Curve that touches three points: Difference between revisions
| Line 86: | Line 86: | ||
x = a-r:0.25:a+r |
x = a-r:0.25:a+r |
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y0 = sqrt.(r^2 .- (x .- a).^2) |
y0 = sqrt.(r^2 .- (x .- a).^2) |
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scene = lines(x, y0 .+ b) |
scene = lines(x, y0 .+ b, color = :red) |
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lines!(scene, x, b .- y0) |
lines!(scene, x, b .- y0, color = :red) |
||
scatter!(scene, [p.x for p in allp], [p.y for p in allp]) |
scatter!(scene, [p.x for p in allp], [p.y for p in allp], color = :black) |
||
</lang>{{out}} |
</lang>{{out}} |
||
<pre> |
<pre> |
||
Revision as of 00:55, 19 March 2019
Draw a curve that touches 3 points (1 starting point, 2 medium, 3 final point)
- Do not use functions of a library, implement the curve() function yourself
- coordinates:(x,y) starting point (10,10) medium point (100,200) final point (200,10)
Go
There are, of course, an infinity of curves which can be fitted to 3 points. The most obvious solution is to fit a quadratic curve (using Lagrange interpolation) and so that's what we do here.
As we're not allowed to use library functions to draw the curve, we instead divide the x-axis of the curve between successive points into equal segments and then join the resulting points with straight lines.
The resulting 'curve' is then saved to a .png file where it can be viewed with a utility such as EOG. <lang go>package main
import "github.com/fogleman/gg"
var p = [3]gg.Point{{10, 10}, {100, 200}, {200, 10}}
func lagrange(x float64) float64 {
return (x-p[1].X)*(x-p[2].X)/(p[0].X-p[1].X)/(p[0].X-p[2].X)*p[0].Y +
(x-p[0].X)*(x-p[2].X)/(p[1].X-p[0].X)/(p[1].X-p[2].X)*p[1].Y +
(x-p[0].X)*(x-p[1].X)/(p[2].X-p[0].X)/(p[2].X-p[1].X)*p[2].Y
}
func getPoints(n int) []gg.Point {
pts := make([]gg.Point, 2*n+1)
dx := (p[1].X - p[0].X) / float64(n)
for i := 0; i < n; i++ {
x := p[0].X + dx*float64(i)
pts[i] = gg.Point{x, lagrange(x)}
}
dx = (p[2].X - p[1].X) / float64(n)
for i := n; i < 2*n+1; i++ {
x := p[1].X + dx*float64(i-n)
pts[i] = gg.Point{x, lagrange(x)}
}
return pts
}
func main() {
const n = 50 // more than enough for this
dc := gg.NewContext(210, 210)
dc.SetRGB(1, 1, 1) // White background
dc.Clear()
for _, pt := range getPoints(n) {
dc.LineTo(pt.X, pt.Y)
}
dc.SetRGB(0, 0, 0) // Black curve
dc.SetLineWidth(1)
dc.Stroke()
dc.SavePNG("quadratic_curve.png")
}</lang>
Julia
To make things more specific, find the circle determined by the points. The curve is then the arc between the 3 points. <lang julia>using Makie
struct Point; x::Float64; y::Float64; end
- Find a circle passing through the 3 points
const p1 = Point(10, 10) const p2 = Point(100, 200) const p3 = Point(200, 10) const allp = [p1, p2, p3]
- set up problem matrix and solve.
- if (x - a)^2 + (y - b)^2 = r^2 then for some D, E, F, x^2 + y^2 + Dx + Ey + F = 0
- therefore Dx + Ey + F = -x^2 - y^2
v = zeros(Int, 3) m = zeros(Int, 3, 3) for row in 1:3
m[row, 1:3] .= [allp[row].x, allp[row].y, 1] v[row] = -(allp[row].x)^2 - (allp[row].y)^2
end q = (m \ v) # [-210.0, -162.632, 3526.32] a, b, r = -q[1] / 2, -q[2] / 2, sqrt((q[1]^2/4) + q[2]^2/4 - q[3])
println("The circle with center at x = $a, y = $b and radius $r.")
x = a-r:0.25:a+r y0 = sqrt.(r^2 .- (x .- a).^2) scene = lines(x, y0 .+ b, color = :red) lines!(scene, x, b .- y0, color = :red) scatter!(scene, [p.x for p in allp], [p.y for p in allp], color = :black)
</lang>
- Output:
The circle with center at x = 105.0, y = 81.31578947368422 and radius 118.78948534384199.
Perl
Hilbert curve task code repeated here, with the addition that the 3 task-required points are marked. Mostly satisfies the letter-of-the-law of task specification while (all in good fun) subverting the spirit of the thing. <lang perl>use SVG; use List::Util qw(max min);
use constant pi => 2 * atan2(1, 0);
- Compute the curve with a Lindemayer-system
%rules = (
A => '-BF+AFA+FB-', B => '+AF-BFB-FA+'
); $hilbert = 'A'; $hilbert =~ s/([AB])/$rules{$1}/eg for 1..6;
- Draw the curve in SVG
($x, $y) = (0, 0); $theta = pi/2; $r = 5;
for (split //, $hilbert) {
if (/F/) {
push @X, sprintf "%.0f", $x;
push @Y, sprintf "%.0f", $y;
$x += $r * cos($theta);
$y += $r * sin($theta);
}
elsif (/\+/) { $theta += pi/2; }
elsif (/\-/) { $theta -= pi/2; }
}
$max = max(@X,@Y); $xt = -min(@X)+10; $yt = -min(@Y)+10; $svg = SVG->new(width=>$max+20, height=>$max+20); $points = $svg->get_path(x=>\@X, y=>\@Y, -type=>'polyline'); $svg->rect(width=>"100%", height=>"100%", style=>{'fill'=>'black'}); $svg->polyline(%$points, style=>{'stroke'=>'orange', 'stroke-width'=>1}, transform=>"translate($xt,$yt)"); my $task = $svg->group( id => 'task-points', style => { stroke => 'red', fill => 'red' },); $task->circle( cx => 10, cy => 10, r => 1, id => 'point1' ); $task->circle( cx => 100, cy => 200, r => 1, id => 'point2' ); $task->circle( cx => 200, cy => 10, r => 1, id => 'point3' );
open $fh, '>', 'curve-3-points.svg'; print $fh $svg->xmlify(-namespace=>'svg'); close $fh;</lang> Hilbert curve passing through 3 defined points (offsite image)
Perl 6
Kind of bogus. There are an infinite number of curves that pass through those three points. I'll assume a quadratic curve. Lots of bits and pieces borrowed from other tasks to avoid relying on library functions.
Saved as a png for wide viewing support. Note that png coordinate systems have 0,0 in the upper left corner.
<lang perl6>use Image::PNG::Portable;
- Solve for a quadratic line that passes through those points
my (\a, \b, \c) =
rref([[10², 10, 1, 10],[100², 100, 1, 200],[200², 200, 1, 10]])[*;*-1];
- General case quadratic line equation
sub f (\x) { a*x² + b*x + c }
- Scale it up a bit for display
my $scale = 2;
my ($w, $h) = (500, 500); my $png = Image::PNG::Portable.new: :width($w), :height($h);
my ($lastx, $lasty) = 8, f(8).round; (9 .. 202).map: -> $x {
my $f = f($x).round; line($lastx, $lasty, $x, $f, $png, [0,255,127]); ($lastx, $lasty) = $x, $f;
}
- Highlight the 3 defining points
dot(|$_, $png, 2) for (10,10,[255,0,0]), (100,200,[255,0,0]), (200,10,[255,0,0]);
$png.write: 'Curve-3-points-perl6.png';
- Assorted helper routines
sub rref (@m) {
return unless @m;
my ($lead, $rows, $cols) = 0, +@m, +@m[0];
for ^$rows -> $r {
$lead < $cols or return @m;
my $i = $r;
until @m[$i;$lead] {
++$i == $rows or next;
$i = $r;
++$lead == $cols and return @m;
}
@m[$i, $r] = @m[$r, $i] if $r != $i;
my $lv = @m[$r;$lead];
@m[$r] »/=» $lv;
for ^$rows -> $n {
next if $n == $r;
@m[$n] »-=» @m[$r] »*» (@m[$n;$lead] // 0);
}
++$lead;
}
@m
}
sub line($x0 is copy, $y0 is copy, $x1 is copy, $y1 is copy, $png, @rgb) {
my $steep = abs($y1 - $y0) > abs($x1 - $x0);
($x0,$y0,$x1,$y1) »*=» $scale;
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;
next if $y < 0;
for $x0 .. $x1 -> $x {
next if $x < 0;
if $steep {
$png.set($y, $x, |@rgb);
} else {
$png.set($x, $y, |@rgb);
}
$error += $Δerror;
if $error >= 0.5 {
$y += $y-step;
$error -= 1.0;
}
}
}
sub dot ($X is copy, $Y is copy, @rgb, $png, $radius = 3) {
($X, $Y) »*=» $scale;
for ($X X+ -$radius .. $radius) X ($Y X+ -$radius .. $radius) -> ($x, $y) {
$png.set($x, $y, |@rgb) if ( $X - $x + ($Y - $y) * i ).abs <= $radius;
}
}</lang> See Curve-3-points-perl6.png (offsite .png image)
Phix
<lang Phix>include pGUI.e
Ihandle dlg, canvas cdCanvas cddbuffer, cdcanvas
enum X, Y constant p = {{10,10},{100,200},{200,10}}
function lagrange(atom x)
return (x - p[2][X])*(x - p[3][X])/(p[1][X] - p[2][X])/(p[1][X] - p[3][X])*p[1][Y] +
(x - p[1][X])*(x - p[3][X])/(p[2][X] - p[1][X])/(p[2][X] - p[3][X])*p[2][Y] +
(x - p[1][X])*(x - p[2][X])/(p[3][X] - p[1][X])/(p[3][X] - p[2][X])*p[3][Y]
end function
function getPoints(integer n)
sequence pts = {}
atom {dx,pt,cnt} := {(p[2][X] - p[1][X])/n, p[1][X], n}
for j=1 to 2 do
for i=0 to cnt do
atom x := pt + dx*i;
pts = append(pts,{x,lagrange(x)});
end for
{dx,pt,cnt} = {(p[3][X] - p[2][X])/n, p[2][X], n+1};
end for
return pts
end function
procedure draw_cross(sequence xy)
integer {x,y} = xy
cdCanvasLine(cddbuffer, x-3, y, x+3, y)
cdCanvasLine(cddbuffer, x, y-3, x, y+3)
end procedure
function redraw_cb(Ihandle /*ih*/, integer /*posx*/, integer /*posy*/)
cdCanvasActivate(cddbuffer)
cdCanvasSetForeground(cddbuffer, CD_BLUE)
cdCanvasBegin(cddbuffer,CD_OPEN_LINES)
atom {x,y} = {p[1][X], p[1][Y]}; -- curve starting point
cdCanvasVertex(cddbuffer, x, y)
sequence pts = getPoints(50)
for i=1 to length(pts) do
{x,y} = pts[i]
cdCanvasVertex(cddbuffer, x, y)
end for
cdCanvasEnd(cddbuffer)
cdCanvasSetForeground(cddbuffer, CD_RED)
for i=1 to length(p) do draw_cross(p[i]) end for
cdCanvasFlush(cddbuffer)
return IUP_DEFAULT
end function
function map_cb(Ihandle ih)
cdcanvas = cdCreateCanvas(CD_IUP, ih) cddbuffer = cdCreateCanvas(CD_DBUFFER, cdcanvas) cdCanvasSetBackground(cddbuffer, CD_WHITE) return IUP_DEFAULT
end function
procedure main()
IupOpen()
canvas = IupCanvas(NULL)
IupSetAttribute(canvas, "RASTERSIZE", "220x220")
IupSetCallback(canvas, "MAP_CB", Icallback("map_cb"))
dlg = IupDialog(canvas,"DIALOGFRAME=YES")
IupSetAttribute(dlg, "TITLE", "Colour wheel")
IupSetCallback(canvas, "ACTION", Icallback("redraw_cb"))
IupCloseOnEscape(dlg)
IupMap(dlg) IupSetAttribute(canvas, "RASTERSIZE", NULL) IupShowXY(dlg,IUP_CENTER,IUP_CENTER) IupMainLoop() IupClose()
end procedure main()</lang>
zkl
Uses Image Magick and the PPM class from http://rosettacode.org/wiki/Bitmap/Bresenham%27s_line_algorithm#zkl <lang zkl>const X=0, Y=1; // p.X == p[X] var p=L(L(10.0, 10.0), L(100.0, 200.0), L(200.0, 10.0)); // (x,y)
fcn lagrange(x){ // float-->float
(x - p[1][X])*(x - p[2][X])/(p[0][X] - p[1][X])/(p[0][X] - p[2][X])*p[0][Y] + (x - p[0][X])*(x - p[2][X])/(p[1][X] - p[0][X])/(p[1][X] - p[2][X])*p[1][Y] + (x - p[0][X])*(x - p[1][X])/(p[2][X] - p[0][X])/(p[2][X] - p[1][X])*p[2][Y]
}
fcn getPoints(n){ // int-->( (x,y) ..)
pts:=List.createLong(2*n+1);
dx,pt,cnt := (p[1][X] - p[0][X])/n, p[0][X], n;
do(2){
foreach i in (cnt){
x:=pt + dx*i; pts.append(L(x,lagrange(x)));
}
dx,pt,cnt = (p[2][X] - p[1][X])/n, p[1][X], n+1;
}
pts
}
fcn main{
var [const] n=50; // more than enough for this
img,color := PPM(210,210,0xffffff), 0; // white background, black curve
foreach x,y in (p){ img.cross(x.toInt(),y.toInt(), 0xff0000) } // mark 3 pts
a,b := p[0][X].toInt(), p[0][Y].toInt(); // curve starting point
foreach x,y in (getPoints(n)){
x,y = x.toInt(),y.toInt();
img.line(a,b, x,y, color); // can only deal with ints
a,b = x,y;
}
img.writeJPGFile("quadraticCurve.zkl.jpg");
}();</lang>
- Output:
Image at quadratic curve
