问题
I have a trapezoid where I know the length of all 4 lines, and I know the coordinates of two of the corners. How do I find the coordinates of the remaining two corners in JavaScript?
NOTE: All 4 lines are of different length, ie this is not an isosceles trapezoid
Here's a little diagram in case it helps:
L3
D ____ C
/ \
L4 / \ L2
--------
A L1 B
I know the coordinates of A and B, and the length of L1, L2, L3, and L4 (Which are all different). I just need to get all of the possible sets of coordinates for D and C!
回答1:
Drop vertical lines from points C and D to the line AB to find their projections on E and F on AB:
Now AED and BFC are right triangles with the same height. Let's call the height h. From Pythagoras:
a² + h² = L4² and b² + h² = L2²
Subtracting one equation from the other you get: a² - b² = L4² - L2²
Also you can divide L1 into segments AE, EF, and FB, so the length of L1 must be:
L1 = a + L3 + b => a + b = L1 - L3
Therefore we have a system of equations in a and b:
a² - b² = L4² - L2²
a + b = L3 - L1
Using the fact that a² - b² = (a+b)(a-b) and the above equation, you get:
(L3 - L1)(a - b) = L4² - L2² => a - b = (L4² - L2²)/(L3 - L1)
(Note that L1 and L3 can not be equal. If L1 = L3, there is an infinite number of solutions.)
So the equations simplify to:
a + b = L3 - L1
a - b = (L4² - L2²)/(L3 - L1)
The solution is:
a = (L3 - L1 + (L4² - L2²)/(L3 - L1)) / 2
b = (L3 - L1 - (L4² - L2²)/(L3 - L1)) / 2
The height of the trapezoid is:
h = sqrt(L4² - a²) = sqrt(L2² - b²)
You could now use a to solve for the angle at point A and b to solve the angle at point B, and use them to calculate the coordinates for C and D. Or you can use a, b, and h directly.
For example: suppose A is at the origin (0,0) and B is at (L1, 0). Then the two solutions are:
- C is at
(a, h)and D is at(L1 - b, h) - C is at
(a, -h)and D is at(L1 - b, -h)
来源：https://stackoverflow.com/questions/61140525/how-to-infer-the-possible-values-for-the-final-two-coordinates-of-a-trapezoid-wh