Question: how to get the coordinate of a swirl

i find equation above

for ii from 0 by 5 to 365 do
eq1 := x^2 + y^2 = arccos(x/r);
eq2 := y = evalf((atan(-1) - ii*Pi/180)/Pi)*(-1.0)*x;
xy := fsolve({eq1, eq2});

but i do not understand how to set r

i use a line scanning 365 degrees to get the path of swirl 

would like to get path data for c# WPF animation

but this method may have multiple solution because an intersection of swirl 

and a line can have multiple points

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