distance between parallel lines

psych 101

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Jun 3, 2005
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OK, so I feel like I should remember this from high school math... I need to know the formula for finding the distance between 2 parallel lines. More specifically, it's 2 parallel runways on the airport surface. How exactly should I go about calculating this? I have access to the lat/long coordinates, so I am able to find the length of the runways and distance between ends.

Thanks!
Danielle
 
You could measure it?

With only Lat and Long, you will need to be careful where those data are taken. Are the "middle of runway" or "edge of runway"?

If they are parallel to the Lat or Long, you can just subtract and then multiply by an appropriate factor, depending on where you are on the planet.

Can you be more specific?
 
Hello, Danielle!
Is this a homework problem . . with latitude/longitude coordinates?
It's pretty messy . . .

I need to know the formula for finding the distance between 2 parallel lines.
More specifically, it's 2 parallel runways on the airport surface.
How exactly should I go about calculating this?
I have access to the lat/long coordinates,
so I am able to find the length of the runways and distance between ends.
I won't make any assumptions about the two runways
. . . like: they are the same length or they are anywhere near each other.
Code:
                               B
                              *(x2,y2)
                       * 
        A       *                            Runway 1 goes from A to B.
         * 
      (x1,y1)                       D        Runway 2 goes from C to D.
                                   *
                            *      
                     * 
               *
  C      *
   *
 (p,q)
I assume you can convert lat/long to cartesian coordinates.

There is a formula for this . . .

The distance from a point C(p,q) to a line: .ax + by + c .= .0

. . . . . . . . . . . . . . . . . . . |ap + bq + c|
. . . is given by: . d . = . ----<u>----------</u>---
. . . . . . . . . . . . . . . . . . . . .√a<sup>2</sup> + b<sup>2</sup>


Find the equation for Runway 1 . . . It is the line through points A and B.
. . . (y<sub>2</sub> - y<sub>1</sub>)x - (x<sub>2</sub> - x<sub>1</sub>)y + [(y<sub>1</sub>(x<sub>2</sub> - x<sub>1</sub>) - x<sub>1</sub>(y<sub>2</sub> - y<sub>1</sub>)] . = . 0

Use any point on Runway 2, say, C(p,q)

And run them through the above formula . . . Good luck!
 
Unfortunately, it's not homework. This is something I need to work out for a project I am involved in at work. The runways are in fact next to eachother, and yes, I have the length of each (they are not the same length). The lat/long points I have are for each end of both runways.

The original formula I have for this involves the area:

[2(area)] / (length rwy1 + length rwy2)

However, including the area in an equation used to find the distance between the 2 doesn't make much sense to me... these runways could presumably go on forever, but the distance between them would remain the same....

Any other thoughts?
 
What are the data you have. Maybe we can make sense of it, since you don't seem to have answered my questions about middle or edge.
 
Sadly, we do not have access to whether the coordinates came from the edge or not, though it is safe to assume it is consistent. Runways are only about 150 feet wide, so this isn't important to us anyway (as long as it remains consistent). Here is one example:

*degree-decimal mins format

Runway1:
(A) 32-58.51980N / 097-19.04110W
(B) 32-59.85552N / 097-19.31455W
Length- 8220 ft

Runway2:
(C) 32-58.53827N / 097-18.92573W
(D) 33-00.09815N / 097-19.24503W
Length- 9600 ft

distance btw A&C=598.6852703 ft
distance btw B&D=1516.216039 ft

_________________
A B
____________________
C D

Thank you so much for your time and consideration!
 
I don't quite understand the results I am getting, but it appears the equations of these runways could be (based on an arbitrary coordinate axis I assigned):

AB: y = 90.566608+.204721*x
CD: y = 90.565469+.204695*x

You can then use the formulas given above for a line and a point. If it were me, I'd use several different combinations to check the result. You have four choices.

A vs CD
B vs CD
C vs AB
D vs AB

They should give the same answer if all is well.

Using an entirely different methodology, I get 598.542477 ft, but I'm not entirely fond of that result.
 
The last result is correct, using the methodology you're not very fond of. As it turns out, I have to treat the runways as trapezoids. So...
2(area) / (length rwy1 + length rwy 2) makes sense after all.

Thanks for all your help!
 
OK, well then that's pretty smart.

The area of a trapezoid is the average of the unequal sides, multiplied by the height.

It the airport example,

height = distance between runways = h
Unequal side #1 = length of first runway = a
Unequal side #2 = length of second runway = b

Then Area = [(a+b)/2]*h

Solving for h = 2*Area/(a+b) -- And there we have it.

Nice to know where it comes from.

Just one last question, at some point we'll have to worry about the curvature of the Earth. I wonder when that point is. I was thinking 9600 ft might be close. Maybe not. A discussion for another day, perhaps.
 
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