How to fit rotated hyperbola accurately? - curve-fitting

Question

My experimental data points looks like pieces of hyperbola. Below I provide a code (Matlab), which generates "dummy" data, which is very similar to original one:
function [x_out,y_out,alpha1,alpha2,ecK,offsetX,offsetY,branchDirection] = dummyGenerator(mu_alpha)
alpha_range=0.1;
numberPoint2Return=100; % number of points to return
ecK=10.^((rand(1)-0.5)*2*2); % eccentricity-related parameter
% slope of the first asimptote (radians)
alpha1 = ((rand(1)-0.5)*alpha_range+mu_alpha);
% slope of the first asimptote (radians)
alpha2 = -((rand(1)-0.5)*alpha_range+mu_alpha);
beta = pi-abs(alpha1-alpha2); % angle between asimptotes (radians)
branchDirection = datasample([0,1],1); % where branch directed
% up: branchDirection==0;
% down: branchDirection==1;
% generate branch
x = logspace(-3,2,numberPoint2Return*100)'; %over sampling
y = (tan(pi/2-beta)*x+ecK./x);
% rotate branch using branchDirection
theta = -(pi/2-alpha1)-pi*branchDirection;
% get rotation matrix
rotM = [ cos(theta), -sin(theta);
sin(theta), cos(theta) ];
% get rotated coordinates
XY1=[x,y]*rotM;
x1=XY1(:,1); y1=XY1(:,2);
% remove possible Inf
x1(~isfinite(y1))=[];
y1(~isfinite(y1))=[];
% add noise
y1=((rand(numel(y1),1)-0.5)+y1);
% downsampling
%x_out=linspace(min(x1),max(x1),numberPoint2Return)';
x_out=linspace(-10,10,numberPoint2Return)';
y_out=interp1(x1,y1,x_out,'nearest');
% randomize offset
offsetX=(rand(1)-0.5)*50;
offsetY=(rand(1)-0.5)*50;
x_out=x_out+offsetX;
y_out=y_out+offsetY;
end
Typical results are presented on figure:
The data has following important property: slopes of both asymptotes comes from the same distribution (just different signs), so for my fitting I have rather goo estimation for mu_alpha.
Now starts the problematic part. I try to fit these data points. The main idea of my approach is to find a rotation to obtain y=k*x+b/x shape and then just fit it.
I use the following code:
function Rsquare = fitFunction(x,y,alpha1,alpha2,ecK,offsetX,offsetY)
R=[];
for branchDirection=[0 1]
% translate back
xt=x-offsetX;
yt=y-offsetY;
% rotate back
theta = (pi/2-alpha1)+pi*branchDirection;
rotM = [ cos(theta), -sin(theta);
sin(theta), cos(theta) ];
XY1=[xt,yt]*rotM;
x1=XY1(:,1); y1=XY1(:,2);
% get fitted values
beta = pi-abs(alpha1-alpha2);
%xf = logspace(-3,2,10^3)';
y1=y1(x1>0);
x1=x1(x1>0);
%x1=x1-min(x1);
xf=sort(x1);
yf=(tan(pi/2-beta)*xf+ecK./xf);
R(end+1)=sum((xf-x1).^2+(yf-y1).^2);
end
Rsquare=min(R);
end
Unfortunately this code works not good, very often I have bad results, even when I use known(from simulation) initial parameters.
Could You help me to find a good solution for such fitting problem?
UPDATE:
I find a solution (see Answer), but
I still have a small problem - my estimation of aparameter is bad, sometimes I did no have good fits because of this reason.
Could You suggest some ideas how to estimate a from experimental point?

Answer 1

I found the main problem (it was my brain as usually)! I did not know about general equation of hyperbola. So equation for my hyperbolas are:
((x-x0)/a).^2-((y-y0)/b).^2=-1
So ,we may not take care about sign, then I may use the following code:
mu_alpha=pi/6;
[x,y,alpha1,alpha2,ecK,offsetX,offsetY,branchDirection] = dummyGenerator(mu_alpha);
% hyperb=#(alpha,a,x0,y0) tan(alpha)*a*sqrt(((x-x0)/a).^2+1)+y0;
hyperb=#(x,P) tan(P(1))*P(2)*sqrt(((x-P(3))./P(2)).^2+1)+P(4);
cost =#(P) fitFunction(x,y,P);
x0=mean(x);
y0=mean(y);
a=(max(x)-min(x))./20;
P0=[mu_alpha,a,x0,y0];
[P,fval] = fminsearch(cost,P0);
hold all
plot(x,y,'-o')
plot(x,hyperb(x,P))
function Rsquare = fitFunction(x,y,P)
%x=sort(x);
yf=tan(P(1))*P(2)*sqrt(((x-P(3))./P(2)).^2+1)+P(4);
Rsquare=sum((yf-y).^2);
end
P.S. LaTex tags did not work for me