Curve that touches three points: Difference between revisions
Line 164: | Line 164: | ||
=={{header|Mathematica}}/{{header|Wolfram Language}}== |
=={{header|Mathematica}}/{{header|Wolfram Language}}== |
||
===Built-in=== |
|||
Built-in function of Mathematica (not a library): |
|||
<lang Mathematica>pts = {{10, 10}, {100, 200}, {200, 10}}; |
<lang Mathematica>pts = {{10, 10}, {100, 200}, {200, 10}}; |
||
cs = Circumsphere[pts] |
cs = Circumsphere[pts] |
||
Line 171: | Line 171: | ||
Outputs the circle: |
Outputs the circle: |
||
<pre>Sphere[{105, 1545/19}, (5 Sqrt[203762])/19]</pre> |
<pre>Sphere[{105, 1545/19}, (5 Sqrt[203762])/19]</pre> |
||
and a graphical representation of the input points and the circle. |
|||
===Built-in=== |
|||
<lang Mathematica>pts = {{10, 10}, {100, 200}, {200, 10}}; |
|||
createCircle[{{x1_, y1_}, {x2_, y2_}, {x3_, y3_}}] := |
|||
With[{a = Det[({{x1, y1, 1}, {x2, y2, 1}, {x3, y3, 1}})], |
|||
d = -Det[({{x1^2 + y1^2, y1, 1}, {x2^2 + y2^2, y2, |
|||
1}, {x3^2 + y3^2, y3, 1}})], |
|||
e = Det[({{x1^2 + y1^2, x1, 1}, {x2^2 + y2^2, x2, 1}, {x3^2 + y3^2, |
|||
x3, 1}})], |
|||
f = -Det[({{x1^2 + y1^2, x1, y1}, {x2^2 + y2^2, x2, |
|||
y2}, {x3^2 + y3^2, x3, y3}})]}, |
|||
Circle[{-(d/(2 a)), -(e/(2 a))}, Sqrt[(d^2 + e^2)/(4 a^2) - f/a]]] |
|||
cs = createCircle[pts] |
|||
Graphics[{PointSize[Large], Point[pts], cs}]</lang> |
|||
{{out}} |
|||
Outputs the circle: |
|||
<pre>Circle[{105, 1545/19}, (5 Sqrt[203762])/19]</pre> |
|||
and a graphical representation of the input points and the circle. |
and a graphical representation of the input points and the circle. |
||
Revision as of 12:50, 11 September 2021
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)
AutoHotkey
<lang AutoHotkey>QuadraticCurve(p1,p2,p3){ ; Y = aX^2 + bX + c x1:=p1.1, y1:=p1.2, x2:=p2.1, y2:=p2.2, x3:=p3.1, y3:=p3.2 m:=x1-x2, n:=x3-x2, m:= ((m*n)<0?-1:1) * m a:=(n*(y1-y2)+m*(y3-y2)) / (n*(x1**2 - x2**2) + m*(x3**2 - x2**2)) b:=((y3-y2) - (x3**2 - x2**2)*a) / (x3-x2) c:=y1 - a*x1**2 - b*x1 return [a,b,c] }</lang> Examples:<lang AutoHotkey>P1 := [10,10], P2 := [100,200], P3 := [200,10] v := QuadraticCurve(p1,p2,p3) a := v.1, b:= v.2, c:= v.3 for i, X in [10,100,200]{ Y := a*X**2 + b*X + c ; Y = aX^2 + bX + c res .= "[" x ", " y "]`n" } MsgBox % "Y = " a " X^2 " (b>0?"+":"") b " X " (c>0?"+":"") c " `n" res</lang>
- for plotting, use code from RosettaCode: Plot Coordinate Pairs
Outputs:
Y = -0.021111 X^2 +4.433333 X -32.222222 [10, 10.000000] [100, 200.000000] [200, 10.000000]
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>
J
NB. coordinates:(x,y) starting point (10,10) medium point (100,200) final point (200,10) X=: 10 100 200 Y=: 10 200 10 NB. matrix division computes polynomial coefficients NB. %. implements singular value decomposition NB. in other words, we can also get best fit polynomials of lower order. polynomial=: (Y %. (^/ ([: i. #)) X)&p. assert 10 200 10 -: polynomial X NB. test Filter=: (#~`)(`:6) Round=: adverb def '<.@:(1r2&+)&.:(%&m)' assert 100 120 -: 100 8 Round 123 NB. test, round 123 to nearest multiple of 100 and of 8 NB. libraries not permitted, character cell graphics are used. GRAPH=: 50 50 $ ' ' NB. is an array of spaces NB. place the axes GRAPH=: '-' [`(([:<0; i.@:#)@:])`]} GRAPH GRAPH=: '|' [`(([:<0;~i.@:#)@:])`]} GRAPH GRAPH=: '+' [`((<0;0)"_)`]} GRAPH NB. origin NB. clip the domain. EXES=: ((<:&(>./X) *. (<./X)&<:))Filter 5 * i. 200 WHYS=: polynomial EXES NB. draw the curve 1j1 #"1 |. 'X' [`((<"1 WHYS ;&>&:([: 1 Round %&5) EXES)"_)`]} GRAPH NB. were we to use a library: load'plot' 'title 3 point fit' plot (j. polynomial) i.201
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], markersize = r / 10)
</lang>
- Output:
The circle with center at x = 105.0, y = 81.31578947368422 and radius 118.78948534384199.
Mathematica /Wolfram Language
Built-in
<lang Mathematica>pts = {{10, 10}, {100, 200}, {200, 10}}; cs = Circumsphere[pts] Graphics[{PointSize[Large], Point[pts], cs}]</lang>
- Output:
Outputs the circle:
Sphere[{105, 1545/19}, (5 Sqrt[203762])/19]
and a graphical representation of the input points and the circle.
Built-in
<lang Mathematica>pts = {{10, 10}, {100, 200}, {200, 10}}; createCircle[{{x1_, y1_}, {x2_, y2_}, {x3_, y3_}}] :=
With[{a = Det[({{x1, y1, 1}, {x2, y2, 1}, {x3, y3, 1}})], d = -Det[({{x1^2 + y1^2, y1, 1}, {x2^2 + y2^2, y2, 1}, {x3^2 + y3^2, y3, 1}})], e = Det[({{x1^2 + y1^2, x1, 1}, {x2^2 + y2^2, x2, 1}, {x3^2 + y3^2, x3, 1}})], f = -Det[({{x1^2 + y1^2, x1, y1}, {x2^2 + y2^2, x2, y2}, {x3^2 + y3^2, x3, y3}})]}, Circle[{-(d/(2 a)), -(e/(2 a))}, Sqrt[(d^2 + e^2)/(4 a^2) - f/a]]]
cs = createCircle[pts] Graphics[{PointSize[Large], Point[pts], cs}]</lang>
- Output:
Outputs the circle:
Circle[{105, 1545/19}, (5 Sqrt[203762])/19]
and a graphical representation of the input points and the circle.
Nim
<lang Nim>import imageman
type
FPoint = tuple[x, y: float] FPoints3 = array[3, FPoint]
func lagrange(p: FPoints3; 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
func points(p: FPoints3; n: int): seq[Point] =
result.setLen(2 * n + 1) var dx = (p[1].x - p[0].x) / float(n) for i in 0..<n: let x = p[0].x + dx * float(i) result[i] = (x.toInt, p.lagrange(x).toInt) dx = (p[2].x - p[1].x) / float(n) for i in n..2*n: let x = p[1].x + dx * float(i - n) result[i] = (x.toInt, p.lagrange(x).toInt)
const N = 50
const P: FPoints3 =[(10.0, 10.0), (100.0, 200.0), (200.0, 10.0)]
var img = initImage[ColorRGBF](210, 210) img.fill(ColorRGBF([float32 1, 1, 1])) # White background. let color = ColorRGBF([float32 0, 0, 0]) # Black. img.drawPolyline(closed = false, color, P.points(N)) img.savePNG("curve.png", compression = 9)</lang>
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)
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", "Quadratic curve") IupSetCallback(canvas, "ACTION", Icallback("redraw_cb"))
IupMap(dlg) IupSetAttribute(canvas, "RASTERSIZE", NULL) IupShowXY(dlg,IUP_CENTER,IUP_CENTER) IupMainLoop() IupClose()
end procedure main()</lang>
Raku
(formerly 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)
Wren
<lang ecmascript>import "graphics" for Canvas, Color, Point import "dome" for Window
class Game {
static init() { Window.title = "Quadratic curve" var width = 210 var height = 210 Window.resize(width, height) Canvas.resize(width, height) Canvas.cls(Color.white) var n = 50 var p = [Point.new(10, 10), Point.new(100, 200), Point.new(200, 10)] var col = Color.black // black curve quadratic(n, p, col) }
static update() {}
static draw(alpha) {}
static lagrange(p, x) { 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 }
static quadratic(n, p, col) { var pts = List.filled(2*n+1, null) var dx = (p[1].x - p[0].x) / n for (i in 0...n) { var x = p[0].x + dx*i pts[i] = Point.new(x, lagrange(p, x)) } dx = (p[2].x - p[1].x) / n for (i in n...2*n+1) { var x = p[1].x + dx*(i-n) pts[i] = Point.new(x, lagrange(p, x)) } var prev = pts[0] for (pt in pts.skip(1)) { Canvas.line(prev.x, prev.y, pt.x, pt.y, col) prev = pt } }
}</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