Haskell中判断四点是否构成正方形的isSquare函数实现问题

Hey there! Let's work through fixing your square-detection function in Haskell. Your initial thought to use distances is a solid starting point, but it misses a few critical details that are tripping you up—let's break this down clearly.

The Problem With Your Initial Approach #

Checking if all pairwise distances are equal only works if all four points are identical (which definitely isn't a square!). A valid square has two distinct pairwise distances:

• 4 instances of the side length (between adjacent vertices)
• 2 instances of the diagonal length (between opposite vertices)
  Plus, the diagonal length squared should be exactly twice the side length squared (thanks to the Pythagorean theorem: (d^2 = s^2 + s^2 = 2s^2)).

A Robust Solution #

We can avoid floating-point precision issues entirely by working with distance squared instead of actual distances. Here's how to implement this:

First, import Data.List for sorting (we'll need it to group our distances):

import Data.List (sort)

Next, write a helper function to calculate the squared distance between two points:

distanceSquared :: (Num a, Ord a) => (a,a) -> (a,a) -> a
distanceSquared (x1, y1) (x2, y2) = (x1 - x2)^2 + (y1 - y2)^2

Now, build the isSquare function. We'll generate all unique pairwise distances, sort them, and verify they match the pattern we expect for a square:

isSquare :: (Num a, Ord a) => (a,a) -> (a,a) -> (a,a) -> (a,a) -> Bool
isSquare p1 p2 p3 p4 = 
  let points = [p1, p2, p3, p4]
      -- Generate all unique pairwise squared distances (6 total for 4 points)
      dists = sort [ distanceSquared a b | a <- points, b <- points, a < b ]
  in case dists of
       -- Check for 4 equal side lengths, 2 equal diagonals, valid proportions, and no overlapping points
       [sideSq, sideSq, sideSq, sideSq, diagSq, diagSq] -> sideSq > 0 && diagSq == 2 * sideSq
       -- Any other pattern means it's not a square
       _ -> False

Why This Works #

• No floating points: Using squared distances avoids precision errors that come with square roots.
• Handles any vertex order: Since we check all pairwise distances, the input order of the four points doesn't matter.
• Eliminates edge cases: We explicitly check that sideSq > 0 to rule out overlapping points, and verify the diagonal/side proportion to avoid mistaking rhombuses or other quadrilaterals for squares.

Example Tests #

To confirm it works, try these cases:

-- A square with side length 1
isSquare (0,0) (0,1) (1,1) (1,0) -- Returns True

-- A rhombus that's not a square
isSquare (0,0) (1,2) (3,2) (2,0) -- Returns False

-- Four identical points
isSquare (5,5) (5,5) (5,5) (5,5) -- Returns False

内容的提问来源于stack exchange，提问作者mathandtic