How to rotate a hyperbola drawn along with the reference line?

I have an equation of calculating the radius of hyperbola for drawing specific maps. However, I could only draw hyperbola at origin.

For instance. I want to draw the hyperbola between two points at some known distance. In the code below, I have used the equation to plot the hyperbola when the line is horizontal and starts at original. For finding the xy coordinates, I use the cosine and sine components and then plot.

However, in cases where I have the same distance between the points but not at origin (I tried xy coordinates with same magnitude but different location and rotation), I cannot construct the rotated hyperbola. Is there a way to draw the rotated parabola?

Additionally, my way is inspired from `excel` and is very unituitive. Is there a more convenient way to construct hyperbola in Matlab?

clc
clear all
close all


%% Input values

D = 250;            % mm distance
c = 4;              % mm/s velocity
del = 10;           % a constant
theta = deg2rad(0:5:360);    % a variable

for i = 1:length(theta)
    angle = theta(i);
    R(i) = 1/2*(D*D - del*del*c*c)/(del*c + D*cos(angle)); % Equation of radius of hyperbola

    x1(i) = 0;
    y1(i) = 0;
    x2(i) = R(i)*cos(angle);
    y2(i) = R(i)*sin(angle);
end

pta = [0 250];
ptb = [0 0];

figure
tiledlayout(2,1)
nexttile
hold on
plot(pta,ptb,'-o');

plot(x2,y2,'.r')
xlim([0 D])


nexttile
hold on
ptc = [50 250];
ptd = [40 190];

plot(ptc,ptd,'-o');
xlim([0 250])
ylim([0 250])

There may be a simple solution, forgive my trignomatry skills and I will appreciate your help in this regard.

All the best.

[Catalytic]

To me, it looks like a hyperbola. Not a parabola.

[UH]

I am sorry, my mistake. Corrected

Answer 1

[Matt J]

%% Input values
D = 250;            % mm distance
c = 4;              % mm/s velocity
del = 10;           % a constant
theta = deg2rad(0:5:360);    % a variable
for i = 1:length(theta)
    angle = theta(i);
    R(i) = 1/2*(D*D - del*del*c*c)/(del*c + D*cos(angle)); % Equation of radius of parabola
    x1(i) = 0;
    y1(i) = 0;
    x2(i) = R(i)*cos(angle);
    y2(i) = R(i)*sin(angle);
end
pta = [0 250];
ptb = [0 0];



[cx,cy]=deal(100,200); %translation

pta=pta+cx; ptb=ptb+cy; x2=x2+cx; y2=y2+cy;

h=plot(pta,ptb,'-o',x2,y2,'.r'); axis equal

rotate(h,[0,0,1],30,[cx,cy,0])

axis([mean(pta),mean(ptb)]+[-D;D])