How to find the vertical height of an ellipse/y axis length of ellipse.

How do I get the vertical height/ y axis length on an ellipse if I have values of the centeroid, orientation, major-axis length and minor axis length?

Answer 1

[Vladimir Sovkov]

In fact, the height and the width of an ellipse can be easily and more accurately found analytically. Here is Matlab program realizing such a computation

function h = elhw(ax,bx,q,ifplot)
% Estimates the spans (width and height) of an ellipse
% ax     - 1st principal axis halflength
% bx     - 2nd principal axis halflength
% q      - rotational angle
% ifplot - set it to true if you want to check the result graphically
% h      - a vector with the calculated spans (width and height) of the ellipse
%%
    st =ax/1000; % sample step for graphical representation; only used with ifplot=true
    %
    R=[[cos(q),-sin(q)];[sin(q),cos(q)]]; % rotation matrix
    D=zeros(2); % ellipse equation in the coordinates of principal axes is [x y] * D^(-1) * [x ; y] = 1
    D(1,1)=ax^2;
    D(2,2)=bx^2;
    DD = R*D*R'; % ellipse equation in the actual coordinates is [x y] * DD^(-1) * [x ; y] = 1
    
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    %% THESE ARE THE VALUES OF INTEREST
    h = sqrt(diag(DD));
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    
    %% check the consistency of the result graphically
    if nargin>3 && ~isempty(ifplot) && ifplot(1)
        a = max(ax,bx);
        xD=(-ax:st:ax);
        yD1=sqrt((1-xD.^2/D(1,1))*D(2,2));
        yD2=-yD1;
        A=R*[xD;yD1];
        xDD1 = A(1,:);
        yDD1=A(2,:);
        A=R*[xD;yD2];
        xDD2 = A(1,:);
        yDD2=A(2,:);
%        figure;
        plot(xD,yD1,'--b',xD,yD2,'--b',xDD1,yDD1,'-k',xDD2,yDD2,'-k',[-a a],[-h(2) -h(2)],'-r',[-a a],[h(2) h(2)],'-r',[-h(1) -h(1)],[-a a],'-r',[h(1) h(1)],[-a a],'-r');
        legend('non-rotated ellipse','','rotated ellipse','','computed spans');
    end
    return;
end

And this is the script to check its functionallity

%% define sample parameters randomly and check elwh.m with them
ax=rand;   % the 1st axis length
bx=rand; % the 2nd axis length
q=rand*pi; % rotation angle
h = elhw(ax,bx,q,true); % estimate the ellipse width, height, and check the results graphically

[Image Analyst]

Very nice! ?

h = elhw(10, 20, 22.5, true)
grid on;
axis equal

Answer 2

[Image Analyst]

Does not seem like a MATLAB question. But anyway, if you digitize it into an x array and y array, you can then find the y range of the digitized values.

height = max(y(:)) - min(y(:));

If you don't have any y, then it's just an analytical geometry problem and you should ask it in some general mathematics forum rather than a MATLAB language forum.