Assuming your question deals with locating the position of a point in
2 dimensions (i.e., on a plane), Mathtalk is correct; in general, you
need the distances to three known points to uniquely determine the
coordinates of the fourth point. If you need to locate a point in
three dimensions, you actually need to know the distance to a fourth
known point. (This is how GPS receivers work to find the latitude,
longitude, and elevation of the receiver.)
For the 2-dimensional case, you can visualize why this is so by
considering that if you know you are a given distance (R1) from a
known point at (x1, y2), you know you are somewhere on the circle with
radius R1 surrounding that point. If, in addition, you know you are a
distance R2 from a second point (x2,y2), you know that you must also
be on a circle of radius R2 from that point. If (R1 + R2) <
sqrt[(x1-x2)^2 + (y1-y2)^2] (this is what Mathtalk means by the
triangle inequality), the two circles do not intersect, and there is
no physically meaningful solution to the location problem. If R1 + R2
> sqrt[(x1-x2)^2 + (y1-y2)^2], the two circles intersect at two
points, and you need the distance to a third point to determine which
of these points is the actual location. If (R1 + R2) = sqrt[(x1-x2)^2
+ (y1-y2)^2], one has the special case that all three points lie on a
single line, and in this case only, you only need the distances to two
known points to determine the coordinates of a third point.
See for http://www.restena.lu/convict/Jeunes/beacon/Positioning.htm
for a nice illustration of these concepts, as well as the equations
you are looking for to determine the coordinates of your unknown
point. |
