Find intersection between line and circle

Hi I currently work with a small project regarding particle behaviour in closed domains.

I want to create a function which takes start point, and start direction for a point. The domain is a circle. How do I calculate the two intersections with the circle?

I know how to do the calculations but I have trouble with the implementation.

px = -0.5; % start x-coordinate of point

py = -0.5; % start y-coordinate of point

rx = 1; % x-direction of vector with the above startcoordinates

ry = 2; % y ------ || ---------

I tried using symbolic toolbox but I can't convert it back to normal variables.

fsolve won't help because the representation of a circle is not a function. What to do?

Thanks in advance

Regards

[Andrew Newell]

Show me what you did using the symbolic toolbox and I'll show you how to convert it back to normal variables.

Answer 1

[Teja Muppirala]

You're pretty close. Except you don't want to solve

'x^2 + y^2 = 0'

You want to solve 'x^2 + y^2 = R^2'

-------------------------------

Try this:

R = sym('R');
x = sym('px + t*rx');
y = sym('py + t*ry');

c = x^2+y^2-R^2;

t_sol = solve(c);

x_sol = subs(x,'t',t_sol)
y_sol = subs(y,'t',t_sol)

Then x_sol gives you the x values, and y_sol has the y values of the solution.

[Andrew Newell]

This is how it would look in a function. Note that once you define t as a symbol, combinations involving t are symbols too.

function [xi,yi] = solveIntersection(px,py,rx,ry,r)

syms t

x = px + t*rx;

y = py + t*ry;

c = x^2+y^2-r^2;

t_sol = solve(c);

xi = subs(x,'t',t_sol);

yi = subs(y,'t',t_sol);

[Kasper]

Sweet. Thanks!

Answer 2

[Kasper]

I insert the parametrisation in the circle equation. And solve it.

x = sym('px + t*rx'); y = sym('py + t*ry');

c = x^2+y^2; solve(c)

ans =

 -(px*rx + py*ry - i*px*ry + i*py*rx)/(rx^2 + ry^2)
 -(px*rx + py*ry + i*px*ry - i*py*rx)/(rx^2 + ry^2)

I need this to respond to the numerical values which were assigned to it in the beginning. But I see there's an imaginary part, which I do not want.

Answer 3

[Kasper]

Thanks but how do I substitute the real values into a 'sym' variable? Since I define all variables as 'sym' the predefined rx = 1,... doesn't really work with those final equations.

[Teja Muppirala]

Option 1: Use the subs command to put in the values of each variable.

subs([x_sol y_sol],{'R' 'rx', 'ry', 'px', 'py'},{1 8 6 0 0})

Or Option 2:

Copy the text of the symbolic expression and then use that just like a regular MATLAB expression where everything is a number.

R = 1;

px = 0;

py = 0;

rx = 8;

ry = 6;

x_sol = [px - (rx*(px*rx + py*ry + (R^2*rx^2 + R^2*ry^2 - px^2*ry^2 + 2*px*py*rx*ry - py^2*rx^2)^(1/2)))/(rx^2 + ry^2); px - (rx*(px*rx + py*ry - (R^2*rx^2 + R^2*ry^2 - px^2*ry^2 + 2*px*py*rx*ry - py^2*rx^2)^(1/2)))/(rx^2 + ry^2)]

y_sol = [px - (rx*(px*rx + py*ry - (R^2*rx^2 + R^2*ry^2 - px^2*ry^2 + 2*px*py*rx*ry - py^2*rx^2)^(1/2)))/(rx^2 + ry^2);

py - (ry*(px*rx + py*ry - (R^2*rx^2 + R^2*ry^2 - px^2*ry^2 + 2*px*py*rx*ry - py^2*rx^2)^(1/2)))/(rx^2 + ry^2)]

[Teja Muppirala]

Oops, I miscopied y_sol, but I think you get the idea.

[Christopher Creutzig]

Instead of copying the symbolic expression manually, you might want to try using matlabFunction(x_sol(1)) etc. At the very least, it will vectorize better/more correctly.

Answer 4

[Kasper]

yes that will work the first option. But that won't make it a function if I manually have to input those numbers. But thanks the subs function helped a great deal.

Answer 5

[Anupama]

find tthe point of intersections of a circle at (1,2) and radius 4 and a straight line 2x+3y =9 using graphical method.