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
