Finding a line in a point cloud segment_plane perpendicular to the z-axis

[sdgoldstein]

I have a point cloud which I've segmented into planes using RANSAC by calling open3d.geometry.PointCloud.segment_plane(). From that, for a given plane, I have:

1. A plane equation, ax+by+cz+d=0
2. A set of inlier points from the point cloud in that plane

Problem: Find the endpoints of a line segment that

1. Lies within the plane
2. Is perpendicular to the z-axis
3. Spans the length of inlier points of the point cloud

I know that I can find the direction of the line segment by taking the cross product of the plane normal and the Z-axis basis vector:

T = k X N = [0, 0, 1] X [a, b, c]

What I'm not sure how to do, is once I have this vector, T, how do I find the endpoint of the line using the inlier points of the point cloud?

Is there a method in Open3D that, given a vector T in the point cloud, find these end points? If not, what are the algebraic equations/processes to arrive at these two endpoints?

Thank you for the help!