Count rectangle intersections

Question

The Challenge

Given an arbitrary amount of rectangles, output the total count of intersections of those when drawn in a 2D plane.

An intersection here is defined as a point P which is crossed by two lines which are orthogonal to each other and are both not ending in P.

Example

Each rectangle here is denoted by a 2-tuple with the coordinates of the upper left corner first and the coordinates of the bottom right corner second.

[(-8,6),(-4,-2)]
[(-4,9),(4,3)]
[(2,10),(14,4)]
[(1,7),(10,-6)]
[(7,4),(10,2)]
[(5,2),(9,-4)]
[(-6,-4),(-2,-6)]

Those rectangles create 6 intersections, which has to be your output.

• As you can see in the image above, touching rectangles will not create intersections here and are not counted.
• You can encode the rectagles in any format you want. Make it clear which format you use.
• If multiple rectangles intersect at the same point, it only counts as one intersection.
• The coordinates will always be integers.
• There won't be any duplicate rectangles in the input.
• You will always get at least one rectangle as input.
• You may not use any builtins which solve this problem directly. Additionally you may not use builtins that solve equations. All other builtins are allowed.
• The output has to be a single integer indicating the intersection count.

Rules

• Function or full program allowed.
• Default rules for input/output.
• Standard loopholes apply.
• This is code-golf, so lowest byte-count wins. Tiebreaker is earlier submission.

Test cases

Same format as in the example above. The rectangles are wrapped in a list.

[[(-8,6),(-4,-2)],[(-4,9),(4,3)],[(2,10),(14,4)],[(1,7),(10,-6)],[(7,4),(10,2)],[(5,2),(9,-4)],[(-6,-4),(-2,-6)]] -> 6
[[(-2,2),(6,-4)]] -> 0
[[(-12,10),(-8,6)],[(-14,6),(-10,2)],[(-10,6),(-6,2)]] -> 0
[[(-4,10),(6,2)],[(-2,8),(4,3)],[(1,6),(8,4)],[(2,11),(5,5)]] -> 10
[[(8,2),(12,-2)],[(10,0),(14,-4)]] -> 2
[[(0,2),(2,0)],[(0,1),(3,0)]] -> 1
[[(-10,-2),(-6,-6)],[(-6,-2),(-2,-6)],[(-8,-4),(-4,-8)]] -> 3

Happy Coding!

Answer 1

JavaScript (ES6), 186 bytes

a=>a.map(([a,b,c,d])=>h.push([b,a,c],[d,a,c])&v.push([a,b,d],[c,b,d]),h=[],v=[])|h.map(([d,a,e])=>v.map(([c,f,b])=>a<c&c<e&b<d&d<f&t.every(([a,b])=>a-c|b-d)&&t.push([c,d])),t=[])|t.length

Splits each rectangle into its component lines, then intersects the horizontal and vertical lines, building up a list of intersections to avoid duplicates.

Answer 2

Mathematica 138 bytes

Not finished! This works for all cases except [[(0,0),(1,2)],[(0,0),(2,1)]]

---

Length@Union[Join@@(Cases[RegionIntersection@@# &/@Subsets[Line[{{#,#2},{#3,#2},{#3,#4},{#,#4},{#,#2}}]&@@@Flatten/@#,{2}],Point@a__:> a])]

---

Example

Length@Union[
Join @@ (Cases[RegionIntersection @@ # & /@ Subsets[
Line[{{#, #2}, {#3, #2}, {#3, #4}, {#, #4}, {#, #2}}] & @@@ Flatten /@ #, {2}], 
Point@a__ :> a])] &@{{{-8, 6}, {-4, -2}}, {{-4, 9}, {4, 3}}, {{2, 10}, {14, 4}}, 
{{1, 7}, {10, -6}}, {{7, 4}, {10, 2}}, {{5, 2}, {9, -4}}, {{-6, -4}, {-2, -6}}}

6