Given a rectangle of height H and width W which has the bottom left corner at (0, 0). The task is to count the number of distinct Rhombi that have all points inside or on the border of the rectangle satisfying the following conditions exists:
- Have non-zero area.
- Have diagonals parallel to the x and y axes.
- Have integer coordinates.
Input: H = 2, W = 2
There is only one rhombus possible with coordinates (0, 1), (1, 0), (2, 1) and (1, 2).
Input: H = 4, W = 4
Approach: Since the diagonals are parallel to the axis, let’s try fixing the diagonals and creating rhombi on them. For the rhombus to have integer coordinates, the length of the diagonals must be even. Let’s fix the length of the diagonals to i and j, the number of rhombi we can form with these diagonal lengths inside the rectangle would be (H – i + 1) * (W – j + 1). Thus, we iterate over all possible values of i and j and update the count.
Below is the implementation of the above approach:
Time Complexity: O(H * W)
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- Check if a point lies on or inside a rectangle | Set-2
- Check whether a given point lies on or inside the rectangle | Set 3
- Check whether a given point lies inside a rectangle or not
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- Number of squares of side length required to cover an N*M rectangle
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