Bitmap/Bézier curves/Quadratic

Using the data storage type defined on this page for raster images, and the draw_line function defined in this one, draw a quadratic bezier curve (definition on Wikipedia).

Ada

<lang ada>procedure Quadratic_Bezier

         (  Picture    : in out Image;
            P1, P2, P3 : Point;
            Color      : Pixel;
            N          : Positive := 20
         )  is
  Points : array (0..N) of Point;

begin

  for I in Points'Range loop
     declare
        T : constant Float := Float (I) / Float (N);
        A : constant Float := (1.0 - T)**2;
        B : constant Float := 2.0 * T * (1.0 - T);
        C : constant Float := T**2;
     begin
        Points (I).X := Positive (A * Float (P1.X) + B * Float (P2.X) + C * Float (P3.X));
        Points (I).Y := Positive (A * Float (P1.Y) + B * Float (P2.Y) + C * Float (P3.Y));
     end;
  end loop;
  for I in Points'First..Points'Last - 1 loop
     Line (Picture, Points (I), Points (I + 1), Color);
  end loop;

end Quadratic_Bezier;</lang> The following test <lang ada> X : Image (1..16, 1..16); begin

  Fill (X, White);
  Quadratic_Bezier (X, (8, 2), (13, 8), (2, 15), Black);
  Print (X);</lang>

should produce;

BBC BASIC

Works with: BBC BASIC for Windows

<lang bbcbasic> Width% = 200

     Height% = 200
     
     REM Set window size:
     VDU 23,22,Width%;Height%;8,16,16,128
     
     REM Draw quadratic Bézier curve:
     PROCbezierquad(10,100, 250,270, 150,20, 20, 0,0,0)
     END
     
     DEF PROCbezierquad(x1,y1,x2,y2,x3,y3,n%,r%,g%,b%)
     LOCAL i%, t, t1, a, b, c, p{()}
     DIM p{(n%) x%,y%}
     
     FOR i% = 0 TO n%
       t = i% / n%
       t1 = 1 - t
       a = t1^2
       b = 2 * t * t1
       c = t^2
       p{(i%)}.x% = INT(a * x1 + b * x2 + c * x3 + 0.5)
       p{(i%)}.y% = INT(a * y1 + b * y2 + c * y3 + 0.5)
     NEXT
     
     FOR i% = 0 TO n%-1
       PROCbresenham(p{(i%)}.x%,p{(i%)}.y%,p{(i%+1)}.x%,p{(i%+1)}.y%, \
       \             r%,g%,b%)
     NEXT
     ENDPROC
     
     DEF PROCbresenham(x1%,y1%,x2%,y2%,r%,g%,b%)
     LOCAL dx%, dy%, sx%, sy%, e
     dx% = ABS(x2% - x1%) : sx% = SGN(x2% - x1%)
     dy% = ABS(y2% - y1%) : sy% = SGN(y2% - y1%)
     IF dx% < dy% e = dx% / 2 ELSE e = dy% / 2
     REPEAT
       PROCsetpixel(x1%,y1%,r%,g%,b%)
       IF x1% = x2% IF y1% = y2% EXIT REPEAT
       IF dx% > dy% THEN
         x1% += sx% : e -= dy% : IF e < 0 e += dx% : y1% += sy%
       ELSE
         y1% += sy% : e -= dx% : IF e < 0 e += dy% : x1% += sx%
       ENDIF
     UNTIL FALSE
     ENDPROC
     
     DEF PROCsetpixel(x%,y%,r%,g%,b%)
     COLOUR 1,r%,g%,b%
     GCOL 1
     LINE x%*2,y%*2,x%*2,y%*2
     ENDPROC</lang>

D

This solution uses two modules, from the Grayscale image and the Bresenham's line algorithm Tasks. <lang d>import grayscale_image, bitmap_bresenhams_line_algorithm;

struct Pt { int x, y; } // Signed.

void quadraticBezier(size_t nSegments=20, Color)

                   (Image!Color im, in Pt p1, in Pt p2, in Pt p3,
                    in Color color)

pure nothrow @nogc if (nSegments > 0) {

   Pt[nSegments + 1] points = void;

   foreach (immutable i, ref p; points) {
       immutable double t = i / double(nSegments),
                        a = (1.0 - t) ^^ 2,
                        b = 2.0 * t * (1.0 - t),
                        c = t ^^ 2;
       p = Pt(cast(typeof(Pt.x))(a * p1.x + b * p2.x + c * p3.x),
              cast(typeof(Pt.y))(a * p1.y + b * p2.y + c * p3.y));
   }

   foreach (immutable i, immutable p; points[0 .. $ - 1])
       im.drawLine(p.x, p.y, points[i + 1].x, points[i + 1].y, color);

}

void main() {

   auto im = new Image!Gray(20, 20);
   im.clear(Gray.white);
   im.quadraticBezier(Pt(1,10), Pt(25,27), Pt(15,2), Gray.black);
   im.textualShow();

}</lang>

Output:

....................
....................
...............#....
...............#....
...............#....
................#...
................#...
.................#..
.................#..
.................#..
.#...............#..
..##.............#..
....##...........#..
......#..........#..
.......#.........#..
........###......#..
...........######...
....................
....................
....................

FBSL

Windows' graphics origin is located at the bottom-left corner of device bitmap.

Translation of BBC BASIC using pure FBSL's built-in graphics functions: <lang qbasic>#DEFINE WM_LBUTTONDOWN 513

DEFINE WM_CLOSE 16

FBSLSETTEXT(ME, "Bezier Quadratic") FBSLSETFORMCOLOR(ME, RGB(0, 255, 255)) ' Cyan: persistent background color DRAWWIDTH(5) ' Adjust point size FBSL.GETDC(ME) ' Use volatile FBSL.GETDC below to avoid extra assignments

RESIZE(ME, 0, 0, 235, 235) CENTER(ME) SHOW(ME)

DIM Height AS INTEGER FBSL.GETCLIENTRECT(ME, 0, 0, 0, Height)

BEGIN EVENTS SELECT CASE CBMSG CASE WM_LBUTTONDOWN: BezierQuad(10, 100, 250, 270, 150, 20, 20) ' Draw CASE WM_CLOSE: FBSL.RELEASEDC(ME, FBSL.GETDC) ' Clean up END SELECT END EVENTS

SUB BezierQuad(x1, y1, x2, y2, x3, y3, n) TYPE POINTAPI x AS INTEGER y AS INTEGER END TYPE

DIM t, t1, a, b, c, p[n] AS POINTAPI

FOR DIM i = 0 TO n t = i / n: t1 = 1 - t a = t1 ^ 2 b = 2 * t * t1 c = t ^ 2 p[i].x = a * x1 + b * x2 + c * x3 + 0.5 p[i].y = Height - (a * y1 + b * y2 + c * y3 + 0.5) NEXT

FOR i = 0 TO n - 1 Bresenham(p[i].x, p[i].y, p[i + 1].x, p[i + 1].y) NEXT

SUB Bresenham(x0, y0, x1, y1) DIM dx = ABS(x0 - x1), sx = SGN(x0 - x1) DIM dy = ABS(y0 - y1), sy = SGN(y0 - y1) DIM tmp, er = IIF(dx > dy, dx, -dy) / 2

WHILE NOT (x0 = x1 ANDALSO y0 = y1) PSET(FBSL.GETDC, x0, y0, &HFF) ' Red: Windows stores colors in BGR order tmp = er IF tmp > -dx THEN: er = er - dy: x0 = x0 + sx: END IF IF tmp < +dy THEN: er = er + dx: y0 = y0 + sy: END IF WEND END SUB END SUB</lang> Output:

Factor

Some code is shared with the cubic bezier task, but I put it here again to make it simple (hoping the two version don't diverge) Same remark as with cubic bezier, the points could go into a sequence to simplify stack shuffling <lang factor>USING: arrays kernel locals math math.functions

rosettacode.raster.storage sequences ;

IN: rosettacode.raster.line

! This gives a function

(quadratic-bezier) ( P0 P1 P2 -- bezier )

   [ :> x 
       1 x - sq P0 n*v
       2 1 x - x * * P1 n*v
       x sq P2 n*v
       v+ v+ ] ; inline

! Same code from the cubic bezier task

t-interval ( x -- interval )

   [ iota ] keep 1 - [ / ] curry map ;

points-to-lines ( seq -- seq )

   dup rest [ 2array ] 2map ;

draw-lines ( {R,G,B} points image -- )

   [ [ first2 ] dip draw-line ] curry with each ;

bezier-lines ( {R,G,B} P0 P1 P2 image -- )

   100 t-interval P0 P1 P2 (quadratic-bezier) map
   points-to-lines
   {R,G,B} swap image draw-lines ;

</lang>

Fortran

Works with: Fortran version 90 and later

(This subroutine must be inside the RCImagePrimitive module, see here)

<lang fortran>subroutine quad_bezier(img, p1, p2, p3, color)

 type(rgbimage), intent(inout) :: img
 type(point), intent(in) :: p1, p2, p3
 type(rgb), intent(in) :: color

 integer :: i, j
 real :: pts(0:N_SEG,0:1), t, a, b, c, x, y

 do i = 0, N_SEG
    t = real(i) / real(N_SEG)
    a = (1.0 - t)**2.0
    b = 2.0 * t * (1.0 - t)
    c = t**2.0
    x = a * p1%x + b * p2%x + c * p3%x 
    y = a * p1%y + b * p2%y + c * p3%y 
    pts(i,0) = x
    pts(i,1) = y
 end do

 do i = 0, N_SEG-1
    j = i + 1
    call draw_line(img, point(pts(i,0), pts(i,1)), &
                   point(pts(j,0), pts(j,1)), color)
 end do

end subroutine quad_bezier</lang>

Go

Translation of: C

<lang go>package raster

const b2Seg = 20

func (b *Bitmap) Bézier2(x1, y1, x2, y2, x3, y3 int, p Pixel) {

   var px, py [b2Seg + 1]int
   fx1, fy1 := float64(x1), float64(y1)
   fx2, fy2 := float64(x2), float64(y2)
   fx3, fy3 := float64(x3), float64(y3)
   for i := range px {
       c := float64(i) / b2Seg
       a := 1 - c
       a, b, c := a*a, 2 * c * a, c*c
       px[i] = int(a*fx1 + b*fx2 + c*fx3)
       py[i] = int(a*fy1 + b*fy2 + c*fy3)
   }
   x0, y0 := px[0], py[0]
   for i := 1; i <= b2Seg; i++ {
       x1, y1 := px[i], py[i]
       b.Line(x0, y0, x1, y1, p)
       x0, y0 = x1, y1
   }

}

func (b *Bitmap) Bézier2Rgb(x1, y1, x2, y2, x3, y3 int, c Rgb) {

   b.Bézier2(x1, y1, x2, y2, x3, y3, c.Pixel())

}</lang> Demonstration program:

<lang go>package main

import (

   "fmt"
   "raster"

)

func main() {

   b := raster.NewBitmap(400, 300)
   b.FillRgb(0xdfffef)
   b.Bézier2Rgb(20, 150, 500, -100, 300, 280, raster.Rgb(0x3f8fef))
   if err := b.WritePpmFile("bez2.ppm"); err != nil {
       fmt.Println(err)
   }

}</lang>

Haskell

<lang haskell>{-# LANGUAGE

   FlexibleInstances, TypeSynonymInstances,
   ViewPatterns #-}

import Bitmap import Bitmap.Line import Control.Monad import Control.Monad.ST

type Point = (Double, Double) fromPixel (Pixel (x, y)) = (toEnum x, toEnum y) toPixel (x, y) = Pixel (round x, round y)

pmap :: (Double -> Double) -> Point -> Point pmap f (x, y) = (f x, f y)

onCoordinates :: (Double -> Double -> Double) -> Point -> Point -> Point onCoordinates f (xa, ya) (xb, yb) = (f xa xb, f ya yb)

instance Num Point where

   (+) = onCoordinates (+)
   (-) = onCoordinates (-)
   (*) = onCoordinates (*)
   negate = pmap negate
   abs = pmap abs
   signum = pmap signum
   fromInteger i = (i', i')
     where i' = fromInteger i

bézier :: Color c =>

   Image s c -> Pixel -> Pixel -> Pixel -> c -> Int ->
   ST s ()

bézier

 i
 (fromPixel -> p1) (fromPixel -> p2) (fromPixel -> p3)
 c samples =
   zipWithM_ f ts (tail ts)
 where ts = map (/ top) [0 .. top]
         where top = toEnum $ samples - 1
       curvePoint t =
           pt (t' ^^ 2) p1 +
           pt (2 * t * t') p2 +
           pt (t ^^ 2) p3
         where t' = 1 - t
               pt n p = pmap (*n) p
       f (curvePoint -> p1) (curvePoint -> p2) =
           line i (toPixel p1) (toPixel p2) c</lang>

J

See Cubic bezier curves for a generalized solution.

Mathematica

<lang Mathematica>pts = {{0, 0}, {1, -1}, {2, 1}}; Graphics[{BSplineCurve[pts], Green, Line[pts], Red, Point[pts]}]</lang>

MATLAB

Note: Store this function in a file named "bezierQuad.mat" in the @Bitmap folder for the Bitmap class defined here. <lang MATLAB> function bezierQuad(obj,pixel_0,pixel_1,pixel_2,color,varargin)

   if( isempty(varargin) )
       resolution = 20;
   else
       resolution = varargin{1};
   end

   %Calculate time axis
   time = (0:1/resolution:1)';
   timeMinus = 1-time;

   %The formula for the curve is expanded for clarity, the lack of
   %loops is because its calculation has been vectorized
   curve = (timeMinus.^2)*pixel_0; %First term of polynomial
   curve = curve + (2.*time.*timeMinus)*pixel_1; %second term of polynomial
   curve = curve + (time.^2)*pixel_2; %third term of polynomial

   curve = round(curve); %round each of the points to the nearest integer

   %connect each of the points in the curve with a line using the
   %Bresenham Line algorithm
   for i = (1:length(curve)-1)
       obj.bresenhamLine(curve(i,:),curve(i+1,:),color);
   end
   
   assignin('caller',inputname(1),obj); %saves the changes to the object

end </lang>

Sample usage: This will generate the image example for the Go solution. <lang MATLAB> >> img = Bitmap(400,300); >> img.fill([223 255 239]); >> img.bezierQuad([20 150],[500 -100],[300 280],[63 143 239],21); >> disp(img) </lang>

OCaml

<lang ocaml>let quad_bezier ~img ~color

       ~p1:(_x1, _y1)
       ~p2:(_x2, _y2)
       ~p3:(_x3, _y3) =
 let (x1, y1, x2, y2, x3, y3) =
   (float _x1, float _y1, float _x2, float _y2, float _x3, float _y3)
 in
 let bz t =
   let a = (1.0 -. t) ** 2.0
   and b = 2.0 *. t *. (1.0 -. t)
   and c = t ** 2.0
   in
   let x = a *. x1 +. b *. x2 +. c *. x3
   and y = a *. y1 +. b *. y2 +. c *. y3
   in
   (int_of_float x, int_of_float y)
 in
 let rec loop _t acc =
   if _t > 20 then acc else
   begin
     let t = (float _t) /. 20.0 in
     let x, y = bz t in
     loop (succ _t) ((x,y)::acc)
   end
 in
 let pts = loop 0 [] in

 (*
 (* draw only points *)
 List.iter (fun (x, y) -> put_pixel img color x y) pts;
 *)

 (* draw segments *)
 let line = draw_line ~img ~color in
 let by_pair li f =
   ignore (List.fold_left (fun prev x -> f prev x; x) (List.hd li) (List.tl li))
 in
 by_pair pts (fun p0 p1 -> line ~p0 ~p1);

</lang>

PicoLisp

This uses the 'brez' line drawing function from Bitmap/Bresenham's line algorithm#PicoLisp. <lang PicoLisp>(scl 6)

(de quadBezier (Img N X1 Y1 X2 Y2 X3 Y3)

  (let (R (* N N)  X X1  Y Y1  DX 0  DY 0)
     (for I N
        (let (J (- N I)  A (*/ 1.0 J J R)  B (*/ 2.0 I J R)  C (*/ 1.0 I I R))
           (brez Img X Y
              (setq DX (- (+ (*/ A X1 1.0) (*/ B X2 1.0) (*/ C X3 1.0)) X))
              (setq DY (- (+ (*/ A Y1 1.0) (*/ B Y2 1.0) (*/ C Y3 1.0)) Y)) )
           (inc 'X DX)
           (inc 'Y DY) ) ) ) )</lang>

Test: <lang PicoLisp>(let Img (make (do 200 (link (need 300 0)))) # Create image 300 x 200

  (quadBezier Img 12 20 100 300 -80 260 180)
  (out "img.pbm"                                  # Write to bitmap file
     (prinl "P1")
     (prinl 300 " " 200)
     (mapc prinl Img) ) )

(call 'display "img.pbm")</lang>

PureBasic

<lang PureBasic>Procedure quad_bezier(img, p1x, p1y, p2x, p2y, p3x, p3y, Color, n_seg)

 Protected i
 Protected.f T, t1, a, b, c, d
 Dim pts.POINT(n_seg)
 
 For i = 0 To n_seg
   T = i / n_seg
   t1 = 1.0 - T
   a = Pow(t1, 2)
   b = 2.0 * T * t1
   c = Pow(T, 2)
   pts(i)\x = a * p1x + b * p2x + c * p3x
   pts(i)\y = a * p1y + b * p2y + c * p3y
 Next
 
 StartDrawing(ImageOutput(img))
   FrontColor(Color)
   For i = 0 To n_seg - 1
     BresenhamLine(pts(i)\x, pts(i)\y, pts(i + 1)\x, pts(i + 1)\y)
   Next
 StopDrawing()

EndProcedure

Define w, h, img w = 200: h = 200: img = 1 CreateImage(img, w, h) ;img is internal id of the image

OpenWindow(0, 0, 0, w, h,"Bezier curve, quadratic", #PB_Window_SystemMenu) quad_bezier(1, 80,20, 130,80, 20,150, RGB(255, 255, 255), 20) ImageGadget(0, 0, 0, w, h, ImageID(1))

Define event Repeat

 event = WaitWindowEvent()

Until event = #PB_Event_CloseWindow </lang>

Python

See Cubic bezier curves#Python for a generalized solution.

R

See Cubic bezier curves#R for a generalized solution.

Racket

lang racket

(require racket/draw)

(define (draw-line dc p q)

 (match* (p q) [((list x y) (list s t)) (send dc draw-line x y s t)]))

(define (draw-lines dc ps)

 (void
  (for/fold ([p0 (first ps)]) ([p (rest ps)])
    (draw-line dc p0 p)
    p)))

(define (int t p q)

 (define ((int1 t) x0 x1) (+ (* (- 1 t) x0) (* t x1)))
 (map (int1 t) p q))

(define (bezier-points p0 p1 p2)

 (for/list ([t (in-range 0.0 1.0 (/ 1.0 20))])
   (int t (int t p0 p1) (int t p1 p2))))

(define bm (make-object bitmap% 17 17)) (define dc (new bitmap-dc% [bitmap bm])) (send dc set-smoothing 'unsmoothed) (send dc set-pen "red" 1 'solid) (draw-lines dc (bezier-points '(16 1) '(1 4) '(3 16))) bm </lang>

Ruby

See Cubic bezier curves#Ruby for a generalized solution.

Tcl

See Cubic bezier curves#Tcl for a generalized solution.

TI-89 BASIC

Note: This example does not use a user-defined image type, since that would be particularly impractical, but rather draws on the calculator's graph screen, which has essentially the same operations as an implementation of Basic bitmap storage would, except for being black-and-white.

<lang ti89b>Define cubic(p1,p2,p3,segs) = Prgm

 Local i,t,u,prev,pt
 0 → pt
 For i,1,segs+1
   (i-1.0)/segs → t   © Decimal to avoid slow exact arithetic
   (1-t) → u
   pt → prev
   u^2*p1 + 2*t*u*p2 + t^2*p3 → pt
   If i>1 Then
     PxlLine floor(prev[1,1]), floor(prev[1,2]), floor(pt[1,1]), floor(pt[1,2])
   EndIf
 EndFor

EndPrgm</lang>

Vedit macro language

This implementation uses de Casteljau's algorithm to recursively split the Bezier curve into two smaller segments until the segment is short enough to be approximated with a straight line. The advantage of this method is that only integer calculations are needed, and the most complex operations are addition and shift right. (I have used multiplication and division here for clarity.)

Constant recursion depth is used here. Recursion depth of 5 seems to give accurate enough result in most situations. In real world implementations, some adaptive method is often used to decide when to stop recursion.

<lang vedit>// Daw a Cubic bezier curve // #20, #30 = Start point // #21, #31 = Control point 1 // #22, #32 = Control point 2 // #23, #33 = end point // #40 = depth of recursion

CUBIC_BEZIER:

if (#40 > 0) {

   #24 = (#20+#21)/2;              #34 = (#30+#31)/2
   #26 = (#22+#23)/2;              #36 = (#32+#33)/2
   #27 = (#20+#21*2+#22)/4;        #37 = (#30+#31*2+#32)/4
   #28 = (#21+#22*2+#23)/4;        #38 = (#31+#32*2+#33)/4
   #29 = (#20+#21*3+#22*3+#23)/8;  #39 = (#30+#31*3+#32*3+#33)/8
   Num_Push(20,40)
       #21 = #24; #31 = #34    // control 1
       #22 = #27; #32 = #37    // control 2
       #23 = #29; #33 = #39    // end point
       #40--
       Call("CUBIC_BEZIER")    // Draw "left" part
   Num_Pop(20,40)
   Num_Push(20,40)
       #20 = #29; #30 = #39    // start point
       #21 = #28; #31 = #38    // control 1
       #22 = #26; #32 = #36    // control 2
       #40--
       Call("CUBIC_BEZIER")    // Draw "right" part
   Num_Pop(20,40)

} else {

   #1=#20; #2=#30; #3=#23; #4=#33
   Call("DRAW_LINE")

} return</lang>

XPL0

<lang XPL0>include c:\cxpl\codes; \intrinsic 'code' declarations

proc Bezier(P0, P1, P2); \Draw quadratic Bezier curve real P0, P1, P2; def Segments = 8; int I; real T, A, B, C, X, Y; [Move(fix(P0(0)), fix(P0(1))); for I:= 1 to Segments do

       [T:= float(I)/float(Segments);
       A:= sq(1.-T);
       B:= 2.*T*(1.-T);
       C:= sq(T);
       X:= A*P0(0) + B*P1(0) + C*P2(0);
       Y:= A*P0(1) + B*P1(1) + C*P2(1);
       Line(fix(X), fix(Y), $00FFFF);          \cyan line segments
       ];

Point(fix(P0(0)), fix(P0(1)), $FF0000); \red control points Point(fix(P1(0)), fix(P1(1)), $FF0000); Point(fix(P2(0)), fix(P2(1)), $FF0000); ];

[SetVid($112); \set 640x480x24 video graphics Bezier([0., 0.], [80., 100.], [160., 20.]); if ChIn(1) then []; \wait for keystroke SetVid(3); \restore normal text display ]</lang>

zkl

Uses the PPM class from http://rosettacode.org/wiki/Bitmap/Bresenham%27s_line_algorithm#zkl

Add this to the PPM class: <lang zkl> fcn qBezier(p0x,p0y, p1x,p1y, p2x,p2y, rgb, numPts=500){

     numPts.pump(Void,'wrap(t){ // B(t)
     	 t=t.toFloat()/numPts; t1:=(1.0 - t);

a:=t1*t1; b:=t*t1*2; c:=t*t; x:=a*p0x + b*p1x + c*p2x + 0.5; y:=a*p0y + b*p1y + c*p2y + 0.5; __sSet(rgb,x,y);

     });
  }</lang>

Doesn't use line segments, they don't seem like an improvement. <lang zkl>bitmap:=PPM(300,300,0xff|ff|ff); bitmap.qBezier(10,200, 250,30, 150,280, 0); bitmap.write(File("foo.ppm","wb"));</lang>

Output:

Same as the BBC BASIC image: