Superellipse
A superellipse is a geometric figure defined as the set of all points (x, y) with
where n, a and b are positive numbers.
The task: draw a superellipse with n = 2.5, and a = b = 200
EchoLisp
Link to the super-ellipse image. <lang scheme> (lib 'plot) (define (eaxpt x n) (expt (abs x) n)) (define (Ellie x y) (+ (eaxpt (// x 200) 2.5) (eaxpt (// y 200) 2.5) -1))
(plot-xy Ellie -400 -400)
→ (("x:auto" -400 400) ("y:auto" -400 400))
</lang>
J
We will fill the ellipse so that we do not have to worry about the size and shape of our pixels:
<lang J>selips=: 4 :0
'n a b'=. y 1 >: ((n^~a%~]) +&|/ n^~b%~]) i:x
)
require'viewmat' viewmat 300 selips 2.5 200 200</lang>
rosettacode does not currently support image uploads, so you'll just have to imagine what it looks like (or run the code yourself).
Java
<lang java>import java.awt.*; import java.awt.geom.Path2D; import static java.lang.Math.pow; import java.util.Hashtable; import javax.swing.*; import javax.swing.event.*;
public class SuperEllipse extends JPanel implements ChangeListener {
private double exp = 2.5;
public SuperEllipse() {
setPreferredSize(new Dimension(650, 650));
setBackground(Color.white);
setFont(new Font("Serif", Font.PLAIN, 18));
}
void drawGrid(Graphics2D g) {
g.setStroke(new BasicStroke(2));
g.setColor(new Color(0xEEEEEE));
int w = getWidth();
int h = getHeight();
int spacing = 25;
for (int i = 0; i < w / spacing; i++) {
g.drawLine(0, i * spacing, w, i * spacing);
g.drawLine(i * spacing, 0, i * spacing, w);
}
g.drawLine(0, h - 1, w, h - 1);
g.setColor(new Color(0xAAAAAA));
g.drawLine(0, w / 2, w, w / 2);
g.drawLine(w / 2, 0, w / 2, w);
}
void drawLegend(Graphics2D g) {
g.setColor(Color.black);
g.setFont(getFont());
g.drawString("n = " + String.valueOf(exp), getWidth() - 150, 45);
g.drawString("a = b = 200", getWidth() - 150, 75);
}
void drawEllipse(Graphics2D g) {
final int a = 200; // a = b
double[] points = new double[a + 1];
Path2D p = new Path2D.Double();
p.moveTo(a, 0);
// calculate first quadrant
for (int x = a; x >= 0; x--) {
points[x] = pow(pow(a, exp) - pow(x, exp), 1 / exp); // solve for y
p.lineTo(x, -points[x]);
}
// mirror to others
for (int x = 0; x <= a; x++)
p.lineTo(x, points[x]);
for (int x = a; x >= 0; x--)
p.lineTo(-x, points[x]);
for (int x = 0; x <= a; x++)
p.lineTo(-x, -points[x]);
g.translate(getWidth() / 2, getHeight() / 2);
g.setStroke(new BasicStroke(2));
g.setColor(new Color(0x25B0C4DE, true));
g.fill(p);
g.setColor(new Color(0xB0C4DE)); // LightSteelBlue
g.draw(p);
}
@Override
public void paintComponent(Graphics gg) {
super.paintComponent(gg);
Graphics2D g = (Graphics2D) gg;
g.setRenderingHint(RenderingHints.KEY_ANTIALIASING,
RenderingHints.VALUE_ANTIALIAS_ON);
g.setRenderingHint(RenderingHints.KEY_TEXT_ANTIALIASING,
RenderingHints.VALUE_TEXT_ANTIALIAS_ON);
drawGrid(g);
drawLegend(g);
drawEllipse(g);
}
@Override
public void stateChanged(ChangeEvent e) {
JSlider source = (JSlider) e.getSource();
exp = source.getValue() / 2.0;
repaint();
}
public static void main(String[] args) {
SwingUtilities.invokeLater(() -> {
JFrame f = new JFrame();
f.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
f.setTitle("Super Ellipse");
f.setResizable(false);
SuperEllipse panel = new SuperEllipse();
f.add(panel, BorderLayout.CENTER);
JSlider exponent = new JSlider(JSlider.HORIZONTAL, 1, 9, 5);
exponent.addChangeListener(panel);
exponent.setMajorTickSpacing(1);
exponent.setPaintLabels(true);
exponent.setBackground(Color.white);
exponent.setBorder(BorderFactory.createEmptyBorder(20, 20, 20, 20));
Hashtable<Integer, JLabel> labelTable = new Hashtable<>();
for (int i = 1; i < 10; i++)
labelTable.put(i, new JLabel(String.valueOf(i * 0.5)));
exponent.setLabelTable(labelTable);
f.add(exponent, BorderLayout.SOUTH);
f.pack();
f.setLocationRelativeTo(null);
f.setVisible(true);
});
}
}</lang>
ooRexx
This program draws 5 super ellipses: black 120,120,1.5 blue 160,160,2 red 200,200,2.5 green 240,240,3 black 280,280,4
<lang oorexx>/* REXX ***************************************************************
- Create a BMP file showing a few super ellipses
- /
Parse Version v If pos('Regina',v)>0 Then
superegg='superegga.bmp'
Else
superegg='supereggx.bmp'
'erase' superegg s='424d4600000000000000360000002800000038000000280000000100180000000000'X||,
'1000000000000000000000000000000000000000'x
z.0=0 black='000000'x white='ffffff'x red ='00ff00'x green='ff0000'x blue ='0000ff'x m=80 n=80 hor=m*8 /* 56 */ ver=n*8 /* 40 */ s=overlay(lend(hor),s,19,4) s=overlay(lend(ver),s,23,4) z.=copies('f747ff'x,3192%3) z.=copies('ffffff'x,8*m) z.0=648 u=320 v=320 Call supegg black,120,120,1.5,u,v Call supegg blue,160,160,2,u,v Call supegg red,200,200,2.5,u,v Call supegg green,240,240,3,u,v Call supegg black,280,280,4,u,v
Do i=1 To z.0
s=s||z.i End
Call lineout superegg,s Call lineout superegg Exit
supegg: Parse Arg color,a,b,n,u,v Do y=0 To b
t=(1-rxCalcpower(y/b,n)) x=a*rxCalcpower(t,1/n) Call point color,format(u+x,4,0),format(v+y,4,0) Call point color,format(u-x,4,0),format(v+y,4,0) Call point color,format(u+x,4,0),format(v-y,4,0) Call point color,format(u-x,4,0),format(v-y,4,0) End
Do x=0 To a
t=(1-rxCalcpower(x/b,n)) y=a*rxCalcpower(t,1/n) Call point color,format(u+x,4,0),format(v+y,4,0) Call point color,format(u-x,4,0),format(v+y,4,0) Call point color,format(u+x,4,0),format(v-y,4,0) Call point color,format(u-x,4,0),format(v-y,4,0) End
Return
lend: Return reverse(d2c(arg(1),4))
point: Procedure Expose z.
Call trace 'O'
Parse Arg color,x0,y0
--Say x0 y0
Do x=x0-2 To x0+2
Do y=y0-2 To y0+2
z.y=overlay(copies(color,3),z.y,3*x)
End
End
Return
- requires rxMath library</lang>
Racket
<lang Racket>#lang racket (require plot)
- (plot-new-window? #t)
(define ((superellipse a b n) x y)
(+ (expt (abs (/ x a)) n)
(expt (abs (/ y b)) n)))
(plot (isoline (superellipse 200 200 2.5) 1
-220 220 -220 220))</lang>
REXX
Here you can see a picture: http://austria-forum.org/af/User/Pachl%20Walter
<lang rexx>/* REXX ***************************************************************
- Create a BMP file showing a few super ellipses
- /
Parse Version v If pos('Regina',v)>0 Then
superegg='superegga.bmp'
Else
superegg='supereggo.bmp'
'erase' superegg s='424d4600000000000000360000002800000038000000280000000100180000000000'X||,
'1000000000000000000000000000000000000000'x
z.0=0 black='000000'x white='ffffff'x red ='00ff00'x green='ff0000'x blue ='0000ff'x m=80 n=80 hor=m*8 /* 56 */ ver=n*8 /* 40 */ s=overlay(lend(hor),s,19,4) s=overlay(lend(ver),s,23,4) z.=copies('f747ff'x,3192%3) z.=copies('ffffff'x,8*m) z.0=648 u=320 v=320 Call supegg black,080,080,0.5,u,v Call supegg black,110,110,1 ,u,v Call supegg black,140,140,1.5,u,v Call supegg blue ,170,170,2 ,u,v Call supegg red ,200,200,2.5,u,v Call supegg green,230,230,3 ,u,v Call supegg black,260,260,4 ,u,v Call supegg black,290,290,7 ,u,v Do i=1 To z.0
s=s||z.i End
Call lineout superegg,s Call lineout superegg Exit
supegg: Parse Arg color,a,b,n,u,v Do y=0 To b
t=(1-power(y/b,n)) x=a*power(t,1/n) Call point color,format(u+x,4,0),format(v+y,4,0) Call point color,format(u-x,4,0),format(v+y,4,0) Call point color,format(u+x,4,0),format(v-y,4,0) Call point color,format(u-x,4,0),format(v-y,4,0) End
Do x=0 To a
t=(1-power(x/b,n)) y=a*power(t,1/n) Call point color,format(u+x,4,0),format(v+y,4,0) Call point color,format(u-x,4,0),format(v+y,4,0) Call point color,format(u+x,4,0),format(v-y,4,0) Call point color,format(u-x,4,0),format(v-y,4,0) End
Return
lend: Return reverse(d2c(arg(1),4))
point: Procedure Expose z.
Call trace 'O'
Parse Arg color,x0,y0
--Say x0 y0
Do x=x0-2 To x0+2
Do y=y0-2 To y0+2
z.y=overlay(copies(color,3),z.y,3*x)
End
End
Return
power: Procedure /***********************************************************************
- Return b**x for any x -- with reasonable or specified precision
- 920903 Walter Pachl
- /
Parse Arg b,x,prec If prec<9 Then prec=9 Numeric Digits (2*prec) Numeric Fuzz 3 If b=0 Then Return 0 If b<> Then x=x*ln(b,prec+2) o=1 u=1 r=1 Do i=1 By 1 ra=r o=o*x u=u*i r=r+(o/u) If r=ra Then Leave End Numeric Digits (prec) Return r+0
ln: Procedure /***********************************************************************
- Return ln(x) -- with specified precision
- Three different series are used for the ranges 0 to 0.5
- 0.5 to 1.5
- 1.5 to infinity
- 920903 Walter Pachl
- /
Parse Arg x,prec,b
If prec= Then prec=9
Numeric Digits (2*prec)
Numeric Fuzz 3
Select
When x<=0 Then r='*** invalid argument ***'
When x<0.5 Then Do
z=(x-1)/(x+1)
o=z
r=z
k=1
Do i=3 By 2
ra=r
k=k+1
o=o*z*z
r=r+o/i
If r=ra Then Leave
End
r=2*r
End
When x<1.5 Then Do
z=(x-1)
o=z
r=z
k=1
Do i=2 By 1
ra=r
k=k+1
o=-o*z
r=r+o/i
If r=ra Then Leave
End
End
Otherwise /* 1.5<=x */ Do
z=(x+1)/(x-1)
o=1/z
r=o
k=1
Do i=3 By 2
ra=r
k=k+1
o=o/(z*z)
r=r+o/i
If r=ra Then Leave
End
r=2*r
End
End
If b<> Then
r=r/ln(b)
Numeric Digits (prec)
Return r+0</lang>
zkl
Uses the PPM class from http://rosettacode.org/wiki/Bitmap/Bresenham%27s_line_algorithm#zkl <lang zkl>fcn superEllipse(plot,n,color=0xff0000){ // we'll assume width <= height
a,p:=(plot.w/2).toFloat(), 1.0/n; // just calculate upper right quadrant
foreach x in ([0.0 .. a]){
y:=(a.pow(n) - x.pow(n)).pow(p); // a==b>0 --> y=(a^n - x^n)^(1/n)
//println( (x/a).abs().pow(n) + (y/b).abs().pow(n) ); // sanity check
plot[x,y]=plot[-x,-y]=plot[-x,y]=plot[x,-y]=color; // all 4 quadrants
}
plot
}</lang> <lang zkl>w:=h:=600; plot:=PPM(w+1,h+1,0x909090); plot.cross(w/2,h/2); foreach n in ([0.01..1, 0.14]){ superEllipse(plot,n, 0x0000bb) }// 0-1: blue foreach n in ([1.0.. 2, 0.14]){ superEllipse(plot,n, 0x00ff00) }// 1-2: green foreach n in ([2.0..10, 1.4]) { superEllipse(plot,n, 0xff0000) }// 2+: red
plot.writeJPGFile("superEllipse.jpg");</lang>