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# B-spline

B-spline is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

Generate a B-spline curve with a list of 12 points and plot or save image.

Coordinates of control points:

```   start=171,171
1 185,111,
2 202,109,
3 202,189
4 328,160
5 208,254
6 241,330
7 164,252
8 69,278
9 139,208
10 72,148
end=168,172
```

Rules!!!!

Do not use third party libraries or functions

## ALGOL 68

Translation of: Lua
Which is
Translation of: Wren
Suppresses unused parts of the plot.
`BEGIN # construct a B-Spline                                                 #     # mode to hold a B Spline                                                #    MODE BSPLINE = STRUCT( FLEX[ 1 : 0, 1 : 2 ]INT control points                         , INT n                         , INT k                         , FLEX[ 1 : 0 ]INT t                         );    PROC  uniform knot vector = ( INT lb, ub )[]INT:          BEGIN             [ lb : ub ]INT range;              FOR n FROM lb TO ub DO range[ n ] := n OD;	     range          END # uniform knot vector # ;     PROC calculate bspline = ( REF BSPLINE bs, INT i, k, x )REAL:	 IF   k = 1         THEN ABS ( ( t OF bs )[ i ] <= x AND x < ( t OF bs )[ i + 1 ] )	 ELSE              PROC helper = ( REF BSPLINE bs, INT i, k, x )REAL:                   IF ( t OF bs )[ i + k ] /= ( t OF bs )[ i ]                   THEN ( x - ( t OF bs )[ i ] ) / ( ( t OF bs )[ i + k ] - ( t OF bs )[ i ] )                   ELSE 0                   FI # helper # ;              (     helper( bs, i,     k - 1, x ) ) * calculate bspline( bs, i,     k - 1, x )            + ( 1 - helper( bs, i + 1, k - 1, x ) ) * calculate bspline( bs, i + 1, k - 1, x )         FI # calculate bspline # ;     PROC round = ( REAL n )INT: ENTIER( n + 0.5 );     PROC get bspline points = ( REF BSPLINE bs )[,]INT:         BEGIN            INT p from = ( t OF bs )[     k OF bs ];            INT p to   = ( t OF bs )[ 1 + n OF bs ] - 1;            [ p from : p to, 1 : 2 ]INT points;            FOR x FROM p from TO p to DO                REAL sum x := 0;                REAL sum y := 0;		FOR i TO n OF bs DO                    REAL f = calculate bspline( bs, i, k OF bs, x );                    sum x +:= f * ( control points OF bs )[ i, 1 ];                    sum y +:= f * ( control points OF bs )[ i, 2 ]                OD;		points[ x, 1 ] := round( sum x );                points[ x, 2 ] := round( sum y )            OD;            points         END # get bspline points # ;     PROC raytrace = ( INT x0, y0, x2, y2, REF[,]BOOL plot, PROC( REF[,]BOOL, INT, INT )VOID visit )VOID:         BEGIN            INT x    := x0;            INT y    := y0;	    INT dx   := ABS ( x2 - x );            INT dy   := ABS ( y2 - y );	    INT n     = 1 + dx + dy;	    INT dir x = IF x2 > x THEN 1 ELSE -1 FI;            INT dir y = IF y2 > y THEN 1 ELSE -1 FI;	    INT err  := dx - dy;            dx *:= 2;            dy *:= 2;            FOR i TO n DO                visit( plot, x, y );                IF   err > 0                THEN x +:= dir x; err -:= dy                ELSE y +:= dir y; err +:= dx                FI            OD         END # raytrace # ;     PROC plot line = ( REF[,]BOOL plot, INT x1, y1, x2, y2 )VOID:	raytrace( x1, y1, x2, y2, plot                , ( REF[,]BOOL plot, INT x, INT y )VOID: IF   x >= 0                                                         AND  y >= 0                                                         AND  x < 1 UPB plot                                                         AND  y < 2 UPB plot                                                         THEN plot[ x + 1, y + 1 ] := TRUE		                                         FI                );     PROC plot bspline = ( REF BSPLINE bs, REF[,]BOOL plot, REAL scale x, scale y )VOID:         IF k OF bs > n OF bs OR k OF bs < 1 THEN            print( ( "k (= ", whole( k OF bs, 0 ), ") can't be more than ", whole( n OF bs, 0 ), " or less than 1." ) );            stop         ELSE            [,]INT points = get bspline points( bs );            # Plot the curve. #            FOR i FROM 1 LWB points TO 1 UPB points - 1 DO                INT p1x = points[ i,     1 ], p1y = points[ i,     2 ];                INT p2x = points[ i + 1, 1 ], p2y = points[ i + 1, 2 ];                plot line( plot                         , round( p1x * scale x ), round( p1y * scale y )                         , round( p2x * scale x ), round( p2y * scale y )                         )            OD	 FI # plot bspline # ;     # print the plot - outputs @ or blank depending on whether the point is plotted or not #    PROC print plot = ( [,]BOOL plot )VOID:         FOR row FROM 1 LWB plot         TO BEGIN # find the highest used row #                INT max row := 1 UPB plot;                WHILE IF max row < 1 LWB plot                      THEN FALSE                      ELSE                          BOOL empty row := TRUE;                          FOR column FROM 2 LWB plot TO 2 UPB plot                          WHILE empty row := NOT plot[ column, max row ]                          DO                              SKIP                          OD;                          empty row                      FI                DO                    max row -:= 1                OD;                max row            END         DO             INT max column := 2 UPB plot;             WHILE IF max column < 2 LWB plot THEN FALSE ELSE NOT plot[ max column, row ] FI             DO                 max column -:= 1             OD;             FOR column FROM 2 LWB plot TO max column DO                 print( ( IF plot[ column, row ] THEN "@" ELSE " " FI ) )             OD;             print( ( newline ) )         OD # print plot # ;     # task #    [,]INT  control points        =   ( ( 171, 171 ), ( 185, 111 ), ( 202, 109 ), ( 202, 189 ), ( 328, 160 ), ( 208, 254 )            , ( 241, 330 ), ( 164, 252 ), (  69, 278 ), ( 139, 208 ), (  72, 148 ), ( 168, 172 )            );    INT     k = 4;  # Polynomial degree is one less than this i.e. cubic. #    BSPLINE bs        := BSPLINE( control points                  , UPB control points                  , k                  , uniform knot vector( 1, UPB control points + k )                  );     REAL scale x = 0.4; # Since we print the plot to the console as text let's scale things appropriately. #    REAL scale y = 0.2;    [ 1 : 350, 1 : 350 ]BOOL plot;    FOR r FROM 1 LWB plot TO 1 UPB plot DO FOR c FROM 2 LWB plot TO 2 UPB plot DO plot[ c, r ] := FALSE OD OD;    plot bspline( bs, plot, scale x, scale y );    print plot( plot )END`
Output:

```                                                                          @@@@
@@@@
@@
@@
@@
@@
@@
@@
@@
@@
@@@@@@@@
@@@@@@@@@@@@@@
@@@@@@@@
@@
@@@
@@
@@@
@@
@                                                          @@@
@@                                                         @@
@@                                                        @@@
@                                                        @@
@@                                                      @@@
@@                                                      @@
@@                                                     @@@
@                                                     @@
@@                                                    @@
@@                                                    @@
@@@@@@@                                               @
@@@@@@@@@@@@@@                                 @@
@@@@@@@@@                         @
@@@@                      @
@@@@@                 @@
@@@@@             @
@@@@         @@
@@@@@     @
@@@@ @@
@@@

```

## Julia

Choose BSpline D of 2, ie degree 1.

`using Graphics, Plots Point(t::Tuple) = Vec2(Float64(t[1]), Float64(t[2]))const controlpoints = Point.([(171, 171), (185, 111), (202, 109), (202, 189), (328, 160),    (208, 254), (241, 330), (164,252), (69, 278), (139, 208), (72, 148), (168, 172)])plt = plot(map(a -> a.x, controlpoints), map(a -> a.y, controlpoints))savefig(plt, "BSplineplot.png")`

## Lua

Translation of: Wren
`local function Range(from, to)	local range = {}	for n = from, to do  table.insert(range, n)  end	return rangeend local function Bspline(controlPoints, k)	return {		controlPoints = controlPoints,		n = #controlPoints,		k = k,		t = Range(1, #controlPoints+k), -- Use a uniform knot vector, delta=1.	}end local function helper(bspline, i, k, x)	return (bspline.t[i+k] ~= bspline.t[i])	   and (x - bspline.t[i]) / (bspline.t[i+k] - bspline.t[i])	   or  0end local function calculateBspline(bspline, i, k, x)	if k == 1 then		return (bspline.t[i] <= x and x < bspline.t[i+1]) and 1 or 0	end	return (  helper(bspline, i  , k-1, x)) * calculateBspline(bspline, i  , k-1, x)	     + (1-helper(bspline, i+1, k-1, x)) * calculateBspline(bspline, i+1, k-1, x)end local function round(n)	return math.floor(n+.5)end local function getBsplinePoints(bspline)	local points = {} 	for x = bspline.t[bspline.k], bspline.t[bspline.n+1]-1 do		local sumX = 0		local sumY = 0 		for i = 1, bspline.n do			local f = calculateBspline(bspline, i, bspline.k, x)			sumX    = sumX + f * bspline.controlPoints[i].x			sumY    = sumY + f * bspline.controlPoints[i].y		end		table.insert(points, {x=round(sumX), y=round(sumY)})	end 	return pointsend local function Plot(unscaledWidth,unscaledHeight, scaleX,scaleY)	local plot = {		width  = round(unscaledWidth  * scaleX),		height = round(unscaledHeight * scaleY),		scaleX = scaleX,		scaleY = scaleY,	}	for row = 1, plot.height do		plot[row] = {}	end	return plotend local function raytrace(x,y, x2,y2, visit)	local dx   = math.abs(x2 - x)	local dy   = math.abs(y2 - y)	local n    = 1 + dx + dy	local dirX = (x2 > x) and 1 or -1	local dirY = (y2 > y) and 1 or -1	local err  = dx - dy	dx, dy     = 2*dx, 2*dy 	for n = 1, n do		visit(x, y)		if err > 0 then  x, err = x+dirX, err-dy		else             y, err = y+dirY, err+dx  end	endend local function plotLine(plot, x1,y1, x2,y2)	raytrace(x1,y1, x2,y2, function(x, y)		if x >= 0 and y >= 0 and x < plot.width and y < plot.height then			plot[y+1][x+1] = true		end	end)end local function plotBspline(bspline, plot)	if bspline.k > bspline.n or bspline.k < 1 then		error("k (= "..bspline.k..") can't be more than "..bspline.n.." or less than 1.")	end	local points = getBsplinePoints(bspline) 	-- Plot the curve.	for i = 1, #points-1 do		local p1 = points[i]		local p2 = points[i+1]		plotLine(plot,			round(p1.x*plot.scaleX), round(p1.y*plot.scaleY),			round(p2.x*plot.scaleX), round(p2.y*plot.scaleY)		)	endend local function printPlot(plot)	for row = 1, plot.height do		for column = 1, plot.width do			io.write(plot[row][column] and "@" or " ")		end		io.write("\n")	endend local controlPoints = {	{x=171, y=171}, {x=185, y=111}, {x=202, y=109}, {x=202, y=189}, {x=328, y=160}, {x=208, y=254},	{x=241, y=330}, {x=164, y=252}, {x= 69, y=278}, {x=139, y=208}, {x= 72, y=148}, {x=168, y=172},}local k       = 4 -- Polynomial degree is one less than this i.e. cubic.local bspline = Bspline(controlPoints, k) local scaleX = .4 -- Since we print the plot to the console as text let's scale things appropriately.local scaleY = .2local plot   = Plot(350,350, scaleX,scaleY)plotBspline(bspline, plot) printPlot(plot)`
Output:
```                                   @@@@
@@@@
@@
@@
@@
@@
@@
@@
@@
@@
@@@@@@@@
@@@@@@@@@@@@@@
@@@@@@@@
@@
@@@
@@
@@@
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@                                                          @@@
@@                                                         @@
@@                                                        @@@
@                                                        @@
@@                                                      @@@
@@                                                      @@
@@                                                     @@@
@                                                     @@
@@                                                    @@
@@                                                    @@
@@@@@@@                                               @
@@@@@@@@@@@@@@                                 @@
@@@@@@@@@                         @
@@@@                      @
@@@@@                 @@
@@@@@             @
@@@@         @@
@@@@@     @
@@@@ @@
@@@
```

## Mathematica/Wolfram Language

`Graphics[ BSplineCurve[{{171, 171}, {185, 111}, {202, 109}, {202, 189}, {328,     160}, {208, 254}, {241, 330}, {164, 252}, {69, 278}, {139,     208}, {72, 148}, {168, 172}}, SplineClosed -> True,   SplineDegree -> 2]]`
Output:

Outputs a graphical representation of a B-spline.

## Perl

Translation of: Raku
`use strict;use warnings;use Class::Struct;use Cairo; { package Line;    struct( A => '@', B => '@');} my (\$WIDTH,  \$HEIGHT,  \$W_LINE,  \$CURVE_F,  \$DETACHED,            \$OUTPUT  ) =   (   400,      400,        2,      0.25,          0,  'run/b-spline.png' ); my @pt = (   [171, 171], [185, 111], [202, 109], [202, 189], [328, 160], [208, 254],   [241, 330], [164, 252], [ 69, 278], [139, 208], [ 72, 148], [168, 172]);my \$cnt = @pt; sub angle {    my(\$g) = @_;    atan2 \$g->B->[1] - \$g->A->[1], \$g->B->[0] - \$g->A->[0]} sub control_points {    my(\$g, \$l) = @_;     my \$h = Line->new;    my \$lgt = sqrt( (\$g->B->[0] - \$l->A->[0])**2 + (\$g->B->[1] - \$l->A->[1])**2 );     @{\$h->B} = @{\$l->A};    @{\$h->A} = (\$g->B->[0] - \$lgt * cos(angle \$g) , \$g->B->[1] - \$lgt * sin(angle \$g));    my \$a    = angle \$h;    my @p1   = (\$g->B->[0] + \$lgt * cos(\$a) * \$CURVE_F,  \$g->B->[1] + \$lgt * sin(\$a) * \$CURVE_F);     @{\$h->A} = @{\$g->B};    @{\$h->B} = (\$l->A->[0] + \$lgt * cos(angle \$l) , \$l->A->[1] + \$lgt * sin(angle \$l));    \$a       = angle \$h;    my @p2   = (\$l->A->[0] - \$lgt * cos(\$a) * \$CURVE_F,  \$l->A->[1] - \$lgt * sin(\$a) * \$CURVE_F);     \@p1, \@p2} my \$surf = Cairo::ImageSurface->create ('argb32', \$WIDTH, \$HEIGHT);my \$cr   = Cairo::Context->create (\$surf);\$cr->set_line_width(\$W_LINE);\$cr->move_to(\$pt[\$DETACHED - 1 + \$cnt][0], \$pt[\$DETACHED - 1 + \$cnt][1]); my Line (\$g,\$l);for my \$j (\$DETACHED..\$cnt-1) {    \$g = Line->new( A=>\$pt[(\$j + \$cnt - 2) % \$cnt], B=>\$pt[(\$j + \$cnt - 1) % \$cnt]);    \$l = Line->new( A=>\$pt[(\$j + \$cnt + 0) % \$cnt], B=>\$pt[(\$j + \$cnt + 1) % \$cnt]);    my(\$p1,\$p2) = control_points(\$g, \$l);    \$cr->curve_to(\$\$p1[0], \$\$p1[1], \$\$p2[0], \$\$p2[1], \$pt[\$j][0], \$pt[\$j][1]);}\$cr->stroke;\$surf->write_to_png(\$OUTPUT);`

Output: b-spline.png (offsite image)

## Phix

Translation of: Wren
Library: Phix/pGUI
Library: Phix/online

You can run this online here.

```--
-- demo\rosetta\B-spline.exw
-- =========================
--
--  Use +/- to change the order between k = 1 and k = 4.
--
with javascript_semantics
include pGUI.e
include IupGraph.e

constant ctrl_points = {{171, 171}, {185, 111}, {202, 109}, {202, 189}, {328, 160}, {208, 254},
{241, 330}, {164, 252}, { 69, 278}, {139, 208}, { 72, 148}, {168, 172}}
integer k = 2, n
sequence t

function w(integer i, k, x)  // B-spline helper function
return iff(t[i+k]!=t[i] ? (x-t[i])/(t[i+k]-t[i]) : 0 )
end function

function b(integer i, k, x)  // B-spline function
if k==1 then return iff(t[i]<=x and x<t[i+1] ? 1 : 0) end if
return w(i,k-1,x)*b(i,k-1,x) + (1-w(i+1,k-1,x))*b(i+1,k-1,x)
end function

function b_spline(Ihandle graph)
n = length(ctrl_points)
t = tagset(n+1+k) // use a uniform knot vector, delta = 1
assert(k<=n+1 and k>=1,"k (= %d) cannot be more than %d or less than 1.",{k,n+1})

sequence px = {}, py = {}
for x=t[k] to t[n+1] do
atom sumX = 0,
sumY = 0
for i=1 to n do
atom f = b(i,k,x)
sumX += f*ctrl_points[i][1]
sumY += f*ctrl_points[i][2]
end for
px &= round(sumX)
py &= round(sumY)
end for

integer xtick = 40,
ytick = 40,
xmin = trunc(min(px)/xtick)*xtick,
xmax = ceil(max(px)/xtick)*xtick,
ymin = trunc(min(py)/ytick)*ytick,
ymax = ceil(max(py)/ytick)*ytick

IupSetInt(graph,"XTICK",xtick)
IupSetInt(graph,"XMIN",xmin)
IupSetInt(graph,"XMAX",xmax)
IupSetInt(graph,"YTICK",ytick)
IupSetInt(graph,"YMIN",ymin)
IupSetInt(graph,"YMAX",ymax)
sequence graphdata = {{px,py,CD_BLUE}}
return graphdata
end function

procedure set_title(Ihandle dlg)
IupSetStrAttribute(dlg, "TITLE", "B-spline curve (order k = %d)",{k})
end procedure

function key_cb(Ihandle dlg, atom c)
if c=K_ESC then return IUP_CLOSE end if
if c='+' then k = min(k+1,4) end if
if c='-' then k = max(k-1,1) end if
set_title(dlg)
IupRedraw(dlg)
return IUP_IGNORE
end function

procedure main()
IupOpen()
Ihandle graph = IupGraph(b_spline,`RASTERSIZE=600x600`)
Ihandle dlg = IupDialog(graph)
IupSetCallback(dlg, "KEY_CB", Icallback("key_cb"))
set_title(dlg)
IupShow(dlg)
if platform()!=JS then
IupMainLoop()
IupClose()
end if
end procedure

main()
```

## Raku

A minimal translation of this C program, by Bernhard R. Fischer.

`# 20211112 Raku programming solution use Cairo; # class point_t { has Num (\$.x,\$.y) is rw } # get by with two element listsclass line_t  { has (\$.A,\$.B) is rw } my (\WIDTH,  \HEIGHT,  \W_LINE,  \CURVE_F,  \DETACHED,          \OUTPUT  ) =        400,      400,        2,      0.25,          0,  './b-spline.png' ; my \cnt = #`(Number of points) ( my \pt = [   [171, 171], [185, 111], [202, 109], [202, 189], [328, 160], [208, 254],   [241, 330], [164, 252], [ 69, 278], [139, 208], [ 72, 148], [168, 172], ]).elems; sub angle(\g) { atan2(g.B.[1] - g.A.[1], g.B.[0] - g.A.[0]) } sub control_points(\g, \l, @p1, @p2){ #`[ This function calculates the control points. It takes two lines g and l as * arguments but it takes three lines into account for calculation. This is * line g (P0/P1), line h (P1/P2), and line l (P2/P3). The control points being * calculated are actually those for the middle line h, this is from P1 to P2. * Line g is the predecessor and line l the successor of line h. * @param g Pointer to first line (P0 to P1) * @param l Pointer to third line (P2 to P3) * @param p1 Pointer to memory of first control point.  * @param p2 Pointer to memory of second control point. ]    my \h = \$ = line_t.new;    my \lgt = sqrt([+]([ g.B.[0]-l.A.[0], g.B.[1]-l.A.[1] ]>>²));#length of P1 to P2    h.B = l.A.clone;  # end point of 1st tangent   # start point of tangent at same distance as end point along 'g'   h.A = g.B.[0] - lgt * cos(angle g) , g.B.[1] - lgt * sin(angle g);    my \$a = angle h ; # angle of tangent   # 1st control point on tangent at distance 'lgt * CURVE_F'   @p1 = g.B.[0] + lgt * cos(\$a) * CURVE_F,  g.B.[1] + lgt * sin(\$a) * CURVE_F;    h.A = g.B.clone; # start point of 2nd tangent   # end point of tangent at same distance as start point along 'l'   h.B = l.A.[0] + lgt * cos(angle l) , l.A.[1] + lgt * sin(angle l);    \$a = angle h; # angle of tangent   # 2nd control point on tangent at distance 'lgt * CURVE_F'   @p2 = l.A.[0] - lgt * cos(\$a) * CURVE_F,  l.A.[1] - lgt * sin(\$a) * CURVE_F;}  given Cairo::Image.create(Cairo::FORMAT_ARGB32, WIDTH, HEIGHT) {   given Cairo::Context.new(\$_) {      my line_t (\$g,\$l);      my (@p1,@p2);       .line_width = W_LINE;      .move_to(pt[DETACHED - 1 + cnt].[0], pt[DETACHED - 1 + cnt].[1]);       for DETACHED..^cnt -> \j {          \$g = line_t.new: A=>pt[(j + cnt - 2) % cnt], B=>pt[(j + cnt - 1) % cnt];         \$l = line_t.new: A=>pt[(j + cnt + 0) % cnt], B=>pt[(j + cnt + 1) % cnt];          # Calculate controls points for points pt[j-1] and pt[j].         control_points(\$g, \$l, @p1, @p2);          .curve_to(@p1[0], @p1[1], @p2[0], @p2[1], pt[j].[0], pt[j].[1]);      }      .stroke;   };   .write_png(OUTPUT) and die # C return}`

Output: (Offsite image file)

## Wren

Library: DOME

In the absence of any clarification on what to use (see Talk page), the following uses a degree of 3 (i.e order k = 4) and a uniform knot vector from 1 to 16 (as there are 12 control points) with a delta of 1.

If one uses a value for k of 1, then the script will simply plot the control points as in the Julia example.

`import "dome" for Window, Processimport "graphics" for Canvas, Color class BSpline {    construct new(width, height, cpoints, k) {        Window.resize(width, height)        Canvas.resize(width, height)        Window.title = "B-spline curve"        _p = cpoints        _n = cpoints.count - 1        _k = k        _t = (1.._n + 1 + k).toList // use a uniform knot vector, delta = 1    }     // B-spline helper function    w(i, k, x) { (_t[i+k] != _t[i]) ? (x - _t[i]) / (_t[i+k] - _t[i]) : 0 }     // B-spline function    b(i, k, x) {        if (k == 1) return (_t[i] <= x  &&  x < _t[i + 1]) ? 1 : 0        return w(i, k-1, x) * b(i, k-1, x) + (1 - w(i+1, k-1, x)) * b(i+1, k-1, x)    }     // B-spline points    p() {        var bpoints = []        for (x in _t[_k-1]..._t[_n + 1]) {            var sumX = 0            var sumY = 0            for (i in 0.._n) {                var f = b(i, _k, x)                sumX = sumX + f * _p[i][0]                sumY = sumY + f * _p[i][1]            }            bpoints.add([sumX.round, sumY.round])         }         return bpoints    }     init() {        if (_k > _n + 1 || _k < 1) {            System.print("k (= %(_k)) can't be more than %(_n+1) or less than 1.")            Process.exit()        }        var bpoints = p()        // plot the curve        for (i in 1...bpoints.count) {            Canvas.line(bpoints[i-1][0], bpoints[i-1][1], bpoints[i][0], bpoints[i][1], Color.white)        }    }     update() {}     draw(alpha) {}} var cpoints = [    [171, 171], [185, 111], [202, 109], [202, 189], [328, 160], [208, 254],    [241, 330], [164, 252], [ 69, 278], [139, 208], [ 72, 148], [168, 172]]var k = 4 // polynomial degree is one less than this i.e. cubicvar Game = BSpline.new(400, 400, cpoints, k)`