Curve that touches three points

From Rosetta Code
Curve that touches three points 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.

Draw a curve that touches 3 points (1 starting point, 2 medium, 3 final point)

  1.  Do not use functions of a library, implement the curve() function yourself
  2.  coordinates:(x,y) starting point (10,10) medium point (100,200) final point (200,10)



Go[edit]

Library: Go Graphics


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.

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")
}

Perl 6[edit]

Works with: Rakudo version 2018.10

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.

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);
}
myx = $x1 - $x0;
my $Δy = abs($y1 - $y0);
my $error = 0;
my $Δerror = $Δy /x;
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;
}
}

See Curve-3-points-perl6.png (offsite .png image)