Given a set of n points (xi, yi) in 2D coordinate. Each point has some weight wi. The task is to check whether a line at 45 degree can be drawn so that sum of weights of points on each side are equal.
Input : x1 = -1, y1 = 1, w1 = 3 x2 = -2, y2 = 1, w2 = 1 x3 = 1, y3 = -1, w3 = 4 Output : Yes Input : x1 = 1, y1 = 1, w1 = 2 x2 = -1, y2 = 1, w2 = 1 x3 = 1, y3 = -1, w3 = 2 Output : No
First, let’s try to solve above problem for a vertical line i.e if a line x = i can divide the plane into two part such that the sum of weight at each side is equal.
Observe, multiple points with the same x-coordinate can be treated as one point with weight equal to the sum of weights of all points with the same x-coordinate.
Now, traverse through all x-coordinates from the minimum x-coordinate to maximum x-coordinate. So, make an array prefix_sum, which will store the sum of weights till the point x = i.
So, there can be two options for which the answer can be ‘Yes’:
- Either prefix_sum[1, 2, …, i-1] = prefix_sum[i+1, …, n]
- or there exist a point i such that a line passes somewhere in between
x = i and x = i+1 and prefix_sum[1, …, i] = prefix_sum[i+1, …, n],
where prefix_sum[i, …, j] is the sum of weight of points from i to j.
int is_possible = false; for (int i = 1; i < prefix_sum.size(); i++) if (prefix_sum[i] == total_sum - prefix_sum[i]) is_possible = true if (prefix_sum[i-1] == total_sum - prefix_sum[i]) is_possible = true
Now, to solve for a line at 45 degrees, we will rotate each point by 45 degrees.
Refer: 2D Transformation or Rotation of objects
So, point at (x, y), after 45 degree rotation will become ((x – y)/sqrt(2), (x + y)/sqrt(2)).
We can ignore the sqrt(2) since it is the scaling factor. Also, we don’t need to care about y-coordinate after rotation because a vertical line cannot distinguish between the point having the same x-coordinate. (x, y1) and (x, y2) will lie to the right, left or on any line of the form x = k.
- Minimum cuts required to divide the Circle into equal parts
- Count the number of ways to divide an array into three contiguous parts having equal sum
- Check if any square (with one colored cell) can be divided into two equal parts
- Shortest distance between a Line and a Point in a 3-D plane
- Find foot of perpendicular from a point in 2 D plane to a Line
- Program to check whether 4 points in a 3-D plane are Coplanar
- How to check if two given line segments intersect?
- Check if a line passes through the origin
- Check whether the point (x, y) lies on a given line
- Check if a line touches or intersects a circle
- Check if it is possible to draw a straight line with the given direction cosines
- Check if it possible to partition in k subarrays with equal sum
- Check if a large number can be divided into two or more segments of equal sum
- Check whether two strings can be made equal by increasing prefixes
- Count of parallelograms in a plane
If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to firstname.lastname@example.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.
Improved By : princiraj1992