Bitmap/Bézier curves/Cubic
< Bitmap
You are encouraged to solve this task according to the task description, using any language you may know.
Using the data storage type defined on this page for raster images, and the draw_line function defined in this other one, draw a cubic bezier curve (definition on Wikipedia).
11l
T Colour = BVec3
V black = Colour(0, 0, 0)
V white = Colour(255, 255, 255)
T Bitmap
Int width, height
Colour background
[[Colour]] map
F (width = 40, height = 40, background = white)
assert(width > 0 & height > 0)
.width = width
.height = height
.background = background
.map = (0 .< height).map(h -> (0 .< @width).map(w -> @@background))
F fillrect(x, y, width, height, colour = black)
assert(x >= 0 & y >= 0 & width > 0 & height > 0)
L(h) 0 .< height
L(w) 0 .< width
.map[y + h][x + w] = colour
F chardisplay()
V txt = .map.map(row -> row.map(bit -> (I bit == @@.background {‘ ’} E ‘@’)).join(‘’))
txt = txt.map(row -> ‘|’row‘|’)
txt.insert(0, ‘+’(‘-’ * .width)‘+’)
txt.append(‘+’(‘-’ * .width)‘+’)
print(reversed(txt).join("\n"))
F set(x, y, colour = black)
.map[y][x] = colour
F get(x, y)
R .map[y][x]
F line(x0, y0, x1, y1)
‘Bresenham's line algorithm’
V dx = abs(x1 - x0)
V dy = abs(y1 - y0)
V (x, y) = (x0, y0)
V sx = I x0 > x1 {-1} E 1
V sy = I y0 > y1 {-1} E 1
I dx > dy
V err = dx / 2.0
L x != x1
.set(x, y)
err -= dy
I err < 0
y += sy
err += dx
x += sx
E
V err = dy / 2.0
L y != y1
.set(x, y)
err -= dx
I err < 0
x += sx
err += dy
y += sy
.set(x, y)
F cubicbezier(x0, y0, x1, y1, x2, y2, x3, y3, n = 20)
[(Int, Int)] pts
L(i) 0 .. n
V t = Float(i) / n
V a = (1. - t) ^ 3
V b = 3. * t * (1. - t) ^ 2
V c = 3.0 * t ^ 2 * (1.0 - t)
V d = t ^ 3
V x = Int(a * x0 + b * x1 + c * x2 + d * x3)
V y = Int(a * y0 + b * y1 + c * y2 + d * y3)
pts.append((x, y))
L(i) 0 .< n
.line(pts[i][0], pts[i][1], pts[i + 1][0], pts[i + 1][1])
V bitmap = Bitmap(17, 17)
bitmap.cubicbezier(16, 1, 1, 4, 3, 16, 15, 11)
bitmap.chardisplay()- Output:
+-----------------+ | | | | | | | | | @@@@ | | @@@ @@@ | | @ | | @ | | @ | | @ | | @ | | @ | | @ | | @ | | @@@@ | | @@@@| | | +-----------------+
Action!
INCLUDE "H6:RGBLINE.ACT" ;from task Bresenham's line algorithm
INCLUDE "H6:REALMATH.ACT"
RGB black,yellow,violet,blue
TYPE IntPoint=[INT x,y]
PROC CubicBezier(RgbImage POINTER img
IntPoint POINTER p1,p2,p3,p4 RGB POINTER col)
INT i,n=[20],prevX,prevY,nextX,nextY
REAL one,two,three,ri,rn,rt,ra,rb,rc,rd,tmp1,tmp2,tmp3
REAL x1,y1,x2,y2,x3,y3,x4,y4
IntToReal(p1.x,x1) IntToReal(p1.y,y1)
IntToReal(p2.x,x2) IntToReal(p2.y,y2)
IntToReal(p3.x,x3) IntToReal(p3.y,y3)
IntToReal(p4.x,x4) IntToReal(p4.y,y4)
IntToReal(1,one) IntToReal(2,two)
IntToReal(3,three) IntToReal(n,rn)
FOR i=0 TO n
DO
prevX=nextX prevY=nextY
IntToReal(i,ri)
RealDiv(ri,rn,rt) ;t=i/n
RealSub(one,rt,tmp1) ;tmp1=1-t
RealMult(tmp1,tmp1,tmp2) ;tmp2=(1-t)^2
RealMult(tmp2,tmp1,ra) ;a=(1-t)^3
RealMult(three,rt,tmp2) ;tmp2=3*t
RealMult(tmp1,tmp1,tmp3) ;tmp3=(1-t)^2
RealMult(tmp2,tmp3,rb) ;b=3*t*(1-t)^2
RealMult(three,rt,tmp2) ;tmp2=3*t
RealMult(rt,tmp1,tmp3) ;tmp3=t*(1-t)
RealMult(tmp2,tmp3,rc) ;c=3*t^2*(1-t)
RealMult(rt,rt,tmp2) ;tmp2=t^2
RealMult(tmp2,rt,rd) ;d=t^3
RealMult(ra,x1,tmp1) ;tmp1=a*x1
RealMult(rb,x2,tmp2) ;tmp2=b*x2
RealAdd(tmp1,tmp2,tmp3) ;tmp3=a*x1+b*x2
RealMult(rc,x3,tmp1) ;tmp1=c*x3
RealAdd(tmp3,tmp1,tmp2) ;tmp2=a*x1+b*x2+c*x3
RealMult(rd,x4,tmp1) ;tmp1=d*x4
RealAdd(tmp2,tmp1,tmp3) ;tmp3=a*x1+b*x2+c*x3+d*x4
nextX=Round(tmp3)
RealMult(ra,y1,tmp1) ;tmp1=a*y1
RealMult(rb,y2,tmp2) ;tmp2=b*y2
RealAdd(tmp1,tmp2,tmp3) ;tmp3=a*y1+b*y2
RealMult(rc,y3,tmp1) ;tmp1=c*y3
RealAdd(tmp3,tmp1,tmp2) ;tmp2=a*y1+b*y2+c*y3
RealMult(rd,y4,tmp1) ;tmp1=d*y4
RealAdd(tmp2,tmp1,tmp3) ;tmp3=a*y1+b*y2+c*y3+d*y4
nextY=Round(tmp3)
IF i>0 THEN
RgbLine(img,prevX,prevY,nextX,nextY,col)
FI
OD
RETURN
PROC DrawImage(RgbImage POINTER img BYTE x,y)
RGB POINTER ptr
BYTE i,j
ptr=img.data
FOR j=0 TO img.h-1
DO
FOR i=0 TO img.w-1
DO
IF RgbEqual(ptr,yellow) THEN
Color=1
ELSEIF RgbEqual(ptr,violet) THEN
Color=2
ELSEIF RgbEqual(ptr,blue) THEN
Color=3
ELSE
Color=0
FI
Plot(x+i,y+j)
ptr==+RGBSIZE
OD
OD
RETURN
PROC Main()
RgbImage img
BYTE CH=$02FC,width=[70],height=[40]
BYTE ARRAY ptr(8400)
IntPoint p1,p2,p3,p4
Graphics(7+16)
SetColor(0,13,12) ;yellow
SetColor(1,4,8) ;violet
SetColor(2,8,6) ;blue
SetColor(4,0,0) ;black
RgbBlack(black)
RgbYellow(yellow)
RgbViolet(violet)
RgbBlue(blue)
InitRgbImage(img,width,height,ptr)
FillRgbImage(img,black)
p1.x=0 p1.y=3
p2.x=10 p2.y=39
p3.x=69 p3.y=31
p4.x=40 p4.y=8
RgbLine(img,p1.x,p1.y,p2.x,p2.y,blue)
RgbLine(img,p2.x,p2.y,p3.x,p3.y,blue)
RgbLine(img,p3.x,p3.y,p4.x,p4.y,blue)
CubicBezier(img,p1,p2,p3,p4,yellow)
SetRgbPixel(img,p1.x,p1.y,violet)
SetRgbPixel(img,p2.x,p2.y,violet)
SetRgbPixel(img,p3.x,p3.y,violet)
SetRgbPixel(img,p4.x,p4.y,violet)
DrawImage(img,(160-width)/2,(96-height)/2)
DO UNTIL CH#$FF OD
CH=$FF
RETURN- Output:
Screenshot from Atari 8-bit computer
Ada
procedure Cubic_Bezier
( Picture : in out Image;
P1, P2, P3, P4 : 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)**3;
B : constant Float := 3.0 * T * (1.0 - T)**2;
C : constant Float := 3.0 * T**2 * (1.0 - T);
D : constant Float := T**3;
begin
Points (I).X := Positive (A * Float (P1.X) + B * Float (P2.X) + C * Float (P3.X) + D * Float (P4.X));
Points (I).Y := Positive (A * Float (P1.Y) + B * Float (P2.Y) + C * Float (P3.Y) + D * Float (P4.Y));
end;
end loop;
for I in Points'First..Points'Last - 1 loop
Line (Picture, Points (I), Points (I + 1), Color);
end loop;
end Cubic_Bezier;
The following test
X : Image (1..16, 1..16);
begin
Fill (X, White);
Cubic_Bezier (X, (16, 1), (1, 4), (3, 16), (15, 11), Black);
Print (X);
should produce output:
HH
HH HH
H H
H H
H H
H H
H H
H H
H H
H H
H H
H
ALGOL 68
File: prelude/Bitmap/Bezier_curves/Cubic.a68
# -*- coding: utf-8 -*- #
cubic bezier OF class image :=
( REF IMAGE picture,
POINT p1, p2, p3, p4,
PIXEL color,
UNION(INT, VOID) in n
)VOID:
BEGIN
INT n = (in n|(INT n):n|20); # default 20 #
[0:n]POINT points;
FOR i FROM LWB points TO UPB points DO
REAL t = i / n,
a = (1 - t)**3,
b = 3 * t * (1 - t)**2,
c = 3 * t**2 * (1 - t),
d = t**3;
x OF points [i] := ENTIER (0.5 + a * x OF p1 + b * x OF p2 + c * x OF p3 + d * x OF p4);
y OF points [i] := ENTIER (0.5 + a * y OF p1 + b * y OF p2 + c * y OF p3 + d * y OF p4)
OD;
FOR i FROM LWB points TO UPB points - 1 DO
(line OF class image)(picture, points (i), points (i + 1), color)
OD
END # cubic bezier #;
SKIPFile: test/Bitmap/Bezier_curves/Cubic.a68
#!/usr/bin/a68g --script #
# -*- coding: utf-8 -*- #
PR READ "prelude/Bitmap.a68" PR; # c.f. [[rc:Bitmap]] #
PR READ "prelude/Bitmap/Bresenhams_line_algorithm.a68" PR; # c.f. [[rc:Bitmap/Bresenhams_line_algorithm]] #
PR READ "prelude/Bitmap/Bezier_curves/Cubic.a68" PR;
# The following test #
test:(
REF IMAGE x = INIT LOC[16,16]PIXEL;
(fill OF class image)(x, (white OF class image));
(cubic bezier OF class image)(x, (16, 1), (1, 4), (3, 16), (15, 11), (black OF class image), EMPTY);
(print OF class image) (x)
)Output:
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffff000000000000ffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffff000000000000ffffffffffff000000000000ffffffffffffffffffffffffffffff ffffffffffffffffffffffff000000ffffffffffffffffffffffffffffffffffff000000ffffffffffffffffffffffff ffffffffffffffffffffffff000000ffffffffffffffffffffffffffffffffffff000000ffffffffffffffffffffffff ffffffffffffffffff000000ffffffffffffffffffffffffffffffffffffffffff000000ffffffffffffffffffffffff ffffffffffff000000ffffffffffffffffffffffffffffffffffffffffffffffff000000ffffffffffffffffffffffff ffffff000000ffffffffffffffffffffffffffffffffffffffffffffffffffffff000000ffffffffffffffffffffffff ffffff000000ffffffffffffffffffffffffffffffffffffffffffffffffffffff000000ffffffffffffffffffffffff ffffff000000ffffffffffffffffffffffffffffffffffffffffffffffffffffff000000ffffffffffffffffffffffff ffffff000000ffffffffffffffffffffffffffffffffffffffffffffffffffffff000000ffffffffffffffffffffffff 000000ffffffffffffffffffffffffffffffffffffffffffffffffffffff000000ffffffffffffffffffffffffffffff 000000ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ATS
BBC BASIC
Width% = 200
Height% = 200
REM Set window size:
VDU 23,22,Width%;Height%;8,16,16,128
REM Draw cubic Bézier curve:
PROCbeziercubic(160,150, 10,120, 30,0, 150,50, 20, 0,0,0)
END
DEF PROCbeziercubic(x1,y1,x2,y2,x3,y3,x4,y4,n%,r%,g%,b%)
LOCAL i%, t, t1, a, b, c, d, p{()}
DIM p{(n%) x%,y%}
FOR i% = 0 TO n%
t = i% / n%
t1 = 1 - t
a = t1^3
b = 3 * t * t1^2
c = 3 * t^2 * t1
d = t^3
p{(i%)}.x% = INT(a * x1 + b * x2 + c * x3 + d * x4 + 0.5)
p{(i%)}.y% = INT(a * y1 + b * y2 + c * y3 + d * y4 + 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
C
"Interface" imglib.h.
void cubic_bezier(
image img,
unsigned int x1, unsigned int y1,
unsigned int x2, unsigned int y2,
unsigned int x3, unsigned int y3,
unsigned int x4, unsigned int y4,
color_component r,
color_component g,
color_component b );
#include <math.h>
/* number of segments for the curve */
#define N_SEG 20
#define plot(x, y) put_pixel_clip(img, x, y, r, g, b)
#define line(x0,y0,x1,y1) draw_line(img, x0,y0,x1,y1, r,g,b)
void cubic_bezier(
image img,
unsigned int x1, unsigned int y1,
unsigned int x2, unsigned int y2,
unsigned int x3, unsigned int y3,
unsigned int x4, unsigned int y4,
color_component r,
color_component g,
color_component b )
{
unsigned int i;
double pts[N_SEG+1][2];
for (i=0; i <= N_SEG; ++i)
{
double t = (double)i / (double)N_SEG;
double a = pow((1.0 - t), 3.0);
double b = 3.0 * t * pow((1.0 - t), 2.0);
double c = 3.0 * pow(t, 2.0) * (1.0 - t);
double d = pow(t, 3.0);
double x = a * x1 + b * x2 + c * x3 + d * x4;
double y = a * y1 + b * y2 + c * y3 + d * y4;
pts[i][0] = x;
pts[i][1] = y;
}
#if 0
/* draw only points */
for (i=0; i <= N_SEG; ++i)
{
plot( pts[i][0],
pts[i][1] );
}
#else
/* draw segments */
for (i=0; i < N_SEG; ++i)
{
int j = i + 1;
line( pts[i][0], pts[i][1],
pts[j][0], pts[j][1] );
}
#endif
}
#undef plot
#undef line
D
This solution uses two modules, from the Grayscale image and Bresenham's line algorithm Tasks.
import grayscale_image, bitmap_bresenhams_line_algorithm;
struct Pt { int x, y; } // Signed.
void cubicBezier(size_t nSegments=20, Color)
(Image!Color im,
in Pt p1, in Pt p2, in Pt p3, in Pt p4,
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) ^^ 3,
b = 3.0 * t * (1.0 - t) ^^ 2,
c = 3.0 * t ^^ 2 * (1.0 - t),
d = t ^^ 3;
alias T = typeof(Pt.x);
p = Pt(cast(T)(a * p1.x + b * p2.x + c * p3.x + d * p4.x),
cast(T)(a * p1.y + b * p2.y + c * p3.y + d * p4.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(17, 17);
im.clear(Gray.white);
im.cubicBezier(Pt(16, 1), Pt(1, 4), Pt(3, 16), Pt(15, 11),
Gray.black);
im.textualShow();
}
- Output:
................. .............#### .........####.... ........#........ .......#......... ......#.......... ......#.......... .....#........... .....#........... .....#........... .....#........... ......##....####. ........####..... ................. ................. ................. .................
Delphi
{This code would normally be stored in a separate library, but they presented here for clarity}
type T2DVector=packed record
X,Y: double;
end;
type T2DLine = packed record
P1,P2: T2DVector;
end;
type T2DVectorArray = array of T2DVector;
function MakeVector2D(const X,Y: double): T2DVector;
{Create 2D Vector from X and Y}
begin
Result.X:=X;
Result.Y:=Y;
end;
procedure DoCubicSplineLine(Steps: Integer; L1,L2: T2DLine; ClearArray: boolean; var PG: T2DVectorArray);
{Do cubic Bezier spline between L1.P1 and L2.P1 }
{L1.P1 = Point1, L1.P2 = Control1, L2.P1=Control2, L2.P2 = Point2}
var P: Integer;
var V: T2DVector;
var T: double;
var A,B,C,D,E,F,G,H : double;
begin
if ClearArray then SetLength(PG,0);
A := L2.P2.X - (3 * L2.P1.X) + (3 * L1.P2.X) - L1.P1.X;
B := (3 * L2.P1.X) - (6 * L1.P2.X) + (3 * L1.P1.X);
C := (3 * L1.P2.X) - (3 * L1.P1.X);
D := L1.P1.X;
E := L2.P2.Y - (3 * L2.P1.Y) + (3 * L1.P2.Y) - L1.P1.Y;
F := (3 * L2.P1.Y) - (6 * L1.P2.Y) + (3 * L1.P1.Y);
G := (3 * L1.P2.Y) - (3 * L1.P1.Y);
H := L1.P1.Y;
for P:=0 to Steps-1 do
begin
T :=P / (Steps-1);
V.X := (((A * T) + B) * T + C) * T + D;
V.Y := (((E * T) + F) * T + G) * T + H;
SetLength(PG,Length(PG)+1);
PG[High(PG)]:=V;
end;
end;
procedure MarkPoint(Image: TImage; P: TPoint);
begin
Image.Canvas.Pen.Width:=2;
Image.Canvas.Pen.Color:=clRed;
Image.Canvas.MoveTo(Trunc(P.X-3),Trunc(P.Y-3));
Image.Canvas.LineTo(Trunc(P.X+3),Trunc(P.Y+3));
Image.Canvas.MoveTo(Trunc(P.X+3),Trunc(P.Y-3));
Image.Canvas.LineTo(Trunc(P.X-3),Trunc(P.Y+3));
end;
procedure DrawControlPoint(Image: TImage; L: T2DLine);
var P1,P2: TPoint;
begin
Image.Canvas.Pen.Width:=1;
Image.Canvas.Pen.Color:=clBlue;
P1:=Point(Trunc(L.P1.X),Trunc(L.P1.Y));
P2:=Point(Trunc(L.P2.X),Trunc(L.P2.Y));
Image.Canvas.MoveTo(P1.X,P1.Y);
Image.Canvas.LineTo(P2.X,P2.Y);
Image.Canvas.Pen.Color:=clRed;
MarkPoint(Image,P2);
end;
procedure DrawOneSpline(Image: TImage; L1,L2: T2DLine);
var PG: T2DVectorArray;
var I: integer;
begin
DoCubicSplineLine(20,L1,L2,True,PG);
DrawControlPoint(Image,L1);
DrawControlPoint(Image,L2);
Image.Canvas.Pen.Width:=2;
Image.Canvas.Pen.Color:=clRed;
Image.Canvas.MoveTo(Trunc(PG[0].X),Trunc(PG[0].Y));
for I:=1 to High(PG) do
Image.Canvas.LineTo(Trunc(PG[I].X),Trunc(PG[I].Y));
end;
procedure ShowBezierCurve(Image: TImage);
var L1,L2: T2DLine;
begin
L1.P1:=MakeVector2D(50,50);
L1.P2:=MakeVector2D(250,50);
L2.P1:=MakeVector2D(50,250);
L2.P2:=MakeVector2D(250,250);
DrawOneSpline(Image, L1,L2);
L1.P1:=MakeVector2D(250,250);
L1.P2:=MakeVector2D(450,250);
L2.P1:=MakeVector2D(250,50);
L2.P2:=MakeVector2D(450,50);
DrawOneSpline(Image, L1,L2);
Image.Invalidate;
end;
- Output:
Elapsed Time: 1.171 ms.
F#
/// Uses Vector<float> from Microsoft.FSharp.Math (in F# PowerPack)
module CubicBezier
/// Create bezier curve from p1 to p4, using the control points p2, p3
/// Returns the requested number of segments
let cubic_bezier (p1:vector) (p2:vector) (p3:vector) (p4:vector) segments =
[0 .. segments - 1]
|> List.map(fun i ->
let t = float i / float segments
let a = (1. - t) ** 3.
let b = 3. * t * ((1. - t) ** 2.)
let c = 3. * (t ** 2.) * (1. - t)
let d = t ** 3.
let x = a * p1.[0] + b * p2.[0] + c * p3.[0] + d * p4.[0]
let y = a * p1.[1] + b * p2.[1] + c * p3.[1] + d * p4.[1]
vector [x; y])
// For rendering..
let drawPoints points (canvas:System.Windows.Controls.Canvas) =
let addLineToScreen (v1:vector) (v2:vector) =
canvas.Children.Add(new System.Windows.Shapes.Line(X1 = v1.[0],
Y1 = -v1.[1],
X2 = v2.[0],
Y2 = -v2.[1],
StrokeThickness = 2.)) |> ignore
let renderPoint (previous:vector) (current:vector) =
addLineToScreen previous current
current
points |> List.fold renderPoint points.Head
Factor
The points should probably be in a sequence...
USING: arrays kernel locals math math.functions
rosettacode.raster.storage sequences ;
IN: rosettacode.raster.line
! this gives a function
:: (cubic-bezier) ( P0 P1 P2 P3 -- bezier )
[ :> x
1 x - 3 ^ P0 n*v
1 x - sq 3 * x * P1 n*v
1 x - 3 * x sq * P2 n*v
x 3 ^ P3 n*v
v+ v+ v+ ] ; inline
! gives an interval of x from 0 to 1 to map the bezier function
: t-interval ( x -- interval )
[ iota ] keep 1 - [ / ] curry map ;
! turns a list of points into the list of lines between them
: 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 P3 image -- )
! 100 is an arbitrary value.. could be given as a parameter..
100 t-interval P0 P1 P2 P3 (cubic-bezier) map
points-to-lines
{R,G,B} swap image draw-lines ;
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:
#DEFINE WM_LBUTTONDOWN 513
#DEFINE WM_CLOSE 16
FBSLSETTEXT(ME, "Bezier Cubic")
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: BezierCube(160, 150, 10, 120, 30, 0, 150, 50, 20) ' Draw
CASE WM_CLOSE: FBSL.RELEASEDC(ME, FBSL.GETDC) ' Clean up
END SELECT
END EVENTS
SUB BezierCube(x1, y1, x2, y2, x3, y3, x4, y4, n)
TYPE POINTAPI
x AS INTEGER
y AS INTEGER
END TYPE
DIM t, t1, a, b, c, d, p[n] AS POINTAPI
FOR DIM i = 0 TO n
t = i / n: t1 = 1 - t
a = t1 ^ 3
b = 3 * t * t1 ^ 2
c = 3 * t ^ 2 * t1
d = t ^ 3
p[i].x = a * x1 + b * x2 + c * x3 + d * x4 + 0.5
p[i].y = Height - (a * y1 + b * y2 + c * y3 + d * y4 + 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
Output: 
Fortran
This subroutine should go inside the RCImagePrimitive module (see Bresenham's line algorithm)
subroutine cubic_bezier(img, p1, p2, p3, p4, color)
type(rgbimage), intent(inout) :: img
type(point), intent(in) :: p1, p2, p3, p4
type(rgb), intent(in) :: color
integer :: i, j
real :: pts(0:N_SEG,0:1), t, a, b, c, d, x, y
do i = 0, N_SEG
t = real(i) / real(N_SEG)
a = (1.0 - t)**3.0
b = 3.0 * t * (1.0 - t)**2
c = 3.0 * (1.0 - t) * t**2
d = t**3.0
x = a * p1%x + b * p2%x + c * p3%x + d * p4%x
y = a * p1%y + b * p2%y + c * p3%y + d * p4%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 cubic_bezier
FreeBASIC
' version 01-11-2016
' compile with: fbc -s console
' translation from Bitmap/Bresenham's line algorithm C entry
Sub Br_line(x0 As Integer, y0 As Integer, x1 As Integer, y1 As Integer, _
Col As UInteger = &HFFFFFF)
Dim As Integer dx = Abs(x1 - x0), dy = Abs(y1 - y0)
Dim As Integer sx = IIf(x0 < x1, 1, -1)
Dim As Integer sy = IIf(y0 < y1, 1, -1)
Dim As Integer er = IIf(dx > dy, dx, -dy) \ 2, e2
Do
PSet(x0, y0), col
If (x0 = x1) And (y0 = y1) Then Exit Do
e2 = er
If e2 > -dx Then Er -= dy : x0 += sx
If e2 < dy Then Er += dx : y0 += sy
Loop
End Sub
' Bitmap/Bézier curves/Cubic BBC BASIC entry
Sub beziercubic(x1 As Double, y1 As Double, x2 As Double, y2 As Double, _
x3 As Double, y3 As Double, x4 As Double, y4 As Double, _
n As ULong, col As UInteger = &HFFFFFF)
Type point_
x As Integer
y As Integer
End Type
Dim As ULong i
Dim As Double t, t1, a, b, c, d
Dim As point_ p(n)
For i = 0 To n
t = i / n
t1 = 1 - t
a = t1 ^ 3
b = t * t1 * t1 * 3
c = t * t * t1 * 3
d = t ^ 3
p(i).x = Int(a * x1 + b * x2 + c * x3 + d * x4 + .5)
p(i).y = Int(a * y1 + b * y2 + c * y3 + d * y4 + .5)
Next
For i = 0 To n -1
Br_line(p(i).x, p(i).y, p(i +1).x, p(i +1).y, col)
Next
End Sub
' ------=< MAIN >=------
ScreenRes 250,250,32 ' 0,0 in top left corner
beziercubic(160, 150, 10, 120, 30, 0, 150, 50, 20)
' empty keyboard buffer
While Inkey <> "" : Wend
Print : Print "hit any key to end program"
Sleep
End
Go
package raster
const b3Seg = 30
func (b *Bitmap) Bézier3(x1, y1, x2, y2, x3, y3, x4, y4 int, p Pixel) {
var px, py [b3Seg + 1]int
fx1, fy1 := float64(x1), float64(y1)
fx2, fy2 := float64(x2), float64(y2)
fx3, fy3 := float64(x3), float64(y3)
fx4, fy4 := float64(x4), float64(y4)
for i := range px {
d := float64(i) / b3Seg
a := 1 - d
b, c := a * a, d * d
a, b, c, d = a*b, 3*b*d, 3*a*c, c*d
px[i] = int(a*fx1 + b*fx2 + c*fx3 + d*fx4)
py[i] = int(a*fy1 + b*fy2 + c*fy3 + d*fy4)
}
x0, y0 := px[0], py[0]
for i := 1; i <= b3Seg; i++ {
x1, y1 := px[i], py[i]
b.Line(x0, y0, x1, y1, p)
x0, y0 = x1, y1
}
}
func (b *Bitmap) Bézier3Rgb(x1, y1, x2, y2, x3, y3, x4, y4 int, c Rgb) {
b.Bézier3(x1, y1, x2, y2, x3, y3, x4, y4, c.Pixel())
}
Demonstration program:
package main
import (
"fmt"
"raster"
)
func main() {
b := raster.NewBitmap(400, 300)
b.FillRgb(0xffefbf)
b.Bézier3Rgb(20, 200, 700, 50, -300, 50, 380, 150, raster.Rgb(0x3f8fef))
if err := b.WritePpmFile("bez3.ppm"); err != nil {
fmt.Println(err)
}
}
J
Solution:
See the Bernstein Polynomials essay on the J Wiki.
Uses code from Basic bitmap storage, Bresenham's line algorithm and Midpoint circle algorithm.
require 'numeric'
bik=: 2 : '((*&(u!v))@(^&u * ^&(v-u)@-.))'
basiscoeffs=: <: 4 : 'x bik y t. i.>:y'"0~ i.
linearcomb=: basiscoeffs@#@[
evalBernstein=: ([ +/ .* linearcomb) p. ] NB. evaluate Bernstein Polynomial (general)
NB.*getBezierPoints v Returns points for bezier curve given control points (y)
NB. eg: getBezierPoints controlpoints
NB. y is: y0 x0, y1 x1, y2 x2 ...
getBezierPoints=: monad define
ctrlpts=. (/: {:"1) _2]\ y NB. sort ctrlpts for increasing x
xvals=. ({: ,~ {. + +:@:i.@<.@-:@-~/) ({:"1) 0 _1{ctrlpts
tvals=. ((] - {.) % ({: - {.)) xvals
xvals ,.~ ({."1 ctrlpts) evalBernstein tvals
)
NB.*drawBezier v Draws bezier curve defined by (x) on image (y)
NB. eg: (42 40 10 30 186 269 26 187;255 0 0) drawBezier myimg
NB. x is: 2-item list of boxed (controlpoints) ; (color)
drawBezier=: (1&{:: ;~ 2 ]\ [: roundint@getBezierPoints"1 (0&{::))@[ drawLines ]
Example usage:
myimg=: 0 0 255 makeRGB 300 300
]randomctrlpts=: ,3 2 ?@$ }:$ myimg NB. 3 control points - quadratic
]randomctrlpts=: ,4 2 ?@$ }:$ myimg NB. 4 control points - cubic
myimg=: ((2 ,.~ _2]\randomctrlpts);255 0 255) drawCircles myimg NB. draw control points
viewRGB (randomctrlpts; 255 255 0) drawBezier myimg NB. display image with bezier line
Java
Using the BasicBitmapStorage class from the Bitmap task to produce a runnable program.
import java.awt.Color;
import java.awt.Graphics;
import java.awt.Image;
import java.awt.Point;
import java.awt.image.BufferedImage;
import java.awt.image.RenderedImage;
import java.io.File;
import java.io.IOException;
import java.util.ArrayList;
import java.util.List;
import javax.imageio.ImageIO;
public final class BezierCubic {
public static void main(String[] args) throws IOException {
final int width = 200;
final int height = 200;
BasicBitmapStorage bitmap = new BasicBitmapStorage(width, height);
bitmap.fill(Color.YELLOW);
Point point1 = new Point(0, 150);
Point point2 = new Point(30, 50);
Point point3 = new Point(120, 130);
Point point4 = new Point(160, 30);
bitmap.cubicBezier(point1, point2, point3, point4, Color.BLACK, 20);
File bezierFile = new File("CubicBezierJava.jpg");
ImageIO.write((RenderedImage) bitmap.getImage(), "jpg", bezierFile);
}
}
final class BasicBitmapStorage {
public BasicBitmapStorage(int width, int height) {
image = new BufferedImage(width, height, BufferedImage.TYPE_INT_RGB);
}
public void fill(Color color) {
Graphics graphics = image.getGraphics();
graphics.setColor(color);
graphics.fillRect(0, 0, image.getWidth(), image.getHeight());
}
public Color getPixel(int x, int y) {
return new Color(image.getRGB(x, y));
}
public void setPixel(int x, int y, Color color) {
image.setRGB(x, y, color.getRGB());
}
public Image getImage() {
return image;
}
public void cubicBezier(
Point point1, Point point2, Point point3, Point point4, Color color, int intermediatePointCount) {
List<Point> points = new ArrayList<Point>();
for ( int i = 0; i <= intermediatePointCount; i++ ) {
final double t = (double) i / intermediatePointCount;
final double u = 1.0 - t;
final double a = u * u * u;
final double b = 3.0 * t * u * u;
final double c = 3.0 * t * t * u;
final double d = t * t * t;
final int x = (int) ( a * point1.x + b * point2.x + c * point3.x + d * point4.x );
final int y = (int) ( a * point1.y + b * point2.y + c * point3.y + d * point4.y );
points.add( new Point(x, y) );
setPixel(x, y, color);
}
for ( int i = 0; i < intermediatePointCount; i++ ) {
drawLine(points.get(i).x, points.get(i).y, points.get(i + 1).x, points.get(i + 1).y, color);
}
}
public void drawLine(int x0, int y0, int x1, int y1, Color color) {
final int dx = Math.abs(x1 - x0);
final int dy = Math.abs(y1 - y0);
final int xIncrement = x0 < x1 ? 1 : -1;
final int yIncrement = y0 < y1 ? 1 : -1;
int error = ( dx > dy ? dx : -dy ) / 2;
while ( x0 != x1 || y0 != y1 ) {
setPixel(x0, y0, color);
int error2 = error;
if ( error2 > -dx ) {
error -= dy;
x0 += xIncrement;
}
if ( error2 < dy ) {
error += dx;
y0 += yIncrement;
}
}
setPixel(x0, y0, color);
}
private BufferedImage image;
}
- Output:
JavaScript
function draw() {
var canvas = document.getElementById("container");
context = canvas.getContext("2d");
bezier3(20, 200, 700, 50, -300, 50, 380, 150);
// bezier3(160, 10, 10, 40, 30, 160, 150, 110);
// bezier3(0,149, 30,50, 120,130, 160,30, 0);
}
// http://rosettacode.org/wiki/Cubic_bezier_curves#C
function bezier3(x1, y1, x2, y2, x3, y3, x4, y4) {
var px = [], py = [];
for (var i = 0; i <= b3Seg; i++) {
var d = i / b3Seg;
var a = 1 - d;
var b = a * a;
var c = d * d;
a = a * b;
b = 3 * b * d;
c = 3 * a * c;
d = c * d;
px[i] = parseInt(a * x1 + b * x2 + c * x3 + d * x4);
py[i] = parseInt(a * y1 + b * y2 + c * y3 + d * y4);
}
var x0 = px[0];
var y0 = py[0];
for (i = 1; i <= b3Seg; i++) {
var x = px[i];
var y = py[i];
drawPolygon(context, [[x0, y0], [x, y]], "red", "red");
x0 = x;
y0 = y;
}
}
function drawPolygon(context, polygon, strokeStyle, fillStyle) {
context.strokeStyle = strokeStyle;
context.beginPath();
context.moveTo(polygon[0][0],polygon[0][1]);
for (i = 1; i < polygon.length; i++)
context.lineTo(polygon[i][0],polygon[i][1]);
context.closePath();
context.stroke();
if (fillStyle == undefined)
return;
context.fillStyle = fillStyle;
context.fill();
}
Julia
using Images
function cubicbezier!(xy::Matrix,
img::Matrix = fill(RGB(255.0, 255.0, 255.0), 17, 17),
col::ColorTypes.Color = convert(eltype(img), Gray(0.0)),
n::Int = 20)
t = collect(0:n) ./ n
M = hcat((1 .- t) .^ 3, # a
3t .* (1 .- t) .^ 2, # b
3t .^ 2 .* (1 .- t), # c
t .^ 3) # d
p = floor.(Int, M * xy)
for i in 1:n
drawline!(img, p[i, :]..., p[i+1, :]..., col)
end
return img
end
xy = [16 1; 1 4; 3 16; 15 11]
cubicbezier!(xy)
Kotlin
This incorporates code from other relevant tasks in order to provide a runnable example.
// Version 1.2.40
import java.awt.Color
import java.awt.Graphics
import java.awt.image.BufferedImage
import kotlin.math.abs
import java.io.File
import javax.imageio.ImageIO
class Point(var x: Int, var y: Int)
class BasicBitmapStorage(width: Int, height: Int) {
val image = BufferedImage(width, height, BufferedImage.TYPE_3BYTE_BGR)
fun fill(c: Color) {
val g = image.graphics
g.color = c
g.fillRect(0, 0, image.width, image.height)
}
fun setPixel(x: Int, y: Int, c: Color) = image.setRGB(x, y, c.getRGB())
fun getPixel(x: Int, y: Int) = Color(image.getRGB(x, y))
fun drawLine(x0: Int, y0: Int, x1: Int, y1: Int, c: Color) {
val dx = abs(x1 - x0)
val dy = abs(y1 - y0)
val sx = if (x0 < x1) 1 else -1
val sy = if (y0 < y1) 1 else -1
var xx = x0
var yy = y0
var e1 = (if (dx > dy) dx else -dy) / 2
var e2: Int
while (true) {
setPixel(xx, yy, c)
if (xx == x1 && yy == y1) break
e2 = e1
if (e2 > -dx) { e1 -= dy; xx += sx }
if (e2 < dy) { e1 += dx; yy += sy }
}
}
fun cubicBezier(p1: Point, p2: Point, p3: Point, p4: Point, clr: Color, n: Int) {
val pts = List(n + 1) { Point(0, 0) }
for (i in 0..n) {
val t = i.toDouble() / n
val u = 1.0 - t
val a = u * u * u
val b = 3.0 * t * u * u
val c = 3.0 * t * t * u
val d = t * t * t
pts[i].x = (a * p1.x + b * p2.x + c * p3.x + d * p4.x).toInt()
pts[i].y = (a * p1.y + b * p2.y + c * p3.y + d * p4.y).toInt()
setPixel(pts[i].x, pts[i].y, clr)
}
for (i in 0 until n) {
val j = i + 1
drawLine(pts[i].x, pts[i].y, pts[j].x, pts[j].y, clr)
}
}
}
fun main(args: Array<String>) {
val width = 200
val height = 200
val bbs = BasicBitmapStorage(width, height)
with (bbs) {
fill(Color.cyan)
val p1 = Point(0, 149)
val p2 = Point(30, 50)
val p3 = Point(120, 130)
val p4 = Point(160, 30)
cubicBezier(p1, p2, p3, p4, Color.black, 20)
val cbFile = File("cubic_bezier.jpg")
ImageIO.write(image, "jpg", cbFile)
}
}
Lambdatalk
Drawing a cubic bezier curve out of any SVG or CANVAS frame.
1) interpolating 4 points
The Bézier curve is defined as an array of 4 given points,
each defined as an array of 2 numbers.
The bezier function returns the point interpolating the 4 points.
{def bezier
{def bezier.interpol
{lambda {:a0 :a1 :a2 :a3 :t :u}
{round
{+ {* :a0 :u :u :u 1}
{* :a1 :u :u :t 3}
{* :a2 :u :t :t 3}
{* :a3 :t :t :t 1}}}}}
{lambda {:bz :t}
{A.new {bezier.interpol {A.get 0 {A.get 0 :bz}}
{A.get 0 {A.get 1 :bz}}
{A.get 0 {A.get 2 :bz}}
{A.get 0 {A.get 3 :bz}} :t {- 1 :t}}
{bezier.interpol {A.get 1 {A.get 0 :bz}}
{A.get 1 {A.get 1 :bz}}
{A.get 1 {A.get 2 :bz}}
{A.get 1 {A.get 3 :bz}} :t {- 1 :t}}} }}
-> bezier
2) plotting a dot
We don't draw in any SVG or CANVAS frame, but directly in the HTML page,
using div HTML blocks designed as colored circles.
{def dot
{lambda {:p :r :col}
{div {@ style="position:absolute;
left: {- {A.get 0 :p} {/ :r 2}}px;
top: {- {A.get 1 :p} {/ :r 2}}px;
width: :rpx; height: :rpx;
border-radius: :rpx;
border: 1px solid #000;
background: :col;"}}}}
-> dot
3) defining 4 control points
{def Q0 {A.new 150 150}} -> Q0
{def Q1 {A.new 500 300}} -> Q1
{def Q2 {A.new 100 500}} -> Q2
{def Q3 {A.new 300 500}} -> Q3
4) defining 2 curves
We use the same control points but in different orders to define two curves
{def BZ1 {A.new {Q0} {Q1} {Q2} {Q3}}}
-> BZ1
{def BZ2 {A.new {Q0} {Q2} {Q1} {Q3}}}
-> BZ2
5) drawing curves and dots
We map the bezier function on a serie of values in a range [start end step]
{S.map {lambda {:t} {dot {bezier {BZ1} :t} 5 red}}
{S.serie -0.1 1.2 0.02}}
{S.map {lambda {:t} {dot {bezier {BZ2} :t} 5 blue}}
{S.serie -0.1 1.2 0.02}}
{dot {Q0} 20 cyan}
{dot {Q1} 20 cyan}
{dot {Q2} 20 cyan}
{dot {Q3} 20 cyan}
The result can be seen in http://lambdaway.free.fr/lambdawalks/?view=bezier
Lua
Starting with the code from Bitmap/Bresenham's line algorithm, then extending:
Bitmap.cubicbezier = function(self, x1, y1, x2, y2, x3, y3, x4, y4, nseg)
nseg = nseg or 10
local prevx, prevy, currx, curry
for i = 0, nseg do
local t = i / nseg
local a, b, c, d = (1-t)^3, 3*t*(1-t)^2, 3*t^2*(1-t), t^3
prevx, prevy = currx, curry
currx = math.floor(a * x1 + b * x2 + c * x3 + d * x4 + 0.5)
curry = math.floor(a * y1 + b * y2 + c * y3 + d * y4 + 0.5)
if i > 0 then
self:line(prevx, prevy, currx, curry)
end
end
end
local bitmap = Bitmap(61,21)
bitmap:clear()
bitmap:cubicbezier( 1,1, 15,41, 45,-20, 59,19 )
bitmap:render({[0x000000]='.', [0xFFFFFFFF]='X'})
- Output:
............................................................. .X........................................................... .X........................................................... ..X.......................................................... ...X......................................................... ...X.....................................XXXXX............... ....X.................................XXX.....XXXX........... ....X..............................XXX............X.......... .....X...........................XX................X......... .....X.........................XX...................X........ ......X.....................XXX......................XX...... .......X..................XX..........................X...... ........X...............XX.............................X..... .........X...........XXX................................X.... ..........XXXX....XXX...................................X.... ..............XXXX.......................................X... .........................................................X... ..........................................................X.. ..........................................................X.. ...........................................................X. .............................................................
Mathematica / Wolfram Language
points= {{0, 0}, {1, 1}, {2, -1}, {3, 0}};
Graphics[{BSplineCurve[points], Green, Line[points], Red, Point[points]}]
MATLAB
Note: Store this function in a file named "bezierCubic.mat" in the @Bitmap folder for the Bitmap class defined here.
function bezierCubic(obj,pixel_0,pixel_1,pixel_2,pixel_3,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).^3*pixel_0; %First term of polynomial
curve = curve + (3.*time.*timeMinus.^2)*pixel_1; %second term of polynomial
curve = curve + (3.*timeMinus.*time.^2)*pixel_2; %third term of polynomial
curve = curve + time.^3*pixel_3; %Fourth 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
Sample usage: This will generate the image example for the PHP solution.
>> img = Bitmap(200,200);
>> img.fill([255 255 255]);
>> img.bezierCubic([160 10],[10 40],[30 160],[150 110],[255 0 0],110);
>> disp(img)
MiniScript
This GUI implementation is for use with Mini Micro.
Point = {"x": 0, "y":0}
Point.init = function(x, y)
p = new Point
p.x = x; p.y = y
return p
end function
drawLine = function(img, x0, y0, x1, y1, colr)
sign = function(a, b)
if a < b then return 1
return -1
end function
dx = abs(x1 - x0)
sx = sign(x0, x1)
dy = abs(y1 - y0)
sy = sign(y0, y1)
if dx > dy then
err = dx
else
err = -dy
end if
err = floor(err / 2)
while true
img.setPixel x0, y0, colr
if x0 == x1 and y0 == y1 then break
e2 = err
if e2 > -dx then
err -= dy
x0 += sx
end if
if e2 < dy then
err += dx
y0 += sy
end if
end while
end function
cubicBezier = function(img, p1, p2, p3, p4, numPoints, colr)
points = []
for i in range(0, numPoints)
t = i / numPoints
u = 1 - t
a = (1 - t)^3
b = 3 * t * (1 - t)^2
c = 3 * t^2 * (1 - t)
d = t^3
x = floor(a * p1.x + b * p2.x + c * p3.x + d * p4.x)
y = floor(a * p1.y + b * p2.y + c * p3.y + d * p4.y)
points.push(Point.init(x, y))
img.setPixel x, y, colr
end for
for i in range(1, numPoints)
drawLine img, points[i-1].x, points[i-1].y, points[i].x, points[i].y, colr
end for
end function
bezier = Image.create(480, 480)
p1 = Point.init(50, 100)
p2 = Point.init(200, 400)
p3 = Point.init(360, 0)
p4 = Point.init(300, 424)
cubicBezier bezier, p1, p2, p3, p4, 20, color.red
gfx.clear
gfx.drawImage bezier, 0, 0
Nim
We use module “bitmap” for bitmap management and module “bresenham” to draw segments.
import bitmap
import bresenham
import lenientops
proc drawCubicBezier*(
image: Image; pt1, pt2, pt3, pt4: Point; color: Color; nseg: Positive = 20) =
var points = newSeq[Point](nseg + 1)
for i in 0..nseg:
let t = i / nseg
let u = (1 - t) * (1 - t)
let a = (1 - t) * u
let b = 3 * t * u
let c = 3 * (t * t) * (1 - t)
let d = t * t * t
points[i] = (x: (a * pt1.x + b * pt2.x + c * pt3.x + d * pt4.x).toInt,
y: (a * pt1.y + b * pt2.y + c * pt3.y + d * pt4.y).toInt)
for i in 1..points.high:
image.drawLine(points[i - 1], points[i], color)
#———————————————————————————————————————————————————————————————————————————————————————————————————
when isMainModule:
var img = newImage(16, 16)
img.fill(White)
img.drawCubicBezier((0, 15), (3, 0), (15, 2), (10, 14), Black)
img.print
- Output:
................ ................ ................ ................ .......HH....... .....HH..HH..... ....H......H.... ....H......H.... ...H.......H.... ..H........H.... .H.........H.... .H.........H.... .H.........H.... .H.........H.... H.........H..... H...............
OCaml
let cubic_bezier ~img ~color
~p1:(_x1, _y1)
~p2:(_x2, _y2)
~p3:(_x3, _y3)
~p4:(_x4, _y4) =
let x1, y1, x2, y2, x3, y3, x4, y4 =
(float _x1, float _y1,
float _x2, float _y2,
float _x3, float _y3,
float _x4, float _y4)
in
let bz t =
let a = (1.0 -. t) ** 3.0
and b = 3.0 *. t *. ((1.0 -. t) ** 2.0)
and c = 3.0 *. (t ** 2.0) *. (1.0 -. t)
and d = t ** 3.0
in
let x = a *. x1 +. b *. x2 +. c *. x3 +. d *. x4
and y = a *. y1 +. b *. y2 +. c *. y3 +. d *. y4
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);
;;
Phix
Output similar to Mathematica
Requires new_image() from Bitmap, bresLine() from Bresenham's_line_algorithm, write_ppm() from Write_a_PPM_file.
Results may be verified with demo\rosetta\viewppm.exw
-- demo\rosetta\Bitmap_BezierCubic.exw (runnable version)
include ppm.e -- black, green, red, white, new_image(), write_ppm(), bresLine() -- (covers above requirements)
function cubic_bezier(sequence img, atom x1, y1, x2, y2, x3, y3, x4, y4, integer colour, segments)
sequence pts = repeat(0,segments*2)
for i=0 to segments*2-1 by 2 do
atom t = i/segments,
t1 = 1-t,
a = power(t1,3),
b = 3*t*power(t1,2),
c = 3*power(t,2)*t1,
d = power(t,3)
pts[i+1] = floor(a*x1+b*x2+c*x3+d*x4)
pts[i+2] = floor(a*y1+b*y2+c*y3+d*y4)
end for
for i=1 to segments*2-2 by 2 do
img = bresLine(img, pts[i], pts[i+1], pts[i+2], pts[i+3], colour)
end for
return img
end function
sequence img = new_image(300,200,black)
img = cubic_bezier(img, 0,100, 100,0, 200,200, 300,100, white, 40)
img = bresLine(img,0,100,100,0,green)
img = bresLine(img,100,0,200,200,green)
img = bresLine(img,200,200,300,100,green)
img[1][100] = red
img[100][1] = red
img[200][200] = red
img[300][100] = red
write_ppm("Bezier.ppm",img)
PHP
Outputs image to the right directly to browser or stdout.
<?
$image = imagecreate(200, 200);
// The first allocated color will be the background color:
imagecolorallocate($image, 255, 255, 255);
$color = imagecolorallocate($image, 255, 0, 0);
cubicbezier($image, $color, 160, 10, 10, 40, 30, 160, 150, 110);
imagepng($image);
function cubicbezier($img, $col, $x0, $y0, $x1, $y1, $x2, $y2, $x3, $y3, $n = 20) {
$pts = array();
for($i = 0; $i <= $n; $i++) {
$t = $i / $n;
$t1 = 1 - $t;
$a = pow($t1, 3);
$b = 3 * $t * pow($t1, 2);
$c = 3 * pow($t, 2) * $t1;
$d = pow($t, 3);
$x = round($a * $x0 + $b * $x1 + $c * $x2 + $d * $x3);
$y = round($a * $y0 + $b * $y1 + $c * $y2 + $d * $y3);
$pts[$i] = array($x, $y);
}
for($i = 0; $i < $n; $i++) {
imageline($img, $pts[$i][0], $pts[$i][1], $pts[$i+1][0], $pts[$i+1][1], $col);
}
}
PicoLisp
This uses the 'brez' line drawing function from Bitmap/Bresenham's line algorithm#PicoLisp.
(scl 6)
(de cubicBezier (Img N X1 Y1 X2 Y2 X3 Y3 X4 Y4)
(let (R (* N N N) X X1 Y Y1 DX 0 DY 0)
(for I N
(let
(J (- N I)
A (*/ 1.0 J J J R)
B (*/ 3.0 I J J R)
C (*/ 3.0 I I J R)
D (*/ 1.0 I I I R) )
(brez Img
X
Y
(setq DX
(-
(+ (*/ A X1 1.0) (*/ B X2 1.0) (*/ C X3 1.0) (*/ D X4 1.0))
X ) )
(setq DY
(-
(+ (*/ A Y1 1.0) (*/ B Y2 1.0) (*/ C Y3 1.0) (*/ D Y4 1.0))
Y ) ) )
(inc 'X DX)
(inc 'Y DY) ) ) ) )Test:
(let Img (make (do 200 (link (need 300 0)))) # Create image 300 x 200
(cubicBezier Img 24 20 120 540 33 -225 33 285 100)
(out "img.pbm" # Write to bitmap file
(prinl "P1")
(prinl 300 " " 200)
(mapc prinl Img) ) )
(call 'display "img.pbm")Processing
noFill();
bezier(85, 20, 10, 10, 90, 90, 15, 80);
/*
bezier(x1, y1, x2, y2, x3, y3, x4, y4)
Can also be drawn in 3D.
bezier(x1, y1, z1, x2, y2, z2, x3, y3, z3, x4, y4, z4)
*/
A working sketch with movable anchor and control points.
It can be run on line :
here.
float[] x = new float[4];
float[] y = new float[4];
boolean[] permitDrag = new boolean[4];
void setup() {
size(300, 300);
smooth();
// startpoint coordinates
x[0] = x[1] = 50;
y[0] = 50;
y[1] = y[2] = 150;
x[2] = x[3] = 250;
y[3] = 250;
}
void draw() {
background(255);
noFill();
stroke(0, 0, 255);
bezier(x[1], y[1], x[0], y[0], x[3], y[3], x[2], y[2]);
// the bezier handles
strokeWeight(1);
stroke(100);
line(x[0], y[0], x[1], y[1]);
line(x[2], y[2], x[3], y[3]);
// the anchor and control points
stroke(0);
fill(0);
for (int i = 0; i< 4; i++) {
if (i == 0 || i == 3) {
fill(255, 100, 10);
rectMode(CENTER);
rect(x[i], y[i], 5, 5);
} else {
fill(0);
ellipse(x[i], y[i], 5, 5);
}
}
// permit dragging
for (int i = 0; i < 4; i++) {
if (permitDrag[i]) {
x[i] = mouseX;
y[i] = mouseY;
}
}
}
void mouseReleased () {
for (int i = 0; i < 4; i++) {
permitDrag[i] = false;
}
}
void mousePressed () {
for (int i = 0; i < 4; i++) {
if (mouseX >= x[i]-5 && mouseX <= x[i]+10 && mouseY >= y[i]-5 && mouseY <= y[i]+10) {
permitDrag[i] = true;
}
}
}
// hand curser when dragging over points
void mouseMoved () {
cursor(ARROW);
for (int i = 0; i < 4; i++) {
if (mouseX >= x[i]-5 && mouseX <= x[i]+10 && mouseY >= y[i]-5 && mouseY <= y[i]+10) {
cursor(HAND);
}
}
}
PureBasic
Procedure cubic_bezier(img, p1x, p1y, p2x, p2y, p3x, p3y, p4x, p4y, 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, 3)
b = 3.0 * t * Pow(t1, 2)
c = 3.0 * Pow(t, 2) * t1
d = Pow(t, 3)
pts(i)\x = a * p1x + b * p2x + c * p3x + d * p4x
pts(i)\y = a * p1y + b * p2y + c * p3y + d * p4y
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) ;this calls the implementation of a draw_line routine
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, cubic", #PB_Window_SystemMenu)
cubic_bezier(1, 160,10, 10,40, 30,160, 150,110, RGB(255, 255, 255), 20)
ImageGadget(0, 0, 0, w, h, ImageID(1))
Define event
Repeat
event = WaitWindowEvent()
Until event = #PB_Event_CloseWindow
Python
Extending the example given here and using the algorithm from the C solution above:
def cubicbezier(self, x0, y0, x1, y1, x2, y2, x3, y3, n=20):
pts = []
for i in range(n+1):
t = i / n
a = (1. - t)**3
b = 3. * t * (1. - t)**2
c = 3.0 * t**2 * (1.0 - t)
d = t**3
x = int(a * x0 + b * x1 + c * x2 + d * x3)
y = int(a * y0 + b * y1 + c * y2 + d * y3)
pts.append( (x, y) )
for i in range(n):
self.line(pts[i][0], pts[i][1], pts[i+1][0], pts[i+1][1])
Bitmap.cubicbezier = cubicbezier
bitmap = Bitmap(17,17)
bitmap.cubicbezier(16,1, 1,4, 3,16, 15,11)
bitmap.chardisplay()
'''
The origin, 0,0; is the lower left, with x increasing to the right,
and Y increasing upwards.
The chardisplay above produces the following output :
+-----------------+
| |
| |
| |
| |
| @@@@ |
| @@@ @@@ |
| @ |
| @ |
| @ |
| @ |
| @ |
| @ |
| @ |
| @ |
| @@@@ |
| @@@@|
| |
+-----------------+
'''
R
# x, y: the x and y coordinates of the hull points
# n: the number of points in the curve.
bezierCurve <- function(x, y, n=10)
{
outx <- NULL
outy <- NULL
i <- 1
for (t in seq(0, 1, length.out=n))
{
b <- bez(x, y, t)
outx[i] <- b$x
outy[i] <- b$y
i <- i+1
}
return (list(x=outx, y=outy))
}
bez <- function(x, y, t)
{
outx <- 0
outy <- 0
n <- length(x)-1
for (i in 0:n)
{
outx <- outx + choose(n, i)*((1-t)^(n-i))*t^i*x[i+1]
outy <- outy + choose(n, i)*((1-t)^(n-i))*t^i*y[i+1]
}
return (list(x=outx, y=outy))
}
# Example usage
x <- c(4,6,4,5,6,7)
y <- 1:6
plot(x, y, "o", pch=20)
points(bezierCurve(x,y,20), type="l", col="red")
Racket
#lang racket
(require racket/draw)
(define (draw-line dc p q)
(match* (p q) [((list x y) (list s t)) (send dc draw-line x y s t)]))
(define (draw-lines dc ps)
(void
(for/fold ([p0 (first ps)]) ([p (rest ps)])
(draw-line dc p0 p)
p)))
(define (int t p q)
(define ((int1 t) x0 x1) (+ (* (- 1 t) x0) (* t x1)))
(map (int1 t) p q))
(define (bezier-points p0 p1 p2 p3)
(for/list ([t (in-range 0.0 1.0 (/ 1.0 20))])
(int t (int t p0 p1) (int t p2 p3))))
(define bm (make-object bitmap% 17 17))
(define dc (new bitmap-dc% [bitmap bm]))
(send dc set-smoothing 'unsmoothed)
(send dc set-pen "red" 1 'solid)
(draw-lines dc (bezier-points '(16 1) '(1 4) '(3 16) '(15 11)))
bm
Raku
(formerly Perl 6)
Uses pieces from Bitmap, and Bresenham's line algorithm tasks. They are included here to make a complete, runnable program.
class Pixel { has UInt ($.R, $.G, $.B) }
class Bitmap {
has UInt ($.width, $.height);
has Pixel @!data;
method fill(Pixel $p) {
@!data = $p.clone xx ($!width*$!height)
}
method pixel(
$i where ^$!width,
$j where ^$!height
--> Pixel
) is rw { @!data[$i + $j * $!width] }
method set-pixel ($i, $j, Pixel $p) {
self.pixel($i, $j) = $p.clone;
}
method get-pixel ($i, $j) returns Pixel {
self.pixel($i, $j);
}
method line(($x0 is copy, $y0 is copy), ($x1 is copy, $y1 is copy), $pix) {
my $steep = abs($y1 - $y0) > abs($x1 - $x0);
if $steep {
($x0, $y0) = ($y0, $x0);
($x1, $y1) = ($y1, $x1);
}
if $x0 > $x1 {
($x0, $x1) = ($x1, $x0);
($y0, $y1) = ($y1, $y0);
}
my $Δx = $x1 - $x0;
my $Δy = abs($y1 - $y0);
my $error = 0;
my $Δerror = $Δy / $Δx;
my $y-step = $y0 < $y1 ?? 1 !! -1;
my $y = $y0;
for $x0 .. $x1 -> $x {
if $steep {
self.set-pixel($y, $x, $pix);
} else {
self.set-pixel($x, $y, $pix);
}
$error += $Δerror;
if $error >= 0.5 {
$y += $y-step;
$error -= 1.0;
}
}
}
method dot (($px, $py), $pix, $radius = 2) {
for $px - $radius .. $px + $radius -> $x {
for $py - $radius .. $py + $radius -> $y {
self.set-pixel($x, $y, $pix) if ( $px - $x + ($py - $y) * i ).abs <= $radius;
}
}
}
method cubic ( ($x1, $y1), ($x2, $y2), ($x3, $y3), ($x4, $y4), $pix, $segments = 30 ) {
my @line-segments = map -> $t {
my \a = (1-$t)³;
my \b = $t * (1-$t)² * 3;
my \c = $t² * (1-$t) * 3;
my \d = $t³;
(a*$x1 + b*$x2 + c*$x3 + d*$x4).round(1),(a*$y1 + b*$y2 + c*$y3 + d*$y4).round(1)
}, (0, 1/$segments, 2/$segments ... 1);
for @line-segments.rotor(2=>-1) -> ($p1, $p2) { self.line( $p1, $p2, $pix) };
}
method data { @!data }
}
role PPM {
method P6 returns Blob {
"P6\n{self.width} {self.height}\n255\n".encode('ascii')
~ Blob.new: flat map { .R, .G, .B }, self.data
}
}
sub color( $r, $g, $b) { Pixel.new(R => $r, G => $g, B => $b) }
my Bitmap $b = Bitmap.new( width => 600, height => 400) but PPM;
$b.fill( color(2,2,2) );
my @points = (85,390), (5,5), (580,370), (270,10);
my %seen;
my $c = 0;
for @points.permutations -> @this {
%seen{@this.reverse.join.Str}++;
next if %seen{@this.join.Str};
$b.cubic( |@this, color(255-$c,127,$c+=22) );
}
@points.map: { $b.dot( $_, color(255,0,0), 3 )}
$*OUT.write: $b.P6;
See example image here, (converted to a .png as .ppm format is not widely supported).
Ruby
Requires code from the Bitmap and Bitmap/Bresenham's line algorithm#Ruby Bresenham's line algorithm tasks
class Pixmap
def draw_bezier_curve(points, colour)
# ensure the points are increasing along the x-axis
points = points.sort_by {|p| [p.x, p.y]}
xmin = points[0].x
xmax = points[-1].x
increment = 2
prev = points[0]
((xmin + increment) .. xmax).step(increment) do |x|
t = 1.0 * (x - xmin) / (xmax - xmin)
p = Pixel[x, bezier(t, points).round]
draw_line(prev, p, colour)
prev = p
end
end
end
# the generalized n-degree Bezier summation
def bezier(t, points)
n = points.length - 1
points.each_with_index.inject(0.0) do |sum, (point, i)|
sum += n.choose(i) * (1-t)**(n - i) * t**i * point.y
end
end
class Fixnum
def choose(k)
self.factorial / (k.factorial * (self - k).factorial)
end
def factorial
(2 .. self).reduce(1, :*)
end
end
bitmap = Pixmap.new(400, 400)
points = [
Pixel[40,100], Pixel[100,350], Pixel[150,50],
Pixel[150,150], Pixel[350,250], Pixel[250,250]
]
points.each {|p| bitmap.draw_circle(p, 3, RGBColour::RED)}
bitmap.draw_bezier_curve(points, RGBColour::BLUE)
Tcl
This solution can be applied to any number of points. Uses code from Basic bitmap storage (newImage, fill), Bresenham's line algorithm (drawLine), and Midpoint circle algorithm (drawCircle)
package require Tcl 8.5
package require Tk
proc drawBezier {img colour args} {
# ensure the points are increasing along the x-axis
set points [lsort -real -index 0 $args]
set xmin [x [lindex $points 0]]
set xmax [x [lindex $points end]]
set prev [lindex $points 0]
set increment 2
for {set x [expr {$xmin + $increment}]} {$x <= $xmax} {incr x $increment} {
set t [expr {1.0 * ($x - $xmin) / ($xmax - $xmin)}]
set this [list $x [::tcl::mathfunc::round [bezier $t $points]]]
drawLine $img $colour $prev $this
set prev $this
}
}
# the generalized n-degree Bezier summation
proc bezier {t points} {
set n [expr {[llength $points] - 1}]
for {set i 0; set sum 0.0} {$i <= $n} {incr i} {
set sum [expr {$sum + [C $n $i] * (1-$t)**($n - $i) * $t**$i * [y [lindex $points $i]]}]
}
return $sum
}
proc C {n i} {expr {[ifact $n] / ([ifact $i] * [ifact [expr {$n - $i}]])}}
proc ifact n {
for {set i $n; set sum 1} {$i >= 2} {incr i -1} {
set sum [expr {$sum * $i}]
}
return $sum
}
proc x p {lindex $p 0}
proc y p {lindex $p 1}
proc newbezier {n w} {
set size 400
set bezier [newImage $size $size]
fill $bezier white
for {set i 1} {$i <= $n} {incr i} {
set point [list [expr {int($size*rand())}] [expr {int($size*rand())}]]
lappend points $point
drawCircle $bezier red $point 3
}
puts $points
drawBezier $bezier blue {*}$points
$w configure -image $bezier
}
set degree 4 ;# cubic bezier -- for quadratic, use 3
label .img
button .new -command [list newbezier $degree .img] -text New
button .exit -command exit -text Exit
pack .new .img .exit -side top
Results in:
TI-89 BASIC
Define cubic(p1,p2,p3,p4,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^3*p1 + 3t*u^2*p2 + 3t^2*u*p3 + t^3*p4 → pt
If i>1 Then
PxlLine floor(prev[1,1]), floor(prev[1,2]), floor(pt[1,1]), floor(pt[1,2])
EndIf
EndFor
EndPrgmWren
Requires version 1.3.0 of DOME or later.
import "graphics" for Canvas, ImageData, Color, Point
import "dome" for Window
class Game {
static bmpCreate(name, w, h) { ImageData.create(name, w, h) }
static bmpFill(name, col) {
var image = ImageData[name]
for (x in 0...image.width) {
for (y in 0...image.height) image.pset(x, y, col)
}
}
static bmpPset(name, x, y, col) { ImageData[name].pset(x, y, col) }
static bmpPget(name, x, y) { ImageData[name].pget(x, y) }
static bmpLine(name, x0, y0, x1, y1, col) {
var dx = (x1 - x0).abs
var dy = (y1 - y0).abs
var sx = (x0 < x1) ? 1 : -1
var sy = (y0 < y1) ? 1 : -1
var err = ((dx > dy ? dx : - dy) / 2).floor
while (true) {
bmpPset(name, x0, y0, col)
if (x0 == x1 && y0 == y1) break
var e2 = err
if (e2 > -dx) {
err = err - dy
x0 = x0 + sx
}
if (e2 < dy) {
err = err + dx
y0 = y0 + sy
}
}
}
static bmpCubicBezier(name, p1, p2, p3, p4, col, n) {
var pts = List.filled(n+1, null)
for (i in 0..n) {
var t = i / n
var u = 1 - t
var a = u * u * u
var b = 3 * t * u * u
var c = 3 * t * t * u
var d = t * t * t
var px = (a * p1.x + b * p2.x + c * p3.x + d * p4.x).truncate
var py = (a * p1.y + b * p2.y + c * p3.y + d * p4.y).truncate
pts[i] = Point.new(px, py, col)
}
for (i in 0...n) {
var j = i + 1
bmpLine(name, pts[i].x, pts[i].y, pts[j].x, pts[j].y, col)
}
}
static init() {
Window.title = "Cubic Bézier curve"
var size = 200
Window.resize(size, size)
Canvas.resize(size, size)
var name = "cubic"
var bmp = bmpCreate(name, size, size)
bmpFill(name, Color.white)
var p1 = Point.new(160, 10)
var p2 = Point.new( 10, 40)
var p3 = Point.new( 30, 160)
var p4 = Point.new(150, 110)
bmpCubicBezier(name, p1, p2, p3, p4, Color.darkblue, 20)
bmp.draw(0, 0)
}
static update() {}
static draw(alpha) {}
}
XPL0
include c:\cxpl\codes; \intrinsic 'code' declarations
proc Bezier(P0, P1, P2, P3); \Draw cubic Bezier curve
real P0, P1, P2, P3;
def Segments = 8;
int I;
real S1, T, T2, T3, U, U2, U3, B, C, X, Y;
[Move(fix(P0(0)), fix(P0(1)));
S1:= 1./float(Segments);
T:= 0.;
for I:= 1 to Segments-1 do
[T:= T+S1;
T2:= T*T;
T3:= T2*T;
U:= 1.-T;
U2:= U*U;
U3:= U2*U;
B:= 3.*T*U2;
C:= 3.*T2*U;
X:= U3*P0(0) + B*P1(0) + C*P2(0) + T3*P3(0);
Y:= U3*P0(1) + B*P1(1) + C*P2(1) + T3*P3(1);
Line(fix(X), fix(Y), $00FFFF); \cyan line segments
];
Line(fix(P3(0)), fix(P3(1)), $00FFFF);
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);
Point(fix(P3(0)), fix(P3(1)), $FF0000);
];
[SetVid($112); \set 640x480x24 video graphics
Bezier([0., 0.], [30., 100.], [120., 20.], [160., 120.]);
if ChIn(1) then []; \wait for keystroke
SetVid(3); \restore normal text display
]zkl
Image cribbed from XPL0
Uses the PPM class from http://rosettacode.org/wiki/Bitmap/Bresenham%27s_line_algorithm#zkl
Add this to the PPM class:
fcn cBezier(p0x,p0y, p1x,p1y, p2x,p2y, p3x,p3y, rgb, numPts=500){
numPts.pump(Void,'wrap(t){ // B(t)
t=t.toFloat()/numPts; t1:=(1.0 - t);
a:=t1*t1*t1; b:=t*t1*t1*3; c:=t1*t*t*3; d:=t*t*t;
x:=a*p0x + b*p1x + c*p2x + d*p3x + 0.5;
y:=a*p0y + b*p1y + c*p2y + d*p3y + 0.5;
__sSet(rgb,x,y);
});
}Doesn't use line segments, they don't seem like an improvement.
bitmap:=PPM(200,150,0xff|ff|ff);
bitmap.cBezier(0,149, 30,50, 120,130, 160,30, 0);
bitmap.write(File("foo.ppm","wb"));Categories:
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