Efficient algorithm to determine the shape from the inputted four points - c

Question

I am trying to solve Determine the shape problem on Uva. From what I can get after reading the question it that it is an Ad Hoc geometry problem in which we have to use some geometry theorem to determine what shape the four points we take as input form on a 2D plane. After spending many hours I still cannot think of any efficient algorithm which can solve the problem efficiently in the given time limit.I tried using distance formula and slopes but was not of much help.Please suggest some good algorithm or theorem I can use to solve this problem.

Answer 1

My first thought was the following:
Determine the lengths of each side. Use the formula sqrt((a - x)^2 + (b - y)^2).
Determine each of the angles.
Determine which shape:
If all angles 90
If all sides equal, return SQUARE
Else, return RECTANGLE
Else if angle 1 = 3 and angle 2 = 4 equal
If all sides equal, return RHOMBUS
Else, return Parallelogram
Else if angle 1 + 2 == 180, or angle 3 + 4 == 180, return TRAPEZIUM
Else, return ORDINARY QUADRILATERAL
But consider the logic followed here; they've used vector math, and simply calculate whether something is a right angle or whether two lines are parallel, rather than calculating all angles.
Prepare some functions to calculate whether two lines are parallel, and whether an angle is a right angle (using vector mathematics).
Sort the points so they're in the correct rotational order (the actual order is ambiguous for concave shapes, but it shouldn't matter, should still return ORDINARY QUAD).
Determine the lengths of each line.
Determine the shape, given the lengths and whether lines are parallel or angles right.

Answer 2

Firstly, different shapes have the following relationship:
Ordinary Quadrilateral --(with one pair of parallel sides)--> Trapezium
Trapezium --(with additional pair of parallel sides)--> Parallelogram
Parallelogram
--(with four equal straight lines)--> Rhombus--(with four 90 degree angles)-->square
|--(with four 90 degree angles)--> Rectangle --(with four equal lines)--> square
Given four points, A, B, C, D, take random one(say A), and we need to calculate statistics(angle and length) of following four pairs of points(not all), including
(1) AB and CD, A1, L1
(2) AC and BD, A2, L2
(3) AB and AC, A3, L3
(4) AB and AD, A4, L4
And then I think the trick comes to how to organize the branches so that we could have minimal computation and code path. My proposal is listed as follows:
A3 = getAngle(AB, AC)
A4 = getAngle(AB, AD)
if A3 > A4
we know AD is the diagonal line, then use A, B, C to calculate
else
we know AC is the diagonal line, then use A, B, D to calculate
# following suppose we use A, B, C to do the calculation, we could easily do the A, B, D thing if define new variables
L3 = LengthEqual(AB, AC)
A1 = getAngle(AB, AD)
if A3 == 90 && A1 == 0
if L3 == True
Square
else
Rectangle
else
A2 = getAngle(AC, BD)
L2 = LengthEqual(AC, BD)
if A2 == 0 && A1 == 0
if L2 == True
Rhombus
else
Parallelogram
else if A2 == 0 || A1 == 0
Trapezium
else
Ordinary Quadrilateral
In this means, we could achieve relatively less computation to wisely branch to the results we want. Hope this helps.