I have a circle oriented in 3d space defined by the parametric equations:
x(θ)=c1+rcos(θ)a1+rsin(θ)b1
y(θ)=c2+rcos(θ)a2+rsin(θ)b2
z(θ)=c3+rcos(θ)a3+rsin(θ)b3
Where c is the center, r is the radius, and a and b are vectors perpendicular to the directional axis of which the circle is pointing. (Parametric Equation of a Circle in 3D Space?)
C = (591.898, 120.412, -3.498), R=3, A=(1, 1, -4.484), B=(4.493, 0.743, 1.167)
Now I also have this equation written in a non parametric form:
P = R cos(θ) ⃗A + R sin(θ) ⃗A × ⃗N + c
Where A is the vector (a1,a2,a3) and B is the vector (b1,b2,b3), ,N=(-0.203, 0.964, 0.169). These equations represent the same circle in 3d space, so feel free to use either one.
So I have a line defined by the unit vector 'N' (-0.203, 0.964, 0.169) that starts at the point (590.7, 140.3, 0). How would I go about finding the x,y,z coordinates of where this line intersects the circle on its circumference?
Thanks in advance.
If $N$ is a direction vector for the line, then it is perpendicular to the plane of the circle. Find the intersection point and compare its distance from the center of the circle to the radius. If they’re equal, you have your intersection point. If not, the line doesn’t intersect the circle at all.