Rectangle Geometry
Authors: Darren Yao, Michael Cao, Benjamin Qi, Ben Dodge
Contributors: Allen Li, Andrew Wang, Dong Liu, Ryan Chou
Problems involving rectangles whose sides are parallel to the coordinate axes.
Prerequisites
Resources | ||||
---|---|---|---|---|
IUSACO | module is based off this |
Most problems in this category include only two or three squares or rectangles, in which case you can simply draw out cases on paper. This should logically lead to a solution.
Example - Fence Painting
Focus Problem – try your best to solve this problem before continuing!
Slow Solution
Time Complexity:
Solution
Fast Solution
Time Complexity:
Solution
Example - Blocked Billboard
Think of this as the 2D analog of the previous example.
Focus Problem – try your best to solve this problem before continuing!
Slow Solution
Time Complexity:
Solution
Fast Solution
Time Complexity:
Solution
Common Formulas
Certain tasks show up often in rectangle geometry problems. For example, many problems involve finding the overlapping area of two or more rectangles based on their coordinate points, or determining whether two rectangles intersect. Here, we'll discuss these formulas.
Note that these formulas only apply to rectangles which have sides parallel to the coordinate axes.
Coordinates
A rectangle can be represented with points: its top right corner and bottom left corner. We'll label these points (top right) and (bottom left).
In this module, we'll assume that increasing moves to the right and increasing moves up.
Finding area
The formula for finding the area of an individual rectangle is .
is the length of the vertical sides, and is the length of the horizontal sides.
Implementation
C++
long long area(int bl_x, int bl_y, int tr_x, int tr_y) {long long length = tr_y - bl_y;long long width = tr_x - bl_x;return length * width;}
Java
int area(int bl_x, int bl_y, int tr_x, int tr_y) {int length = tr_y - bl_y;int width = tr_x - bl_x;return length * width;}
Python
def area(bl_x: int, bl_y: int, tr_x: int, tr_y: int) -> int:length = tr_y - bl_ywidth = tr_x - bl_xreturn length * width
Checking if two rectangles intersect
Given two rectangles and , there are only two cases where they do not intersect:
- or .
- or .
In all other cases, the rectangles intersect.
Implementation
C++
bool intersect(vector<int> s1, vector<int> s2) {int bl_a_x = s1[0], bl_a_y = s1[1], tr_a_x = s1[2], tr_a_y = s1[3];int bl_b_x = s2[0], bl_b_y = s2[1], tr_b_x = s2[2], tr_b_y = s2[3];// no overlapif (bl_a_x >= tr_b_x || tr_a_x <= bl_b_x || bl_a_y >= tr_b_y ||tr_a_y <= bl_b_y) {return false;} else {return true;}}
Java
boolean intersect(int[] s1, int[] s2) {int bl_a_x = s1[0], bl_a_y = s1[1], tr_a_x = s1[2], tr_a_y = s1[3];int bl_b_x = s2[0], bl_b_y = s2[1], tr_b_x = s2[2], tr_b_y = s2[3];// no overlapif (bl_a_x >= tr_b_x || tr_a_x <= bl_b_x || bl_a_y >= tr_b_y ||tr_a_y <= bl_b_y) {return false;} else {return true;}}
Python
def intersect(s1, s2) -> bool:bl_a_x, bl_a_y, tr_a_x, tr_a_y = s1[0], s1[1], s1[2], s1[3]bl_b_x, bl_b_y, tr_b_x, tr_b_y = s2[0], s2[1], s2[2], s2[3]# no overlapif bl_a_x >= tr_b_x or tr_a_x <= bl_b_x or bl_a_y >= tr_b_y or tr_a_y <= bl_b_y:return Falseelse:return True
Finding area of intersection
We'll assume that the shape formed by the intersection of two rectangles is itself a rectangle.
First, we'll find this rectangle's length and width. . .
If either of these values are negative, the rectangles do not intersect. If they are zero, the rectangles intersect at a single point. Multiply the length and width to find the overlapping area.
Implementation
C++
int inter_area(vector<int> s1, vector<int> s2) {int bl_a_x = s1[0], bl_a_y = s1[1], tr_a_x = s1[2], tr_a_y = s1[3];int bl_b_x = s2[0], bl_b_y = s2[1], tr_b_x = s2[2], tr_b_y = s2[3];return ((min(tr_a_x, tr_b_x) - max(bl_a_x, bl_b_x)) *(min(tr_a_y, tr_b_y) - max(bl_a_y, bl_b_y)));}
Python
def inter_area(s1, s2) -> int:bl_a_x, bl_a_y, tr_a_x, tr_a_y = s1[0], s1[1], s1[2], s1[3]bl_b_x, bl_b_y, tr_b_x, tr_b_y = s2[0], s2[1], s2[2], s2[3]return (min(tr_a_x, tr_b_x) - max(bl_a_x, bl_b_x)) * (min(tr_a_y, tr_b_y) - max(bl_a_y, bl_b_y))
Java
int interArea(int[] s1, int[] s2) {int bl_a_x = s1[0], bl_a_y = s1[1], tr_a_x = s1[2], tr_a_y = s1[3];int bl_b_x = s2[0], bl_b_y = s2[1], tr_b_x = s2[2], tr_b_y = s2[3];return ((Math.min(tr_a_x, tr_b_x) - Math.max(bl_a_x, bl_b_x)) *(Math.min(tr_a_y, tr_b_y) - Math.max(bl_a_y, bl_b_y)));}
Problems
Status | Source | Problem Name | Difficulty | Tags | |
---|---|---|---|---|---|
Bronze | Very Easy | Show TagsRectangle | |||
Bronze | Easy | Show TagsRectangle | |||
CF | Normal | Show TagsRectangle | |||
CF | Normal | Show TagsRectangle |
Module Progress:
Join the USACO Forum!
Stuck on a problem, or don't understand a module? Join the USACO Forum and get help from other competitive programmers!