Count the Unique Rectangles!

Question

In my last challenge, you were asked to find all rectangles given a m x n grid of them. However, it turned out to be very trivial as there actually was a mathematical formula I did not even know about to solve the problem! So now, for a little bit more of a challenge, how about calculating the number of unique rectangles, i.e. find the number rectangles that are all of different dimensions?

For example, consider 4 horizontal, or y lines at [-250,-50,50,250] and 4 vertical, or x lines at [-250,-70,70,250]. Graphing these on a coordinate plane with infinite dimensions results in the following 500 x 500 pixel closed grid, in which the length of each segment in pixels and the lines corresponding to their respected values from the arrays are shown:

Example

which contains the 16 unique rectangles shown in this animated GIF:

All unique Rectangles in Example

However, if the topmost line (y = 250) were to be removed, there would only be 12 unique rectangles, as the top 3 rectangles would be factored out since they each won't be fully closed without the y = 250 line.

Task

So, as shown above, the task is counting the number of rectangles rectangles with different dimensions. In other words, given an input of 2 arrays, with the first one containing all equations of all x lines, and the latter containing those of all y lines, output the total number of rectangles of different dimensions created when the lines corresponding to those equations are graphed on a coordinate plane.

Rules

• The use of any built-ins that directly solve this problem is explicitly disallowed.

• If either of the arrays have less than 2 elements, the output should be 0, since if there are less than 4 lines on the plane, there are no closed, 4 sided figures.

• The input arrays are not guaranteed to be sorted.

• You can assume that there are not any repeated values in either of the the input arrays.

• A n x m rectangle is the same as a m x n rectangle. For example, a 300 x 200 rectangle is the same as a 200 x 300 one.

• Standard loopholes are prohibited.

Test Cases

Given in the format Comma Separated Arrays Input -> Integer output:

[],[] -> 0
[-250,-50,50,250],[-250,-70,70,250] -> 16 (Visualized above)
[-250,-50,50,250],[-250,-70,70] -> 12
[-40, 40],[-80, 50] -> 1
[10],[10] -> 0
[60, -210, -60, 180, 400, -400], [250, -150] -> 12
[0,300,500],[0,300,500] -> 6
[0,81,90],[0,90,100] -> 9

Remember, this is code-golf, so shortest code wins!

• Sp3000: [[0,300,500],[0,300,500]] should probably be a test case
• R. Kap: @Sp3000 All right. I'll add that one as soon as possible.
• R. Kap: @Sp3000 It's added.
• R. Kap: @PeterTaylor Yeah, you're right. There are 6. I forgot to count the entire square itself! It's updated now.
• R. Kap: @PeterTaylor Okay. Added.

Answer 1

JavaScript (ES6), 99

(X,Y,Z=new Set)=>X.map(v=>X.map(t=>t>v&&Y.map(w=>Y.map(u=>u>w&&Z.add([u-w,t-v].sort()+0)))))|Z.size

Note: sort is not numeric but lexicographic, but in this specific case I don't care

Less golfed

(X, Y, Z = new Set) =>
 X.map(
 v => X.map(
 t => t>v && Y.map(
 w => Y.map(
 u => u>w && Z.add([u-w, t-v].sort() + 0)
 )
 )
 )
 ) | Z.size

Test

F=(X,Y,Z=new Set)=>X.map(v=>X.map(t=>t>v&&Y.map(w=>Y.map(u=>u>w&&Z.add([u-w,t-v].sort()+0)))))|Z.size
 
;[
 [[],[], 0]
,[[-250,-50,50,250],[-250,-70,70,250], 16]
,[[-250,-50,50,250],[-250,-70,70], 12]
,[[-40, 40],[-80, 50], 1]
,[[10],[10], 0]
,[[60, -210, -60, 180, 400, -400], [250, -150], 12]
,[[0,300,500],[0,300,500], 6]
].forEach(t=>{
 var a=t[0],b=t[1],k=t[2],r=F(a,b)
 
 console.log(r==k?'OK':'KO','['+a+']','['+b+']',r)
})

Answer 2

Jelly, 13 bytes

ṢIẆS€μ€ŒpṢ€QL

Try it online!

A rectangle is uniquely defined by its height and its width.

ṢIẆS€μ€ŒpṢ€QL Main chain, argument: z
 μ€ For each subarray:
Ṣ sort
 I compute consecutive increments
 Ẇ yield all substrings
 S€ compute sum of each substring
 Œp Cartesian product
 Ṣ€ Sort each
 Q Remove duplicates
 L Length

• R. Kap: Great job! +1 from me! :)
• Leaky Nun: @PeterTaylor But it does work for that input...
• Peter Taylor: @R.Kap, was this just because the test case was wrong?
• R. Kap: Actually, it wasn't producing any output when I entered that test case at the time, but that was because I entered it in the wrong format. My bad. It works really well for that test case. Good job.

Answer 3

CJam (22 bytes)

q~{2m*::-:z0-}/m*:$_&,

Takes input as an array of arrays. Online demo, test suite

Dissection

q~ e# Parse input
{ e# For each of the two elements in the top-level array
 2m* e# Take its Cartesian self-product
 ::- e# Map fold subtraction, giving the separations between lines
 :z e# Map absolute value
 0- e# Remove 0, since trivial rectangles don't count
}/
m* e# Cartesian product of the two sets of separations
:$ e# Sort so that mxn === nxm
_& e# Deduplicate
, e# Count

Answer 4

MATL, 22 bytes

"@gd&Xfo!s|]Z*!S!Xuz2/

Try it online! Or verify all test cases.

Explanation

" % Implicitly input cell array of two numerical arrays. For each cell
 @g % Push cell contents
 d % Difference of its two entries
 &Xf % Cell array of all substrings
 o % Convert to numerical array, padding with zeros
 !s % Sum of each row. Gives a row vector
 | % Absolute value of each entry
] % End for
Z* % Cartesian product
!S! % Sort each row
Xu % Unique rows
z2/ % Number of unique rows excluding zeros. Implicitly display

Answer 5

Python, 208 bytes

import itertools as i
def h(x,y):
 s=[]
 k=[p for p in i.product(x,y)]
 for j,(v,w) in enumerate(k):
 for a,b in k[j+1:]:
 q=sorted([abs(a-v),abs(b-w)])
 if 0not in q and q not in s:s+=[q]
 return len(s)