Parametric Line intersections

In summary: The equation you are looking for is t = x3-x1 In summary, the person is having a problem intersecting parametric lines. They are using the equations x=x1+t(x2-x1), y=y1+t(y2-y1), and x=x3+t(x4-x3). They calculate the intersection, using this: t = (x3-x1) / ( (x2-x1) - (x4-x3)). However, it doesn't work. They think it might be because they used lines of equal length and intersected in the middle, but now they understand that the equation is t = x3-x
  • #1
Pazil
8
0
Hello!

I'm having a pretty big problem intersecting parametric lines. I'm using the equations:
x = x1 + t(x2 - x1)
y = y1 + t(y2 - y1)

Given the 2 lines:
x = x1 + t(x2 - x1)
y = y1 + t(y2 - y1)

x = x3 + t(x4 - x3)
y = y3 + t(y4 - y3)

I calculate the intersection, using this:
t = ( x3 - x1 ) / ( (x2 - x1) - (x4 - x3) )

And then I would plug it into the first line...

For some odd reason...it doesn't work! And the funny thing is, that it used to work in my program. I'm not sure if I made any changes, but it's really screwing things up...

Please help!

P.
 
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  • #2
Why would you think that would work? There is no reason to thing that the same value of the parameter in each line gives the point of intersection. Think of "t" as representing the time some object is at each point on the line. The same t giving the point of interesection would be equivalent to the two objects being at that point at the same time which is not necessary for a point of intersection. To make it clear that the parameters in the two different lines are not necessarily the same, use different letters, say t and s:
x = x1 + t(x2 - x1)
y = y1 + t(y2 - y1)
and
x = x3 + s(x4 - x3)
y = y3 + s(y4 - y3)

set x and y equal:
x1+ t(x2-x1)= x3+ s(x4- x3) and y1+ t(y2-y1)= y3+ s(y4-y3)
and you have two equations to solve for the two parameters s and t.
 
  • #3
ARRRG!

I was wondering why it worked before! In my first tests I had used lines of equal length and intersecting in the middle!
I now understand. I knew that you of course couldn't plug "t" into any of the two...I thought it depended to which equation you plugged it into.
Anywho, I found a complete equation finding the "t" variable for the first line:
http://local.wasp.uwa.edu.au/~pbourke/geometry/lineline2d/

Thanks for your help and explaining anyhow! The more math stuff I start getting through my head, the more I'll be able to learn on my own!

P.
 
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1. What is a parametric line intersection?

A parametric line intersection is a mathematical concept that refers to the point at which two parametric lines (lines defined by equations with one or more parameters) intersect or cross each other.

2. How is the intersection point of two parametric lines determined?

The intersection point of two parametric lines is determined by solving the equations for the two lines simultaneously. This typically involves setting the equations equal to each other and solving for the values of the parameters that make the equations equal.

3. Can two parametric lines intersect at more than one point?

Yes, it is possible for two parametric lines to intersect at more than one point. This can occur when the equations for the lines have multiple solutions, or when the lines are parallel and never truly intersect.

4. How does the number of parameters in the equations affect the intersection of parametric lines?

The number of parameters in the equations can affect the intersection of parametric lines in various ways. For example, if both equations have the same number of parameters, there may be a unique solution for the intersection point. However, if one equation has more parameters than the other, there may be an infinite number of solutions for the intersection point.

5. What are some real-world applications of parametric line intersections?

Parametric line intersections have many practical applications in fields such as engineering, computer graphics, and physics. They can be used to determine the position of objects in motion, calculate the trajectory of projectiles, and create realistic 3D models of objects and environments.

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