# Thread: Geometric locus

1. ## Geometric locus

Hi,
Given two crossed lines in the space,what is the geometric locus of all the midpoints of segments connecting a point of one line with a point on the other line?
(I solved it for intersecting lines and found that it is the entire plane determined by the two lines.am I right?)

2. ## Re: Geometric locus

No. Where ever the two points on a pair of intersecting lines are, the midpoint of the line connecting them lies on the bisector of the angle between the lines. The locus is the two lines bisecting the four angles formed by the original intersecting lines.

3. ## Re: Geometric locus Originally Posted by hedi Given two crossed lines in the space,what is the geometric locus of all the midpoints of segments connecting a point of one line with a point on the other line?
(I solved it for intersecting lines and found that it is the entire plane determined by the two lines.am I right?)
hedi, one again there must be something lost in translation of this into English.
Are you talking about SKEW LINES https://en.wikipedia.org/wiki/Skew_lines For any two skew lines, there is a unique perpendicular between them, Therefore it makes sense to speak of its midpoint. BUT "locus of all the midpoints of segments connecting a point of one line with a point on the other line" is far too vague. Any two points determine a line segment. Perhaps you can explain what you did for intersecting lines??

4. ## Re: Geometric locus

For every point in the plane there exsist two points on the two lins having this point as a middle poit.

5. ## Re: Geometric locus

The statement of halsofivy is not true ...

6. ## Re: Geometric locus Originally Posted by hedi For every point in the plane there exsist two points on the two lins having this point as a middle poit. Originally Posted by hedi This is not true.
Now that is correct.
Have you drawn a two such lines?

7. ## Re: Geometric locus Originally Posted by hedi For every point in the plane there exsist two points on the two lins having this point as a middle poit.
call the point of intersection $(0,0)$

then the two lines are

$(u,~m_u u),~(v,~m_v v)$

given some $(u,v)$

The midpoint is found at $\left(\dfrac{u+v}{2}, \dfrac{m_u u + m_v v}{2}\right)$

pick a point in the plane $(x,y)$

$x = \dfrac{u+v}{2},~y = \dfrac{m_u u + m_v v}{2}$

$u = -\dfrac{2 \left(x m_v-y\right)}{m_u-m_v},~v = \dfrac{2 \left(x m_v-y\right)}{m_u-m_v}+2 x$

gets you a point along each line such that their midpoint corresponds to your point of choice.

Note the denominator will never be zero as long as the two lines are not the same line.

If they are the same line then the locus of midpoints is just that line.

8. ## Re: Geometric locus

I wrote two linear equations with the parameters t,s of the lines as the unknown and any given point in the lane as the nnhomogeneous part on the right side and saw that there is always a solution'since the determinant is not zero.

9. ## Re: Geometric locus

So you agree that in case of intercecting lines the geometric locus is the entire plane determined by the two lines,rirht?

10. ## Re: Geometric locus Originally Posted by hedi So you agree that in case of intercecting lines the geometric locus is the entire plane determined by the two lines,rirht?
yes

11. ## Re: Geometric locus

In case of cross lines what is the geometric locus?

12. ## Re: Geometric locus Originally Posted by hedi In case of cross lines what is the geometric locus?
what's the difference between cross lines and intersecting lines?

13. ## Re: Geometric locus Originally Posted by romsek what's the difference between cross lines and intersecting lines?
Skew lines? These lines are in $\mathbb{R}^3$ so each midpoint is a triple. How does that effect your reply #10?

14. ## Re: Geometric locus

there might not be a solution for any arbitrary point in the space.I need to think more about it,but anywayI will appreciate some help. Originally Posted by hedi 