The number of integer points inside a circle given the radius and centre of circle
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Welcome to our exploration of counting integer points inside a circle. Given a circle with center coordinates h comma k and radius r, we want to find how many points with integer coordinates lie strictly inside the circle. These points must satisfy the inequality: x minus h squared plus y minus k squared is less than r squared. Let's visualize this with an example circle centered at zero point five, zero point five with radius two point two. The green dots represent integer points that are strictly inside the circle.
Now let's understand the algorithm step by step. First, we determine the range of possible x coordinates by taking the floor of h minus r for the minimum, and ceiling of h plus r for the maximum. Then, for each integer x in this range, we calculate dx squared equals x minus h squared, and find the y limit squared equals r squared minus dx squared. Finally, we count the valid integer y coordinates within the range from k minus dy limit to k plus dy limit. Watch as we scan through each x value and identify the integer points inside the circle.
Let's work through a detailed calculation for x equals 1. First, we calculate dx squared equals 1 minus 0.5 squared, which equals 0.25. Then dy limit squared equals 2.2 squared minus 0.25, which equals 4.59. Taking the square root gives us dy limit equals 2.14. This means y values must be between negative 1.64 and positive 2.64. The integer y values in this range are negative 1, 0, 1, and 2. So for x equals 1, we have exactly 4 integer points inside the circle.
Now let's see the complete algorithm in action. We scan through each x value from negative 2 to 3. For x equals negative 2, we find 1 point. For x equals negative 1, we find 3 points. For x equals 0, we find 4 points. For x equals 1, we find 4 points. For x equals 2, we find 3 points. And for x equals 3, we find 1 point. Adding all these up: 1 plus 3 plus 4 plus 4 plus 3 plus 1 equals 16 total integer points inside the circle.
To summarize what we've learned: The row-by-row scanning method efficiently counts integer points inside any circle. We first calculate the x-coordinate range, then find valid y-coordinates for each x value. The key inequality x minus h squared plus y minus k squared less than r squared determines point inclusion. This algorithm works for any circle center and radius, with time complexity proportional to r squared for practical applications.