I’m working on a geocache puzzle that requires me to find 4 different points on a map which form a sort of large kite-shaped quadralateral. I basically have the X and Y coords for each of the four vertices and I now need to find the X,Y value of the point in which the diagonals between vertices intersect.
I’m not even sure if I’m explaining this right, but to make it clearer - if you take a kite, and draw a line from the top to bottom points, and then from side to side (as if each line were a stick in the kite), where would they intersect?
An important point… the sides aren’t of equal length, so it’s sort of a lopsided kite shape.
I know there is a formula to determine this point but I’m damned if I can figure out what it is. It’s been a very very long time since I’ve had to take a math class.
Do the diagonals of the kite bisect the angles they split? If so, this is pretty easy. Position your kite so that the leftmost vertex is at (0, 0) and the rightmost is at (a, 0). Then the top is at (a/2, b) and the bottom at (a/2, c), and the diagonals intersect at the point (a/2, 0).
If not, you should still make one diagonal be a segment of the x-axis. Having zeros in there will make the math easier.
If you remember enough algebra to do the following (and, if you do remember, it’s a fairly straightforward exercise), you can do it thusly:
Find an equation for the line through two opposite vertices.
Find an equation for the line through the remaining pair of opposite vertices.
Considering these two equations to be a system, find the solution to the system.
Alternatively, points along the diagonal from the “top” of the “kite” (x[sub]t[/sub], y[sub]t[/sub]) to the bottom (x[sub]b[/sub], y[sub]b[/sub]) have coordinates
x = x[sub]b[/sub] + t(x[sub]t[/sub] - x[sub]b[/sub])
y = y[sub]b[/sub] + t(y[sub]t[/sub] - y[sub]b[/sub])
for t from 0 to 1.
Points on the diagonal from “left” to “right” of the “kite” (x[sub]l[/sub], y[sub]l[/sub]) to (x[sub]r[/sub], y[sub]r[/sub]) have coordinates
x = x[sub]l[/sub] + s(x[sub]r[/sub] - x[sub]l[/sub])
y = y[sub]l[/sub] + s(y[sub]r[/sub] - y[sub]l[/sub])
for s from 0 to 1.
If you have the numerical x and y corrdinates for all 4 points, you can plug them into the above expressions, set the two x’s equal to each other:
x[sub]b[/sub] + t(x[sub]t[/sub] - x[sub]b[/sub]) = x[sub]l[/sub] + s(x[sub]r[/sub] - x[sub]l[/sub])
and the y’s equal to each other
y[sub]b[/sub] + t(y[sub]t[/sub] - y[sub]b[/sub]) = y[sub]l[/sub] + s(y[sub]r[/sub] - y[sub]l[/sub])
and solve for s and t, then plug them back into either expression to get x and y.
There may be other methods that are simpler to describe or give a formula for, but, assuming I’m understanding the question correctly, that’s the best I can come up with.
Wow, thanks! This looks like it might do the trick. I’m going to start plugging in some numbers and see what I come up with. I’ll let you know how it goes.