Causing a rotating arc to touch another curve and not intersect it (Matlab code) - matlab

Question

I want to write a code where an arc stops rotating as soon as it comes in contact with a semi-circle.
I have written a code to do so, my arc does not just touch the circle but it slightly intersects it.
I have put rotation of arc inside a while loop by using linspace() to change theta. I used polyxpoly() for finding intersection. Condition for the while loop is that as long as I have empty array the loop continues, but as soon as I get a value from polyxpoly() my loop stops.
However, at the place of touch the theta value exceeds what I needed, so as a result I get an intersection.
How do I modify the code so that the arc will touch the semi-circle and not intersect it?
Here is the output. Click the link below
Image of intersection but I need touch and not intersection
clc,clear
R = 5; % radius of a circle
r = 10; % radius of arc
aa = 60*pi/180; % arc angle
ap = 0*pi/180; % arc position angle
% defining the semi-circle about the origin
t = linspace(0,pi);
[x,y] = pol2cart(t,R); % circle data
% Shifting circle centre to (3.5,0)
x=x+3.5;
y=y+0;
% defining the arc about the origin
t1 = linspace(0,aa)-aa/2+ap;
[x1,y1] = pol2cart(t1,r); % arc data
% shifting arc-lower-end to (14,0)
delx=14-x1(1); % Finding the x difference between arc-lower-end x-coordinate & 14
dely=0-y1(1); % Finding the y difference between arc-lower-end y-coordinate & 0
x1=x1+delx;
y1=y1+dely;
theta =linspace(0,pi,1000);
i=1;
xc=[];
yc=[];
while isempty(xc)&& isempty(yc)
% create a matrix of these points, which will be useful in future calculations
v = [x1;y1];
% choose a point which will be the center of rotation
x_center = 14;
y_center = 0;
% create a matrix which will be used later in calculations
center = repmat([x_center; y_center], 1, length(x1));
% define a 60 degree counter-clockwise rotation matrix
R = [cos(theta(i)) -sin(theta(i)); sin(theta(i)) cos(theta(i))];
% do the rotation...
s = v - center; % shift points in the plane so that the center of rotation is at the origin
so = R*s; % apply the rotation about the origin
vo = so + center; % shift again so the origin goes back to the desired center of rotation
% this can be done in one line as:
% vo = R*(v - center) + center
% pick out the vectors of rotated x- and y-data
x_rotated = vo(1,:);
y_rotated = vo(2,:);
[xc,yc] = polyxpoly(x_rotated,y_rotated,x,y)
[xc1,yc1] = polyxpoly(x1,y1,x,y)
i=i+1;
end
% make a plot
plot(x,y)
hold on
plot(x1, y1, 'k-', x_rotated, y_rotated, 'r-', x_center, y_center, 'bo');
axis equal
I need to find way it contacts a circle and does not intersect it.
The code is in matlab.
Any suggestions are welcome.

Answer 1

This is the problem of collision detection. Most, if not all, methods I know of in collision detection requires the computer to check for intersections of some sort. It's very difficult (if not impossible) to have two objects "just touch" in simulations, because you'll need the precise (analytically solved) location of the boundaries of those objects.
polyxpoly() is a function to return the intersection of two polygons. So unfortunately, if you insist that the arc cannot touch, then you cannot use polyxpoly(). In that extreme case, you'll have to solve some mathematical equation for when the tip of the arc coincides with a point on the circle perfectly, then simulate up until that point in time.
But realistically, what you need is a finer simulation (although I personally think what you have alone is good enough). So in every simulation step, you calculate a smaller movement, so that when the arc eventually intersects, only a very small amount of the arc intersects.