Bitmap/Bézier curves/Cubic
From Rosetta Code
< 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).
Contents |
[edit] 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
[edit] ALGOL 68
File: prelude/Bitmap/Bezier_curves/Cubic.a68# -*- coding: utf-8 -*- #File: test/Bitmap/Bezier_curves/Cubic.a68
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 #;
SKIP
#!/usr/bin/a68g --script #Output:
# -*- 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)
)
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffff000000000000ffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffff000000000000ffffffffffff000000000000ffffffffffffffffffffffffffffff ffffffffffffffffffffffff000000ffffffffffffffffffffffffffffffffffff000000ffffffffffffffffffffffff ffffffffffffffffffffffff000000ffffffffffffffffffffffffffffffffffff000000ffffffffffffffffffffffff ffffffffffffffffff000000ffffffffffffffffffffffffffffffffffffffffff000000ffffffffffffffffffffffff ffffffffffff000000ffffffffffffffffffffffffffffffffffffffffffffffff000000ffffffffffffffffffffffff ffffff000000ffffffffffffffffffffffffffffffffffffffffffffffffffffff000000ffffffffffffffffffffffff ffffff000000ffffffffffffffffffffffffffffffffffffffffffffffffffffff000000ffffffffffffffffffffffff ffffff000000ffffffffffffffffffffffffffffffffffffffffffffffffffffff000000ffffffffffffffffffffffff ffffff000000ffffffffffffffffffffffffffffffffffffffffffffffffffffff000000ffffffffffffffffffffffff 000000ffffffffffffffffffffffffffffffffffffffffffffffffffffff000000ffffffffffffffffffffffffffffff 000000ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
[edit] 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
[edit] 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
[edit] 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 if (nSegments > 0) {
Pt[nSegments + 1] points = void;
foreach (immutable i, ref p; points) {
immutable double t = i / cast(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:
................. .............#### .........####.... ........#........ .......#......... ......#.......... ......#.......... .....#........... .....#........... .....#........... .....#........... ......##....####. ........####..... ................. ................. ................. .................
[edit] 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
[edit] 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: 
[edit] 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 ;
[edit] 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)**2.0
c = 3.0 * (1.0 - t) * t**2.0
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
[edit] 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)
}
}
[edit] 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
[edit] Mathematica
points= {{0, 0}, {1, 1}, {2, -1}, {3, 0}};
Graphics[{BSplineCurve[points], Green, Line[points], Red, Point[points]}]
[edit] 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)
[edit] 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);
;;
[edit] 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);
}
}
[edit] 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")
[edit] 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
[edit] 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 :
+-----------------+
| |
| |
| |
| |
| @@@@ |
| @@@ @@@ |
| @ |
| @ |
| @ |
| @ |
| @ |
| @ |
| @ |
| @ |
| @@@@ |
| @@@@|
| |
+-----------------+
'''
[edit] 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")
[edit] 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
[edit] 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)
[edit] 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:
[edit] 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
EndPrgm
[edit] XPL0
include c:\cxpl\codes; \intrinsic 'code' declarations
func real Power(X, Y); \X raised to the Y power; (X > 0.0)
real X, Y;
return Exp(Y * Ln(X));
proc Bezier(P0, P1, P2, P3); \Draw cubic Bezier curve
real P0, P1, P2, P3;
def Segments = 8;
int I;
real T, A, B, C, D, X, Y;
[Move(fix(P0(0)), fix(P0(1)));
for I:= 1 to Segments-1 do
[T:= float(I)/float(Segments);
A:= Power((1.-T), 3.);
B:= 3.*T*Power((1.-T), 2.);
C:= 3.*Power(T, 2.)*(1.-T);
D:= Power(T, 3.);
X:= A*P0(0) + B*P1(0) + C*P2(0) + D*P3(0);
Y:= A*P0(1) + B*P1(1) + C*P2(1) + D*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
]
