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

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### Accepted Answer

Vladimir Sovkov
on 25 Dec 2019

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

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### More Answers (1)

Image Analyst
on 25 Dec 2019

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.

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