Line which intersects two planes

You should convince yourself that a graph of a single equation cannot be a line in three dimensions. Instead, to describe a line, you need to find a parametrization of the line. How can we obtain a parametrization for the line formed by the intersection of these two planes?

We start be attemping to solve this system of two equations. But this is consistent with our above conclusion that the intersection is a line, not a point. We'll use the method of elimination. Home Threads Index About. It only takes a minute to sign up. Connect and share knowledge within a single location that is structured and easy to search. Here, since a line that isn't parallel to any coordinate plane passes through all three, you can check if it is parallel to one by using the directional vector.

While this problem has a great textbook answer, as walcher explained, I don't think it's very elegant. This is because, the solution depends on picking an arbitrary point, which lacks geometric intuition. Ideally, we'd like this point to have some meaning, such as being close to the planes, or the line or etc. For that, I'd like to remind you of a solution by John Krumm , which remains unnoticed by many. I think this is a pretty neat approach giving a nice and simple method, with a geometrically interpretable results.

Then we can simultaneously solve the the two planes equation by putting this point in it. Now, we can find the direction of the line we need to find by taking cross product of normal vectors of two given planes. I tried the systems of equations approach posted by multiple people, but dealing with division by zero made things really messy.

So I came up with a more intuitive approach. Start with the cross product of the normal vectors of the 2 planes Normal1 and Normal2 to get a direction of the intersection line Normal3 :. Now if we look at the existing planes from the perspective of that direction, our 2 planes look like 2 lines, because we're viewing them both edge-on. So we want to calculate what those 2 lines are. We can get the direction of each line as the cross product of our new plane's normal with the original normal:.

Now if we find the intersection of those two lines, it will give a point which occurs on both lines, which means it also occurs on both planes:. Once you have the intersection point, combine it with the Normal3 we calculated at the beginning to get the intersection line. Sign up to join this community. The best answers are voted up and rise to the top.

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Ask Question. Asked 8 years, 2 months ago. Active 5 months ago. Viewed k times. That you can do by setting one of the variables to 0 and solving it.