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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.

mathematics2x2life
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Help Me
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  • Your text is a bit confuse. As far as I can tell, what you call "non parametric form" is the same as the equations above, written in vector form. But I think only $\vec B$ should appear there, instead of $\vec A\times\vec B$. – Intelligenti pauca Jan 26 '17 at 16:51
  • Aretino, you were correct in B being off, and i have updated the question. I am converting my confusing in a code, into readable letters, which is what caused this. Hope this helps – Help Me Jan 26 '17 at 17:19

1 Answers1

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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.

amd
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  • I already know that the intersects the circle. However, i need to know where, in x,y,z coordinated, does this happen. – Help Me Jan 27 '17 at 20:52
  • @HelpMePlease I’ve explained how: find the intersection of the line and the plane of the circle. You have everything you need to do that (a point on the plane, its normal vector, and an equation for the line), which requires solving a set of linear equations. If you don’t know how to do that, a simple Internet search will give you literally hundreds of web pages that explain it. – amd Jan 27 '17 at 21:35