Triangle(Sender: TObject);

  begin

  end;

end.

 

Protoe budeme chtt pracovat s body v rovin, deklarujeme v sti Interface  typ TPoint jako uspodanou dvojici relnch sel (pedefinovali jsme tedy typ TPoint, kter je v DELPHI deklarovn. Dvody budou zejm pozdji). Dle do odstavce public typu TForm1 zapeme hlaviku procedury Scale(x1,x2,y1,y2:Double). Parametry tto procedury jsou uivatelsk rozmry kreslic plochy tak, jak bylo popsno ve. Procedura bude realizovat vpoty dle vzorc (1), (2). Dle dv funkce Function XCoor (x:Double):Integer; Function YCoor(y:Double):Integer. Funkce jsou zvltn typy procedur, kter maj prv jeden vstup (procedury mohou mt vstup vce, nebo tak dn). Tyto funkce budou reln uivatelsk souadnice x resp. y pepotvat na svtov celoseln souadnice podle vzorc (3). Dle je teba deklarovat promnn, kter budou pouvny v cel jednotce (globln promnn). Jsou to reln sla Unit_X a Unit_Y  - dlky uivatelskch jednotek na jedn. osch,  dle bod O (uivatelsk potek) - promnn nmi definovanho typu TPoint a konen x1,x2,y1,y2 - uivatelsk rozmry kreslic plochy. Do sti Implemantation za zpis {$R *.DFM} (co je direktiva pekladae, kterou zapsalo DELPHI) musme nyn tyto procedury a funkce specifikovat. Vimnte si, e zde je v hlavikch ped nzvem specifikovn typ,  kter procedura i funkce obohacuje procedure TForm1.Scale(x1,x2,y1,y2:Double);].

 

Pklad 1: Ve uveden st kdu by samozejm jet trojhelnk nevykreslila. Doplme jej tak, abychom na vstupu trojhelnk dostali. V eenm pkladu jsou podobn jako ve uveden ti procedury a funkce doplnny dal:

 

          procedure PutPixel(X:TPoint;Color:DWord);

          procedure MoveTo(X:TPoint);

          procedure LineTo(X:TPoint;Color:DWord);

          procedure Line(X,Y:TPoint;Color:DWord);

          procedure XAxis(x1,x2,y:Double;Color:DWord);

          procedure YAxis(y1,y2,x:Double;Color:DWord);

 

Procedury PutPixel, MoveTo a LineTo funguj analogicky, jako metody obsaen v TCanvas ve standardnm DELPHI, pracuj vak v uivatelskch souadnicch (v argumentu jsou promnn typu TPoint) a dle promnn definujc barvu, kterou se m kreslit. Dle je zde procedura Line, kter spoj dva uivatelsk body X, Y sekou. Konen procedury XAxis resp. YAxis sestrojuj souadn osy, pesnji eeno vodorovn resp. svisl pmky. Nap. u XAxis uruj parametry x1,x2 x ov souadnice krajnch bod, parametr y pak y ovou souadnici prseku s osou y (YAxis analogicky). Vechny tyto procedury vyuvaj standardnch metod objektu TCanvas. Napklad procedura LineTo vypad takto:

 

procedure TForm1.LineTo(X:TPoint;Color:DWord);

  begin

   With Image1.Canvas do

     Begin

      Pen.Color:=Color;LineTo(XCoor(X[1]),YCoor(X[2]));

     end;

  end;

 

V tchto procedurch tedy nen vyuit pstup do bitmapy pes obrazov dky tak, jak bylo uvedeno v matematickch podrobnostech pedchoz kapitoly. V dalm textu budeme prozatm pouvat osmibitov barvy, resp. peddeklarovan konstanty - nap. slRed, slGreen atd., kter byly popsny v matematickch podrobnostech pedchoz kapitoly, a vlastn proceduru Line pro seku, kter je popsna v matematickch podrobnostech tto kapitoly  .

 

Jsou-li k dispozici ve uveden procedury, pak trojhelnk vykreslme velmi jednodue:

 

procedure TForm1.Triangle(Sender: TObject);

var A,B,C:TPoint;

begin

    A[1]:=-2;A[2]:=6; B[1]:= 4;B[2]:=-1; C[1]:= 5;C[2]:=5;             {definice vrchol}

    x1:=-4;x2:=7;y1:=-3;y2:=8;

    Scale(x1,x2,y1,y2);        {definice uivatelsk kreslic plochy}

    Line(A,B,slRed);Line(B,C,slRed);Line(A,C,slRed);          {vlastn konstrukce}

end;

 

Zde najdete kompletn              zdrojov kd         a zde               spustiteln kd

 

Pklad 2:  Pedchoz pklad vylepme o cejchovn souadnch os, kter obstarvaj procedury XGauge (x1,x2,y:Double;Color:Byte); resp. YGauge (y1,y2,x:Double; Color: Byte). Podle velikosti mtka procedury zvol vhodnou velikost jednotky na popis, k popisu pipoj vzdlenost sousednch fous. Funguj nsledujcm zpsobem (jako pklad uveme popis osy x):

 

procedure TForm1.XScale(x1,x2,y:Double;Color:DWord);

var   AxisUnit       :Double;

      Text           :String;

      Lx,Exponent    :Integer;

begin

Image1.Canvas.Font.Color:=Color;

AxisUnit:=1E30;Lx:=0;Exponent:=30;

While (x2-x1)/AxisUnit<1 do

begin

  AxisUnit:=AxisUnit/10;Exponent:=Exponent-1;

end;                                                         {nalezen vhodn velikosti jednotky}

if x1<0                                                                              {nalezen nejmen popisky}

     then Repeat Lx:=Lx-1 Until AxisUnit*(Lx-1)<x1                

     else Repeat Lx:=Lx+1 Until AxisUnit*(Lx+1)>x1;

With Image1.Canvas do

  repeat

      MoveTo(XCoor(AxisUnit*Lx),YCoor(y)-3);     {cejchovn osy}

      LineTo(XCoor(AxisUnit*Lx),YCoor(y)+3);

Str(Lx,Text);TextOut(XCoor(AxisUnit*Lx),YCoor(y)+5,Text);Lx:=Lx+1;                                                                                                                                {popisky}

  until AxisUnit*Lx>x2;

Str(Exponent,Text);                {funkce Str pevd selnou hodnotu na etzec}          

Image1.Canvas.TextOut(XCoor(x2)+15,YCoor(y)+5,'*10');

                                                                                                          {jednotka a exponent}  

Image1.Canvas.TextOut(XCoor(x2)+35,YCoor(y),Text);                

end;

 

Zde najdete kompletn              zdrojov kd         a zde               spustiteln kd

 

Pedchoz pklady neumouj mnit parametry za bhu programu. Tento nedostatek napravme v nsledujcm pkladu.

 

Pklad 3: K ovldn naeho formule Form1 zalome cel nov formul. V menu File klikneme na New Form. Objev se nm nov formul Form2 spolu s novou jednotkou Unit2. Tomuto formuli zmnme titulek na Control Panel. a umstme na nj ti tlatka, kter vybereme ze strnky Standard vcestrnkov lity. Tmto tlatkm zmnme titulky na Start, Clear Image a Exit. Dle nkolik objekt Static Text, tento typ najdeme v lit Additional, jejich titulky budou Triangle, Scale, dal budou slouit pro popis vrchol - A=, B=, C= a uivatelskch rozmr kreslic plochy x, y. Dle zde umstme Check Box (lita Standard) s titulkem Show Axses, s jeho pomoc bude uivatel vybrat monost zobrazen souadnch os. O tom, zda se souadn osy maj i nemaj zobrazit, rozhoduje vlastnost Checked typu Boolean.  Konen pidme potebn poet objekt  Edit (opt v lit Standard), do kterch bude uivatel zadvat hodnoty. Zmnme jejich jmna (vlastnost Name v Object Inspectoru) na EditA1, EditA2, , resp EditX1, EditX2, a Text, tak, aby pi sputn programu tato okna obsahovala vchoz parametry. N formul vypad tak, jak vidme na piloenm obrzku. Procedury tkajc se rovinn uivatelsk souadn soustavy jsou v samostatn jednotce - pvodn Unit1 pejmenujeme na Graph2D pomoc File/SaveAs. Do tto jednotky jsme tak umstili objekt typu TImage, kter jsme nazvali Draw2D a vechny konstrukce probhaj v tomto objektu. Nap. procedura pro mazn kreslic plochy kresl pes celou tuto plochu bl obdlnk:

 

procedure TDraw2D.ClearImage;

begin

  with Image1.Canvas do

    begin Pen.Color:=clWhite;Brush.Color:=clWhite;end;

    with Image1 do Canvas.Rectangle(0,0,Width-1,Height-1);

end;

 

V Control Panelu nejdve zadme ten parametr z Edit-oknek, a to procedurou Setting:

 

Procedure TControl_Panel.Setting;

var Code:Integer;

begin

   With Draw2D do

   begin

      Val(EditA1.Text,A[1],Code);

      Val(EditA2.Text,A[2],Code);

      Val(EditY2.Text,y2,Code);

      Scale(x1,x2,y1,y2);

  end;

end;

 

Tlatka v Control Panelu spojme pes udlost OnClick poad s procedurami Triangle, ClearImage a Exit:

 

procedure TControl_Panel.Triangle(Sender: TObject);

begin

    With Draw2D do

      begin

      ClearImage;Setting;

      if CheckBox1.Checked then CoorSystem(slRed);

      Line(A,B,slBlue);Line(B,C,slBlue);Line(A,C,slBlue);

     end;

end;

 

procedure TControl_Panel.ClearImage(Sender: TObject);

begin  Draw2D.ClearImage;end;

 

procedure TControl_Panel.Exit(Sender: TObject);

begin Halt; end;

 

V pkladu je navc eeno zadvn vrchol trojhelnka my

 

Zde najdete kompletn              zdrojov kd         a zde               spustiteln kd

 

Kivky typu

 

Z hlediska konstrukce se jedn o nejjednodu kivky. Lze je konstruovat jako lomen ry sloen z seek dle pipojenho obrzku:

 

Pklad 1: Sestrojme proceduru Explicit_Curve, kter sestroj graf funkce   (viz pipojen vpis). Nejdve opt sestavme ovldac panel ze znmch komponent (v panelu NumberOfSegments bude uivatel zadvat poet segment, z kterch m bt kivka sestrojena. Stejn jako v pedchoz kapitole sestrojme uivatelskou kreslic plochu, souadn osy a definujeme barvu sestrojovan kivky. Vidme, e funkce   je definovna uvnit procedury.  Vlastn konstrukce spov v nastaven kroku hx a v postupn konstrukci seek :

 

 

Procedure  TControl_Panel.Explicit_Curve(Sender:TObject);

var x,hx:Double;

        A,B :TPoint;

function f(x:Double):Double;

begin if x=0 then f:=0

                           else f:=x+10*x*x*sin(1/x); end;

begin

  With Draw2D do                                                            {Uivatelsk kreslic plocha}

  begin

    if CheckBox1.Checked then

begin                                   {souadn osy a jejich popis}

 XScale(x1,x2,0,clRed);             {sestroj stupnici rovnobnou s osou x, resp. y.}

 YScale(y1,y2,0,clRed);     { Pro uren barvy je pouita standardn konstanta }

end;                    { kvli snazmu popisu os pomoc procedury TextOut}

hx:=(x2-x1)/NumberOfSegments;x:=x1;A[1]:=x;A[2]:=f(x);    

Repeat                            {konstrukce kivky jako lomen ry}

  B[1]:=x+hx;B[2]:=f(x+hx); Line(A,B,slBlue);A:=B;x:=x+hx

Until x>x2

end;

end;

 

 

Zde najdete kompletn              zdrojov kd         a zde               spustiteln kd

 

V ppad kivek bv vhodn posloupnost bod, ktermi kivka prochz, nejdve celou spotat a teprve potom proloit kivku.  Delphi m sice metodu PolyLine, kter umouje proloit polygon posloupnost bod. Tato metoda vak pracuje jen se svtovmi souadnicemi, kdeto my pracujeme ji v souadnicch uivatelskch. Tuto metodu si tedy opt pedefinujeme.

 

Pklad 2: Pedefinujme metodu PolyLine a funkci z p. 1. sestrojme pomoc tto nov metody. Nae metoda bude pracovat s posloupnost bod, musme tedy nejdve deklarovat nov typ, nejlpe hned za TPoint:

 

TPoint            = array [1..2] of Double;

TArrayOfPoints    =  array [0..800] of TPoint;

 

TPoint je tedy jeden bod - uspodan dvojice sel, TArrayOfPoints je posloupnost nejve osmi set bod. Do promnn tohoto typu se budou ukldat body, ktermi m prochzet nae kivka. S nstroji, kter ji mme k dispozici, je jej konstrukce jednoduch:

procedure Draw2D.PolyLine(Q:TArrayOfPoints; n:Word;Color:Byte);

var i:Word;

begin

   MoveTo(Q[1]);

   for i:=1 to n do LineTo(Q[i],Color);

end;

 

Vimnme si nejdve hlaviky: Q je posloupnost bod, n definuje poet bod, kter maj bt spojeny, Color pak barvu, kterou m bt kivka sestrojena. Procedura nejdve umst pero do prvnho bodu pomoc na metody MoveTo, do nsledujcch pak kresl seky na metodou LineTo. Samozejm je teba upravit i nai proceduru Explicit_Curve (uvdm pouze st kdu, kter se od pedchozho pkladu li).

 

procedure TControl_Panel.Explicit_Curve(Sender:TObject);

var x,hx:Double;     Q     :TArrayOfPoints;     i      :Word;

function f.....

begin

   .......

  hx:=(x2-x1)/NumberOfSegments;i:=0;x:=x1;

  Repeat

   Q[i,1]:=x;Q[i,2]:=f(x);       {konstrukce kivky jako lomen ry}

   x:=x+hx;i:=succ(i);

  Until x>x2+hx;

  PolyLine(Q,i,Color);

end;

 

Zde najdete kompletn              zdrojov kd         a zde               spustiteln kd

 

Je zde jedna ponkud nepjemn vc: V promnn typu TArrayOfPoints nm pibyl jeden index, kter udv poad bodu v posloupnosti. Ten je teba uvdt vdy jako prvn. Zpis Q [i,2] tedy znamen, e se jedn o druhou souadnici  i - tho bodu v posloupnosti Q. Nai metodu PolyLine pouijeme i pi konstrukci kivek zadanch parametricky a polrn. Zde bude ponkud odlin pouze zpsob naplnn pole Q.

 

Z hlediska uivatele zstv jin nepjemnost vc: z ovldacho panelu nen mon mnit zadn funkce.

 

V pkladu 3, jeho                 zdrojov kd je zde                 a          spustiteln kd je zde,

 

 

je tento nedostatek odstrann. etezec zadvan do okna EditFunction je vyslovn v jednotce Graph2D_256 funkc Calc(Text:String; ErrorReport:String). V zadvanm etzci je mono zadvat vechny bn funkce vetn cyklometrickch, k oznaen promnnch lze pout x, y, t, k. ErrorReport vrac chybov hlen. Mme-li k dispozici tuto proceduru, odpad zadvn funkce f ve zdrojovm kdu. Promnnou f deklarujeme jako etzec a pkaz A[i,2]:=f(x) nahradme pkazem A[i,2]:=Calc(f, ErrorReport). etzec ErrorReport je pi korektnm vpotu sprvn. Pi chybnm vyhodnocen je vrcen popis chyby, eho je mono vyut k chybovmu hlen a peruen procedury, nap:

 

A[i,2]:=Calc(f, ErrorReport)

if ErrorReport<>

     then begin

         ShowMessage(ErrorReport);

         exit;

         end;

 

Samotn vyslen etezce (funkce Calc) vak nen jednoduchou zleitost, bezprostedn nesouvis pmo s grafickmi problmy a nebudeme se jm proto podrobnji zabvat.

 

Dve, ne postoupme dle, je teba uinit jet jednu poznmku. Barvy zadvme v Delphi pomoc peddeklarovanch konstant (clRed, clYellow). Tchto konstant je vak k dispozici pouze estnct pestoe jsou tyiadvacetibitov. Mme tedy k dispozici 2563 barev, kter lze zadat selnou hodnotou. Tyto barvy pracuj v systmu RGB popsanmu dve. Prozatm jsme pouvali jednotku Graph2D_256.pas, kter pracuje v reimu 256 barev, kter jsou deklarovny pomoc osmibitovch konstant - slRed, slYellow(scanlineRed) - Pstup k ostatnm barvm v prostoru RGB zajiuje jednotka Graph2D.pas, kter s barvami zachz ponkud jinak. Grafickm procedurm je teba barvu zadvat pomoc sloek Red, Green Blue. Dal odlinost spov v tom, e tyto procedury pracuj bu v objektu Canvas, anebo pmo v bitmap. Procedury pracujc z Canvasem jsou odlieny zkratkou cv nap. procedura cvPolyLine(Q,i,255,0,0) sestroj polygon napotan v poli Q.  Procedury pistupujc pmo do bitmapy tuto pedponu nemaj, ped jejich pouitm je vak teba bitmapu vytvoit. To lze provst metodou InitImage, kter je v Graph2D k dispozici. Bitmapu vytvome nejlpe na udlost onActivate ovldacho panelu. Sestrojen grafu funkce pmm pstupem do bitmapy pak vypad takto:

 

procedure TControl_Panel.FormActivate(Sender:TObject);

begin

   Draw2D.InitImage(Image1.Width,Image1.Height);

end;

 

procedure TControl_Panel.Explicit_Curve(Sender:TObject);

var x,hx:Double;     Q     :TArrayOfPoints;     i      :Word;

function f.....

begin

   .......

  hx:=(x2-x1)/NumberOfSegments;i:=0;x:=x1;

  Repeat

   Q[i,1]:=x;Q[i,2]:=f(x);       {konstrukce kivky jako lomen ry}

   x:=x+hx;i:=succ(i);

  Until x>x2+hx;

  PolyLine(Q,i,Color);

end;

 

Odkazy na kdy pracujc v True Color reimu budeme uvdt takto:

 

True Color:                  zdrojov kd                                      spustiteln kd

 

 

Kivky zadan parametricky a polrn

 

Konstrukce tchto kivek je velmi podobn, ukeme si ji na konkrtnm pkladu:

 

Pklad 1: Sestrojte kivku urenou obecn parametrickmi  rovnicemi ;  (jedn se o tzv. Lissajousovy kivky, kter opisuje kyvadlo rozkmitan ve dvou navzjem kolmch rovinch a amplitudami ,  a hlovmi frekvencemi ;). N kol e procedura Param_Curve. Nejprve jsou deklarovny konstanty t1, t2, kter definuj rozsah parametru. Msto promnnch x, hx je deklarovn parametr t a jeho krok ht. Na deklaraci parametrickch rovnic ve zdrojovm kdu musme msto funkce pout proceduru, nebo jako vstup potebujeme dv promnn - x, y.

 

procedure TControl_Panel.Param_Curve (Sender: TObject);

var i:Word;

procedure Lissajous(t:double;var x,y:double);

  begin

        x:=3*sin(3*t);  y:=3*cos(5*t);

     end;

begin

With Draw2D do

begin

  ClearImage(slWhite);          {vyplnn plochy blou barvou}

  Scale(x1,x2,y1,y2);

  ht:=(t2-t1)/NumberOfSegments;

  t:=t1;i:=0;

  Repeat

     Lissajous(t,Q[i,1],Q[i,2]);

     t:=t+ht;i:=succ(i);

  Until t>t2+ht;

  PolyLine(Q,i,slBlue);

  end;

end;

 

Vstupn parametry od vstupnch je teba v hlavice procedury oddlit klovm slovem var. Nsleduj parametrick rovnice (s konkrtnmi volbami za a, b, j, y). Ve vlastn procedue jsem se omezil opt vhradn na konstrukci kivky. Nejdve je poteba nastavit krok parametru ht a jeho poten hodnotu. V cyklu pak naplujeme pole Q  parametrickmi rovnicemi, dokud parametr t  krokem ht neprobhne cel interval t1;t2ñ (indexem i  potme body v posloupnosti Q). Nakonec kivku vykreslme opt pomoc PolyLine.

 

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Najdeme-li dv takov strany, pak metodou plen intervalu zjistme prseky A B s obvodem s pesnost na jeden fyzick pixel. Tmito body pak prolome seku, m vlastn kivku v tomto tverci interpolujeme. Pi realizaci programu musme rozliit, zda plme svislou i vodorovnou stranu obdlnka. Uveme pklad vodorovnho plen:

 

Procedure HorizHalf(x1,x2,y:Double; var x:Double);

{vstupn parametry v hlavice maj stejn vznam, jako pi konstrukci osy x, vstupn parametr udv x- ovou souadnici nalezenho prseku.}  

var i      :Integer;

        a,b,c:Double;                                                                                                                       

begin

  a:=x1;b:=x2;                                                             

  For i:=1 to Trunc(Step) do                                        

  begin                     {plen intervalu a vbr intervalu s prsekem}

     c:=(a+b)/2;if f(a,y)*f(c,y)<0 then b:=c else a:=c;                                                         

  end;

  x:=c;                                                                                          {nastaven vstupn hodnoty}

end;

 

Dle je teba zjistit, jak (a zda vbec) kivka testovan tverec protn. Kivka  protn testovan tverec zejm tehdy,  kdy v jeho vrcholech m funkce  rzn znamnka. Ohodnome vrcholy tverce  postupn hodnotami ; ; ; , a to prv tehdy, kdy v pslunm vrcholu je , jinak vrcholu piame nulu. Cel tverec tak me bt ohodnocen hodnotami   - z hlediska na konstrukce existuje prv estnct zpsob, jak me sestrojovan kivka testovan tverec protnat. tverec na obrzku vlevo je ohodnocen slem , tverec vpravo pak slem . Souet tchto ohodnocen je  a je zejm, e postup konstrukce bude v obou ppadech stejn - v obou ppadech je teba hledat prseky na sekch , . Akoli je tedy celkov poet ppad, kter je teba eit, trnct (tverce s ohodnocenmi 0 resp.15 kivka neprotn), program sta vtvit pouze na sedm vtv. Zajmav vyjden ohodnocen je ve dvojkov soustav, kdy bod je ohodnocen pouze nulou, i jednikou. Nap. pro lev obrzek mme:

       

 

Konen je teba si uvdomit, e v ppad  resp.  nelze jednoznan rozhodnout, jak interpolan seky vlastn sestrojit (viz monosti 5, 10 v pipojen sekvenci kdu - obrzek na dal stran). V tom ppad je teba vybrat jen jednu z obou monost.

 

begin

hx:=Step*(x2-x1)/Image1.Width;      {po obvyklm peten hodnot z Control}

hy:=Step*(y2-y1)/Image1.Height;   {Panelu a nastaven mtka nastavme }

x:=x1;           {kroky hx, hy tak,aby pesn odpovdaly uivatelskm rozmrm}

Repeat                         {fyzickho pixelu a zanme prochzet s}

y:=y1;                                                                     {Promnn h bude urovat, na kterch}

Repeat                         {stranch tverce byly nalezeny prseky}

  h:=0;

  if f(x,y)      >0 then h:=h+8;

  if f(x+hx,y)   >0 then h:=h+4;

  if f(x+hx,y+hy)>0 then h:=h+2;

  if f(x   ,y+hy)>0 then h:=h+1;

  Case h of

    1,14:begin

           K[1]:=x;VertHalf(y,y+hy,x,K[2]);

           HorizHalf(x,x+hx,y+hy,L[1]);L[2]:=y+hy;

           Line(K,L,Red,Green,Blue);

         end;

    5,10:begin

           K[1]:=x   ;VertHalf(y,y+hy,x+hx,K[2]);

           HorizHalf(x,x+hx,y,L[1]);L[2]:=y;

           Line(K,L,Red,Green,Blue);

           K[1]:=x   ;VertHalf(y,y+hy,x,K[2]);

           HorizHalf(x,x+hx,y+hy,L[1]);L[2]:=y+hy;

           Line(K,L,Red,Green,Blue);

          end;

    ..          {ostatn ppady analogicky}

      end;

  y:=y+hy;

Until y>y2;

x:=x+hx;

until x>x2;

 

          

 

V pkladu 5 je pedchoz algoritmus zpracovn.

 

Zde najdete kompletn  zdrojov kd               a zde                spustiteln kd

 

Podobnm zpsobem lze konstruovat plochy uren rovnic  v prostoru. Zde se podobnm zpsoben hledaj prseky plochy s hranami krychle a plocha se interpoluje pomoc trojhelnk. Rznch monost vak nen trnct resp. sedm, ale 254 resp. 127. Tmto problmem se budeme zabvat dle