Q about line intersection

newage_lightbulb

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Hello, I was reading the line intersection algorithm posted on geomalgorithms.com (previously known as SoftSurfer) and I am stuck on something. Here is the quote:

"In any n-dimensional space, the two lines L1 and L2 are closest at unique points PC = P(sC) and QC = Q(tC) for which w(sC, tC) is the unique minimum for w(s,t). Further, if L1 and L2 are not parallel and do not intersect each other, then the segment PCQC joining these points is uniquely simultaneously perpendicular to both lines."

I don't get that last part. If two lines are not parallel and not intersecting, how can a segment between any two points on them be simultaneously perpendicular to both? I'm not the brightest math guy but can't a line only be simultaneously perpendicular to two others if those two others are parallel?

Thanks for any help.
 
Hello, I was reading the line intersection algorithm posted on geomalgorithms.com (previously known as SoftSurfer) and I am stuck on something. Here is the quote:

"In any n-dimensional space, the two lines L1 and L2 are closest at unique points PC= P(sC) and QC= Q(tC) for which w(sC, tC) is the unique minimum for w(s,t). Further, if L1 and L2 are not parallel and do not intersect each other, then the segment PCQC joining these points is uniquely simultaneously perpendicular to both lines."

I don't get that last part. If two lines are not parallel and not intersecting, how can a segment between any two points on them be simultaneously perpendicular to both? I'm not the brightest math guy but can't a line only be simultaneously perpendicular to two others if those two others are parallel?

Thanks for any help.
Possibly the easiest way to envision this is to think of a two space, i.e. just two regular lines. If the lines intersect in a single unique coexistent point, for example
y = 3*(x-5) + 7
and
y = 5*(x-5) + 7
which intersect at (x,y)=(5,7), or whether they are coexistent lines, for example
y = 3*(x-15)
and
y = 3*(x-23) + 24
which are the same lines written two different ways, the 'line segment' joining the two points closest to one another is of zero length and so is not perpendicular to anything. If the two lines are parallel and not coexistent, for example
y = x
and
y = x + 1,
they are always the same distance apart and so there is no unique minimum for w(s, t).
 
Hi, thank you for replying. I'm still not really understanding the 'perpendicular' part. A friend of mine who understands math better than me tried explaining it this way:
1. The theorem is discussing infinite lines, not line segments or rays.
2. The only way two infinite lines can be non-parallel and non-intersecting is if we're looking at dimensions greater than 2D (all 2D non-parallel infinite lines will eventually intersect; two non-parallel 3D infinite lines can not intersect simply by existing in different planes of space, such as XY and XZ)
3. In 3-space the theorem is pretty easy to 'prove' using fingers or terrible white board drawings

This seems correct, so I thought I would leave this here in case anybody else has similar difficulties.

Thanks!
 
Hi, thank you for replying. I'm still not really understanding the 'perpendicular' part. A friend of mine who understands math better than me tried explaining it this way:
1. The theorem is discussing infinite lines, not line segments or rays.
2. The only way two infinite lines can be non-parallel and non-intersecting is if we're looking at dimensions greater than 2D (all 2D non-parallel infinite lines will eventually intersect; two non-parallel 3D infinite lines can not intersect simply by existing in different planes of space, such as XY and XZ)
3. In 3-space the theorem is pretty easy to 'prove' using fingers or terrible white board drawings

This seems correct, so I thought I would leave this here in case anybody else has similar difficulties.

Thanks!

As for the perpendicular part, lay a ruler/pencil/straightedge/... on the floor representing a line. Now lay another on a (possibly tilted) board on a table to represent another. Use another ruler/broom/... to measure the distance between them by holding the ruler at an angle to the one line and moving along it to the 'closest point of approach'. Notice that that 'closest point of approach' is when the ruler is perpendicular to both 'lines'.

If the lines are parallel then the 'closest point of approach' occurs everywhere along the two lines. That is a line parallel to a line, say L1, can be defined (loosely) as 'given a distance d>0, line L2 is parallel to line L1 if for every point P on L1, there exists a point Q on L2 such that the distance from P to Q is d and the distance between P and Q is the minimum distance from P to L2'. It also turns out that the line segment PQ is normal to both L1 and L2. The original statement doesn't hold only because the original statement had a uniqueness criterion.
 
"In any n-dimensional space, the two lines L1 and L2 are closest at unique points PC= P(sC) and QC= Q(tC) for which w(sC, tC) is the unique minimum for w(s,t). Further, if L1 and L2 are not parallel and do not intersect each other, then the segment PCQC joining these points is uniquely simultaneously perpendicular to both lines."
It is hard to know where to begin with this. I find a great deal of confusion in the whole thread.
First, if we have dimensions higher than three special considerations are in order.

If 1=A+tM & 2=B+tN\displaystyle \ell_1=A+tM~\&~\ell_2=B+tN are two lines then let C=M×N\displaystyle C=M\times N.
If C=0\displaystyle C=0 then the two lines are parallel.
Otherwise, if 22=\displaystyle \ell_2\cap\ell_2=\emptyset then the two lines are skew lines.
The distance between the two skew lines is d=AB(M×N)M×N\displaystyle d = \dfrac{{\left| {\overrightarrow {AB} \cdot \left( {M \times N} \right)} \right|}}{{\left\| {M \times N} \right\|}}

If d=0\displaystyle d=0 the lines intersect and are not skew.
Moreover P1 & Q2\displaystyle \exists P\in\ell_1~\&~\exists Q\in\ell_2 such that the segment PQ\displaystyle \overline{PQ} is perpendicular to both lines and has length d\displaystyle d.

The very hard part is finding the coordinates of P & Q\displaystyle P~\&~Q. Fifteen plus years ago, I was doing some work with the educational component of MathSoft. I wrote a MathCad program called SkewDist. It nicely calculated those ordinates. I no longer have access to the coding. But as a rough outline: first the distance between the lines; next find a point if one line which has the exact distance to the other line; finally using a perpendicular find the point on the other line.
 
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