parametric equations

loveroflight

Hi

Had a problem with finding point of section of two lines given the parametric equations of both.
Why can't the point of intersection be found by allowing the x of one lin in parametric form be equated to the x of the other line in parametric form, thus finding the value of t and using this to find the point of intersection?

Would it have something to do with the rate of change of each t being different?

Hope someone can help

LeixlipRed

Given one line: x=f_1(t), y=g_1(t) and another line: x=f_2(t), y=g_2(t) you wish to find their point of intersection. Simply isolate t in both the x functions and equate. And do similarly for the y functions. This will give you explicit values for x and y if such values exist.

rjt

loveroflight wrote: »

Hi

Had a problem with finding point of section of two lines given the parametric equations of both.
Why can't the point of intersection be found by allowing the x of one lin in parametric form be equated to the x of the other line in parametric form, thus finding the value of t and using this to find the point of intersection?

Would it have something to do with the rate of change of each t being different?

Hope someone can help

The reason your method doesn't work is that it assumes that there is some value of t for which both parametric forms give the same point (x,y). This is not always the case, for example:

We have two parametric equations:
f(t) = (t,t)
g(t) = (t-1,1-t)
So f is the line x=y (the line with slope 1 going through the origin) and g is the line perpindicular to this through the origin. They both go through the origin, so this is their point of intersection. Let's try your method of letting the x coordinates and y coordinates be equal:

t = t-1
t = 1-t

But this doesn't have any solutions... So why didn't this work? Well, letting the x coordinates and y coordinates be equal is the same as saying f(t) = g(t). But this is never the case, because the only point they share is the origin, and f(0) = (0,0), g(1) = (0,0). So, they share a point, but the corresponding values of t are different.

So instead of saying f(t) = g(t) (which gave us the two equations above), try f(s) = g(t). This gives:

s = t-1
s = 1+t

Which has solution s=0, t=1. So we know that the point of intersection is
f(0) = g(1) = (0,0)

loveroflight

Thanks for that