I understand the concept better now thanks to the link provided

the function xfm1 rotates a point on the locus of the hyperbola about 60 degrees from P(x',y') to P(x",y") around (xo,yo).

but I am stuck on how xfm1 below is incorporated into create_hyperbola as in the code box below;

Can someone please help

% Program to plot the hyperbola
% y^2/a^2 - x^2/b^2 = 1
% The hyperbolae are open up/down, so that x is the independent variable
% for plotting. (Using the form x^2/a^2 - y^2/b^2 = 1 requires that y be
% the independent variable, which is awkward programming-wise.)
clear % all variables
figure(1), hold off % start a new figure
set(gca,'FontSize',14) % adjust fontsize
xmax = 30; ymax = 20;
x = linspace(-xmax,xmax,1001); % array of x values for plot : This function generates vector with 1001 points equally distributed between -xmax and xmax.
%1001 is probably to have 500 points on one side (>0), 500 on another side (<0) and to include 0 itself. So you will have
%more natural numbers.
a = 5; b = 3;
y=sqrt(((x.^2)./(b^2)+1).*a^2); % corresponding y values
plot(x,y)
hold on % add to current plot
plot(x,-y) % Plot other half of hyperbola
% plot the elipse
% y^2/a^2 - x^2/b^2 = 1
y=sqrt(((-x.^2)./(b^2)+1).*a^2); % corresponding y values
plot(x,y)
plot(x,-y) % Plot other half of elipse
axis([-xmax xmax -ymax ymax]) % specify axis limits
xlabel('x')
ylabel('y')
title(['Hyperbola and elipse $y^2/a^2 - x^2/b^2 = 1$;$y^2/a^2 + x^2/b^2 = 1$; $a$ = ', num2str(a), ...
', $b$ = ', num2str(b),';'],'Interpreter','latex')
% Add axes
plot([0 0],[-ymax ymax],'k') % y axis (black line - k)
plot([-xmax xmax],[0 0],'k') % x axis
f1=sqrt(a.^2+b.^2);
f2=-(sqrt(a.^2+b.^2));
plot([0 0],[f1 f2],'.')
text(0,f1,'\leftarrow +d','VerticalAlignment','middle','HorizontalAlignment','left')
text(0,f2,'\leftarrow -d','VerticalAlignment','middle','HorizontalAlignment','left')
f3=sqrt(a.^2-b.^2);
f4=-(sqrt(a.^2-b.^2));
plot([0 0],[f3 f4],'.')
text(0,f3,'\leftarrow +c','VerticalAlignment','middle','HorizontalAlignment','left')
text(0,f4,'\leftarrow -c','VerticalAlignment','middle','HorizontalAlignment','left')
text(0,a, '\leftarrow +a')
text(0,-a, '\leftarrow -a')
function [xout,yout] = xfm1(xin,yin,theta,x_offset,y_offset)
% Program to rotate and translate x,y values from x",y" to x,y space.
% Written to plot hyperbolas for time of arrival code.
% theta value assumed to be in radians.
% rotation matrix
xfm = [cos(theta) sin(theta); ...
-sin(theta) cos(theta)];
% make x,y values into a column vector
r_in = [xin; yin];
% rotate
r_out = xfm*r_in;
x = r_out(1,:); y = r_out(2,:);
xout = x + x_offset;
yout = y + y_offset;

Of course, you didn't put the parameters you need.
Theta=60 deg, Xo=10, Yo=5.
Basically now before drawing anything you need to transform x and y coordinates with your function. Then plot new coordinates.

You cannot put function and script into one m-file. Function should be in a separate m-file with the same name as function name. So create xfm1.m file and paste your function code there. Keep this file in the same directory with your script.
You can read more about m-files, scripts and functions here: http://www.mathworks.com/access/helpdesk/help/techdoc/matlab_prog/f7-38085.html

Add something like this to your code. x and y are coordinates for original hyperbola, you need to apply the function to every part of your figure.
When you call function in Matlab, input and output parameters names do not have to match those in function implementation.

BTW I think you made a wrong rotation matrix. No, it's correct, but rotate in not usual direction. It should be
xfm = [cos(theta) -sin(theta); ...
sin(theta) cos(theta)];

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Ok, I added this at the end of code but the plot does not seem to be correct as per page 4 of the tutorial... am I doing this wrong?

% Program to plot the hyperbola
% y^2/a^2 - x^2/b^2 = 1
% The hyperbolae are open up/down, so that x is the independent variable
% for plotting. (Using the form x^2/a^2 - y^2/b^2 = 1 requires that y be
% the independent variable, which is awkward programming-wise.)
clear % all variables
figure(1), hold off % start a new figure
set(gca,'FontSize',14) % adjust fontsize
xmax = 30; ymax = 20;
x = linspace(-xmax,xmax,1001); % array of x values for plot : This function generates vector with 1001 points equally distributed between -xmax and xmax.
%1001 is probably to have 500 points on one side (>0), 500 on another side (<0) and to include 0 itself. So you will have
%more natural numbers.
a = 5; b = 3;
y=sqrt(((x.^2)./(b^2)+1).*a^2); % corresponding y values
plot(x,y)
hold on % add to current plot
plot(x,-y) % Plot other half of hyperbola
% plot the elipse
% y^2/a^2 - x^2/b^2 = 1
y=sqrt(((-x.^2)./(b^2)+1).*a^2); % corresponding y values
plot(x,y)
plot(x,-y) % Plot other half of elipse
axis([-xmax xmax -ymax ymax]) % specify axis limits
xlabel('x')
ylabel('y')
title(['Hyperbola and elipse $y^2/a^2 - x^2/b^2 = 1$;$y^2/a^2 + x^2/b^2 = 1$; $a$ = ', num2str(a), ...
', $b$ = ', num2str(b),';'],'Interpreter','latex')
% Add axes
plot([0 0],[-ymax ymax],'k') % y axis (black line - k)
plot([-xmax xmax],[0 0],'k') % x axis
f1=sqrt(a.^2+b.^2);
f2=-(sqrt(a.^2+b.^2));
plot([0 0],[f1 f2],'.')
text(0,f1,'\leftarrow +d','VerticalAlignment','middle','HorizontalAlignment','left')
text(0,f2,'\leftarrow -d','VerticalAlignment','middle','HorizontalAlignment','left')
x1=10;
y1=sqrt(((x1.^2)./(b^2)+1).*a^2);
line([0 x1], [f1 y1],'color','r')
line([0 x1], [f2 y1],'color','r')
f3=sqrt(a.^2-b.^2);
f4=-(sqrt(a.^2-b.^2));
plot([0 0],[f3 f4],'.')
text(0,f3,'\leftarrow +c','VerticalAlignment','middle','HorizontalAlignment','left')
text(0,f4,'\leftarrow -c','VerticalAlignment','middle','HorizontalAlignment','left')
text(0,a, '\leftarrow +a')
text(0,-a, '\leftarrow -a')
theta=1; % in radians, 1 radian = 180/pi degrees
xoff=0;
yoff=0;
[xout,yout] = xfm1(x,y,theta,xoff,yoff);
plot(xout,yout)

Of course, you didn't put the parameters you need.
Theta=60 deg, Xo=10, Yo=5.
Basically now before drawing anything you need to transform x and y coordinates with your function. Then plot new coordinates.
I didn't put focus points on the figure. Hope you can do it.

clear % all variables
figure(1), hold off % start a new figure
set(gca,'FontSize',14) % adjust fontsize
xmax = 30; ymax = 20;
x = linspace(-xmax,xmax,1001);
a = 5; b = 3;
theta=60*pi/180; % in radians, 1 radian = 180/pi degrees
xoff=10;
yoff=5;
y=sqrt(((x.^2)./(b^2)+1).*a^2); % corresponding y values
[xout,yout] = xfm1(x,y,theta,xoff,yoff); % rotation and shift
plot(xout,yout)
hold on % add to current plot
[xout,yout] = xfm1(x,-y,theta,xoff,yoff); % rotation and shift of negative part
plot(xout,yout) % Plot other half of hyperbola
axis([-xmax xmax -ymax ymax]) % specify axis limits
xlabel('x')
ylabel('y')
% Add axes
plot([0 0],[-ymax ymax],'k') % y axis
plot([-xmax xmax],[0 0],'k') % x axis

Brilliant... yes I understand you invoke it through the plot function after defining the variables above...
I know how to get the focus points as well
thanks a million :)

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