Problem 193. Smallest distance between a point and a rectangle

Created by Alfonso Nieto-Castanon
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Given two points x and y placed at opposite corners of a rectangle, find the minimal euclidean distance between another point z and every point within this rectangle.

For example, the two points

     x = [-1,-1];
     y = [1,1];

define a square centered at the origin. The distance between the point

   z = [4,5];

and this square is

   d = 5;

(the closest point in the square is at [1,1])

The distance between the point z = [0,2] and this same square is d = 1 (closest point at [0,1])

The distance between the point z = [0,0] and this same square is d = 0 (inside the square)

Notes:

you can always assume that x < y (element-wise)
The function should work for points x,y,z in an arbitrary n-dimensional space (with n>1)

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830 Solutions
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Last Solution submitted on Feb 21, 2025

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3 Comments

3 Comments

Sam on 8 Oct 2014

For the n dimensional case it would be better to say that x and y lie on opposite vertices of the n-hypercuboid such that each edge is parallel to a coordinate axis.

Brandon on 3 May 2023

Two points do not define a rectangle. This is especially true in 3D space. A correct answer to this question would be 0 or the point closest to the circle defined by the two points on the diameter, depending on the rectangle you chose to make. (Every rectangle formed from two points defining opposite corners makes a circle, and in 3D, a sphere). I genuinely do not know how you want me to handle the 3D cases.

DH on 8 Mar 2024

I agree to Brandon's comment. In a 2D space, there should be infinite numbers of rectangles, which makes a circle. In a 3D space, the infinite numbers of rectangle makes a sphere.

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