How to plot an Ellipse

Asked by Dimitris Arapidis on 8 Sep 2013

Latest activity Commented on by Sandy M on 27 Jul 2019

3.303 views (last 30 days)

I want to plot an Ellipse. I have the verticles for the major axis: d1(0,0.8736) d2(85.8024,1.2157) (The coordinates are taken from another part of code so the ellipse must be on the first quadrant of the x-y axis) I also want to be able to change the eccentricity of the ellipse.

1 Comment

muhammad arfan on 18 Jun 2019

dears!!!

i have asigned to write a matlab code for 8 point to fit it in ellipse by using least square method..

i am new in using matlab and try my best but my points are not fit on ellipse. i use annealing method so that i have satisfied my teacher by my work. please chk my work and help me.

thanks

arfan khan

clc;
clear all;
close all;
r1 = rand(1);
r2 = [1+rand(1)];  % r2>r1
x0 = 0;
y0 = 0;
N = 8;
n= 100;
x1 = 1;
x2 = 2;
y1 = 1;
y2 = 2;
for i = 1:n
    x = x1 +(x2-x1).*rand(N,1);
    y = y1 +(y2-y1).*rand(N,1);
    f = ((((x./r1).^2) +(y./r2).^2)-1).^2;
    [m,l] = min(f);
    z =.001* exp(10*(1-i/n));
    v = z/2;
%     disp('v');
%     disp(v)
    x1 = x(l)*v;
    x2 = x(l)*v;
%     disp('x1')
%     disp(x1)
%     disp('x2')
%     disp(x2)
%      ay = v./y(l);
%      by = v./y(l);
%     disp(v);
%     
% %     hold on;
       disp('f');
       disp(f);
end
% plot(f,'or')
   plot(x,y, '*b');
x=((x(i)-x0)*cos(z)) - ((y(i)-y0)*sin(z))
    y=(x(i)-x0)*sin(z)-(y(i)-y0)*cos(z)
    xa(i)=rand(1)
    x(i)= a+(b-a)*rand(1);
    y(i)= rand(1);
    for 
        m(i) = ((((x).^2)/a^2) + (((y).^2)/b^2)-1).^2
        end
         hold on;

4 Answers

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Answer by Roger Stafford on 8 Sep 2013

Edited by Cris LaPierre on 5 Apr 2019

Accepted Answer

Let (x1,y1) and (x2,y2) be the coordinates of the two vertices of the ellipse's major axis, and let e be its eccentricity.

 a = 1/2*sqrt((x2-x1)^2+(y2-y1)^2);
 b = a*sqrt(1-e^2);
 t = linspace(0,2*pi);
 X = a*cos(t);
 Y = b*sin(t);
 w = atan2(y2-y1,x2-x1);
 x = (x1+x2)/2 + X*cos(w) - Y*sin(w);
 y = (y1+y2)/2 + X*sin(w) + Y*cos(w);
 plot(x,y,'y-')
 axis equal

9 Comments

Show 6 older comments

Walter Roberson on 26 Jul 2018

If you already have the minor axis length then you can use that directly in b. If you do not have the minor axis length then you need to calculate it, and you need the information to calculate it; the main other way of characterizing it is to specify the eccentricity. If you do not know the minor axis length and you do not know the eccentricity, then what do you know?

Kaleesh Bala on 27 Jul 2018

ok fine let me put it in parametric type having two foci as x1 y1, x2 y2

determining r1,r2 to get the elliptical form? I think from r1 can get r2.

So how to determine r1.

xt = r1 * cos(t) + xc;

yt = r2 * sin(t) + yc;

Walter Roberson on 27 Jul 2018

The foci are not enough information to determine the ellipse.

http://www.purplemath.com/modules/ellipse3.htm

Link

Answer by Azzi Abdelmalek on 8 Sep 2013

Edited by Azzi Abdelmalek on 12 Jun 2015

a=5; % horizontal radius
b=10; % vertical radius
x0=0; % x0,y0 ellipse centre coordinates
y0=0;
t=-pi:0.01:pi;
x=x0+a*cos(t);
y=y0+b*sin(t);
plot(x,y)

3 Comments

Ivailo Ivanov on 14 Jan 2016

It was very simple and comprehensible.

Cynthia Dickerson on 27 Jun 2018

Thanks! This code worked for me perfectly. :)

Sandy M on 27 Jul 2019

Hi, I do my ellipse graph

A=10; 
B=7.5;
X=-10:.1:10;
Y=(7.5/10)*(1-x^2)^(1/2)
z=-(7.5/10)*(1-x^2)^(1/2)
Plot(x,y,x,z)

Its ok but i need it in cm units cause if i change properties of figure and paper to cm i get deference’s about 3 or 5 mm How can I justify the unit

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Answer by Kate on 24 Feb 2014

how would you plot a ellipse with only knowing some co-ordinates on the curve and not knowing the y radius and x radius?

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Roger Stafford on 24 Feb 2014

That involves more work, Kate. I'll show you a first, easy step and let you struggle through the rest.

It requires a minimum of five points to uniquely determine a conic section - that is, an ellipse, a parabola, or a hyperbola. One way to approach the problem is to find a set of values A, B, C, D, E, and F such that

A*x^2 + B*x*y + C*y^2 + D*x + E*y + F

is as close to zero as possible for the points (assuming the six constants have been normalized by having the sum of their squares be unity.) With five points it should be essentially zero for each of the points. For more points, it depends on how close they all are to a valid conic section. If they actually lie on some conic, then with the appropriate constants A, B, etc., the above would also be zero at each point.

These constants can be found using matlab's 'svd' singular value decomposition function. Let x be a column vector of all the points' x-coordinates and y a column vector of their corresponding y-coordinates. Do this:

[U,S,V] = svd([x.^2,x.*y,y.^2,x,y,ones(size(x))]);

Then the desired constants will be in the vector V(:,6), namely, A = V(1,6), B = V(2,6), C = V(3,6), D = V(4,6), E = V(5,6), and F = V(6,6). These are associated with the smallest singular value in S. If that value is zero, then there is a precise match with all points.

Probably the easiest way to proceed from here on in plotting the conic section, would be to solve for y as a function of x, (or possibly x as a function of y,) in the quadratic equation

A*x^2 + B*x*y + C*y^2 + D*x + E*y + F = 0

The type of conic section is determined by the discriminant, B^2-4*A*C. If it is negative, the curve is an ellipse, if it is zero, the curve is a parabola, and if it is positive, it is a hyperbola. Recalling high school algebra, there will in general be two y solutions or none for each x, (and two x solutions or none for each y,) using the famous quadratic formula.

Making a successful plot would require a determination of the range of x (or of y) for which there are solutions and then generating a set of x values (or y values,) maybe with linspace, and making two separate plots of the two possible solutions for y (or for x).

Another approach to handling such a quadratic equation involves making the appropriate translation and rotations so as to place the equation in a standard form from which plots can readily be made. This, however, probably involves more effort than the above, though it is more informative.

Image Analyst on 25 Feb 2014

Perhaps of some interest, if you need to find the ellipse points: http://www.ecse.rpi.edu/homepages/qji/Papers/ellipse_det_icpr02.pdf

Devi Satya Cheerla on 12 Jun 2015

in the equation of ellipse X2/a2 + Y2/b2 = 1. knowing the points on ellipse, can find a and b. then enter the code below to mathematically compute y and to plot x,y.

code:

x=(0:.01:a); # x value is from 0 to 'a' and discrete with 0.01 scale#

i=1:(a*100+1); # i is to calculate y at every discrete value. it should be for 1 i.e first x value to the last x value.. as it does not have a zero, add 1#

clear y # to clear any previous y value#

for i=1:(a*100+1)

y(i)=(b^2*(1-(x(i)^2)/a^2))^.5; #from the ellipse equation y=sqrt(b2(1-(x2/a2))#

end

plot(x,y)

hold on

plot(x,-y)

hold on

plot(-x,y)

hold on

plot(-x,-y)

Sandy M on 27 Jul 2019

hi why u product the nmber with 100?

and, if i want the graph with cm units, what i do? cause i change garaph and paper properties but i still defreces about 4 mm when i prented it

Link

Answer by Omar Maaroof on 13 May 2019

you can use

Ellipse2d

1 Comment

Walter Roberson on 13 May 2019

MATLAB does not offer Ellipse2d plotting directly. Instead, the Symbolic Toolbox's engine, MuPAD, offers plot::Ellipse2d https://www.mathworks.com/help/symbolic/mupad_ref/plot-ellipse2d.html which can only be used from within a MuPAD notebook . R2018b was intended to be the last release that included the MuPAD notebook, but it was carried on to R2019a as well.

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