Find the shortest distance between 2 non-paralle lines (skew lines) in 3-space

In summary, the shortest distance between 2 non-parallel lines is found by taking the cross product of the two lines, constructing a perpendicular plane, and projecting the lines onto this plane. The point of intersection is found, and the distance between the points is found.
  • #1
yanyin
21
0
explain how you go about finding the shortest distance between 2 non-parallel lines (i.e. skew lines).
for example:
x=3-2t, y=1-4t, z = t and (x+2)/3=y+1=(z+1)/-2
 
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  • #2
You can work out the details.

1. Take the cross product of the two lines.

2. Construct a plane perpendicular to this vector.

3. Project the two original lines onto this plane.

4. Find the point of intersection of the two projections.

5. Find the points on the original two lines which projected onto the intersection point.

6. Find the distance between these points. That's what you are looking for.
 
  • #3
Since this is the calculus forum, here is an alternate approach to my previous note.

1. Represent both lines in parameter form (s and t).

2. Set up the distance squared (H) between arbitrary points p on line 1 and q on line 2 as a function of the two parameters.

3, Take partial derivatives of H with respect to t and with respect to s. This will be two linear equations in two unknowns s and t.

4. Solve for s and t. Find the points on the 2 lines corresponding to these values of s and t and compute the distance between them.

5. If you are not interested in the points on the lines but only the distance, plug the found values of s and t into the equation for H and take a square root.
 
  • #4
i found a very simple, basic and easy to understand solution on the web, take a look if you like:
http://www.mis.coventry.ac.uk/jtm/slides/8/sld8p5.pdf [Broken]
 
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  • #5
I like this approach for its simplicity and symmetry:
1. Normalize-Cross-Prod the directions of L1 and L2 to obtain the direction of their adjoining lineSeg dL
2. Extrude L1 along direction dL (= plane PL1)
3. Extrude L2 along direction dL (= plane PL2)
4. The endpoints of the shortest LineSeg bridging between L1 and L2 are:
intersection( PL1 , L2 )
intersection( PL2 , L1 )

from Pierre Bierre, author, "Flexing the Power of Algorithmic Geometry"
 

1. What is the definition of skew lines?

Skew lines are two lines that do not intersect and are not parallel. They lie in different planes and are not in the same direction.

2. How do you determine if two lines are skew?

To determine if two lines are skew, you can use the dot product of their direction vectors. If the dot product is equal to zero, the lines are perpendicular and therefore not skew. If the dot product is not equal to zero, the lines are skew.

3. Can two skew lines ever intersect?

No, by definition, skew lines do not intersect. They can only be parallel or intersecting.

4. How do you find the shortest distance between two skew lines?

To find the shortest distance between two skew lines, you can use the cross product of their direction vectors to find the shortest distance between their closest points. This will give you the distance between the two lines along the shortest path.

5. Is there a formula for finding the shortest distance between two skew lines?

Yes, the formula is: d = |(a1 - a2) * (b1 x b2)| / |b1 x b2|, where a1 and a2 are points on the first line, b1 and b2 are direction vectors of the first and second lines, and x represents the cross product.

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