Finding a line that is perpendicular to the z-axis and passes through a set of points that form a plane

[sgoldstein527]

Suppose I have a set of points, {P}, in 3D space. Those points lie in a plane defined by the equation ax + by + cz + d = 0.

Problem: Find the endpoints of a line segment that

1. Lies within this plane
2. Is perpendicular to the z-axis
3. Spans the length of points of in {P}

I know that I can find a vector indicating 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 endpoints of the line using the points in {P}.

Thank you for the help!

 

[Dr.Peterson]

Please clarify what this means:

3. Spans the length of points of in {P}

I think "of" isn't meant to be there; points don't have length; a set of points still doesn't have a length; and "span" could mean several different things.

It may help if you explain at least part of the context. What do these points represent?

 

[sgoldstein527]

Yeah, I agree it's a bit abstract. Suppose the points represent one side of a wall of a building. The length or span, in this case, is the width of the wall.

Scott

 

[khansaheb]

Find the endpoints of a line segment that

What do the lines represent?

 

[sgoldstein527]

The ultimate goal is to find the base of the wall. However, finding a line anywhere within the wall perpendicular to the z-axis would be okay

 

[blamocur]

Lines orthogonal to z-axis are horizontal. Lines lying in the plane satisfy the equation of the plane. I.e., to get such a line you simply fix 'z', e.g.
[imath]z=z_0[/imath], then equations [imath]z=z_0, ax+by + cz_0 + d=0[/imath] define your line.

 

[Dr.Peterson]

Yeah, I agree it's a bit abstract. Suppose the points represent one side of a wall of a building. The length or span, in this case, is the width of the wall.

Scott

Possibly what you want could be to project each point onto the line, and delimit a segment by the extreme points (the farthest projection in each direction. This length would be the "span" of the points in that particular direction, which sounds like a "width". (Otherwise, I would say the "span" would be the greatest distance between any two points in the set, without regard to direction.)