How to draw a smooth, closed curve given 4 points

Is there an easy way to draw a smooth, closed curve from 4 points? Two points will be on the x axis, and two on the y axis. Of each set of two points, one will be on the positive axis and one will be on the negative axis. I would like to have the final closed curve be smooth like an egg shaped curve. An example data set might be:

x = [0 560 0 -60];  
y = [310 0 -185 0];

Thanks.

[Jan]

There is an infinite number of smooth closed curves through the points. Do you have any further restrictions? How smooth should the curve be? Would a cubic spline be sufficient?

[David]

I don't really have any further restrictions currently. There is some curve that would be "correct", but I don't currently have a way to calculate it. This is just to have an approximate way to visualize what it would look like that is more visually appealing than just plotting x and y with straight lines connecting them. So I'm not sure about smoothness or a cubic spline. Is there a quick way for me to try it out and see what it looks like?

Answer 1

[Image Analyst]

See if you can adapt this snippet:

clc;  % Clear command window.
clear;  % Delete all variables.
close all;  % Close all figure windows except those created by imtool.
workspace;  % Make sure the workspace panel is showing.
fontSize = 20;

x = [0, 560, 0, -60];  
y = [310, 0, -185, 0];
% Append first point to last to close the curve
x = [x, x(1)];
y = [y, y(1)];
plot(x, y, 'r*');
grid on;

set(gcf, 'Position', get(0,'Screensize')); % Enlarge figure to full screen.
set(gcf,'name','Spline Image Analysis Demo','numbertitle','off') 

% Calculate the area within the blue spline curve.
% You do not need to connect the last point back to the first point.
knots = [x; y];
areaOfPolygon = polyarea(x,y);

numberOfPoints = length(x);
% Interpolate with a spline curve and finer spacing.
originalSpacing = 1 : numberOfPoints;
% Make 9 points in between our original points that the user clicked on.
finerSpacing = 1 : 0.1 : numberOfPoints;
% Do the spline interpolation.
splineXY = spline(originalSpacing, knots, finerSpacing);

% Plot the interpolated curve.
hold off;
plot(knots(1, :), knots(2, :), 'ro', 'LineWidth', 2, 'MarkerSize', 16);
hold on;
plot(splineXY(1, :), splineXY(2, :), 'b+-', 'LineWidth', 2, 'MarkerSize', 16);
title('Blue Spline Between Red Knots', 'FontSize', fontSize);
legend('Knots', 'Spline');
xlabel('X', 'FontSize', fontSize);
ylabel('Y', 'FontSize', fontSize);
grid on;
hold off;
% Calculate the area within the blue spline curve.
% You do not need to connect the last point back to the first point.
x = splineXY(1, :);
y = splineXY(2, :);
areaInsideSplineCurve = polyarea(x,y);

% Give the area calculations.
message = sprintf('The area inside the polygon you drew is %.2f.\nThe area inside the blue spline curve is %.2f', ...
  areaOfPolygon, areaInsideSplineCurve);
fprintf(1, '%s', message); % Print to command window.
msgbox(message); % Show user via a popup message box.

[David]

This is pretty close to what I want! However, I end up with a plot with a point at the top that makes it look like a Hershey's kiss. Is there any way to have the slope of the contour be continuous so there are no sharp points?

[Image Analyst]

I tried to extend the curve by doing

x = [x, x(1:2)];
y = [y, y(1:2)];

and it made it smooth going past the first point, but we had two non-overlapping smooth curves between point 1 and 2. Not sure what to do off the top of my head - I'd have to think about it.

For 4 points, you can exactly fit a circle to them, if you'd be happy with a circle. See the FAQ: http://matlab.wikia.com/wiki/FAQ#How_can_I_fit_a_circle_to_a_set_of_XY_data.3F

[David]

I tried to adapt the example from the documentation that uses spline for interpolation of angular data (See below for the code from the doc. My MATLAB is on a separate computer with no internet, so I can't easily share it...), but I ended up with the same curve as before. I feel like I'm close, but missing something simple because I don't fully understand what is going on inside the function... Does this give you any ideas?

x = pi*[0:.5:2]; 
  y = [0  1  0 -1  0  1  0; 
       1  0  1  0 -1  0  1];
  pp = spline(x,y);
  yy = ppval(pp, linspace(0,2*pi,101));
  plot(yy(1,:),yy(2,:),'-b',y(1,2:5),y(2,2:5),'or'), axis equal

[Image Analyst]

For a circle, it works fine, but the curve is not closed in general, like

x = pi*[0:.5:2];
y = [0  4  0 -1  0  .3  0;
  5  -4  .6  0 -1  0  3];
pp = spline(x,y);
yy = ppval(pp, linspace(0,2*pi,101));
plot(yy(1,:),yy(2,:),'-b',y(1,2:5),y(2,2:5),'or'), axis equal

[David]

I got a pretty decent curve using the method described here ...

[Image Analyst]

If that's smooth enough for you, fine. But it does give sharp points sometimes. Just run a modified version of the code I made that does not give such a round starting set of points:

n = 20;
r = 1 + rand(n-1,1)*2.1;%noisy r's
theta = sort(2*pi*rand(n-1,1));
% closing the circle
r(end+1) = r(1);
theta(end+1) = theta(1);
% convert to cartesian
[x,y] = pol2cart(theta,r);
% interpolate with parameter t
t = (1:n)';
v = [x,y];
tt = linspace(1,n,100);
X = interp1(t,v,tt,'pchip');
% plot
plot(x,y,'o');
hold on
plot(X(:,1),X(:,2));