Pick's theorem says that if a polygon has I vertices/points inside it and B vertices or points on the border of it, then the AREA of this polygon is : AREA = I + ( B / 2 ) -1 There some conditions for applying this theorem. The shape must be a polygon. It can be concave or convex, but not self-intersecting. No circles apply here. Each vertex of the polygon must fall on the board. The polygon must be complete. No holes. Although, if it does have a hole, we can still compute the area by subtracting the smaller area from the larger area if the interior hole is also a lattice polygon. Pick's formula assumes unit measures and unit squares for the area. You can make your unit whatever you desire though if sides of your shape aren't whole numbers. However, you must adjust the value of your unit squares accordingly. So, if you made your lattice units 0.5, your interior squares would each be 0.5*0.5=0.25 square units each. A problem related to Picks Theorem is LightOJ ...
Using Euclid's algorithm A much more efficient method is the Euclidean algorithm , which uses a division algorithm such as long division in combination with the observation that the gcd of two numbers also divides their difference. To compute gcd(48,18), divide 48 by 18 to get a quotient of 2 and a remainder of 12. Then divide 18 by 12 to get a quotient of 1 and a remainder of 6. Then divide 12 by 6 to get a remainder of 0, which means that 6 is the gcd. Note that we ignored the quotient in each step except to notice when the remainder reached 0, signalling that we had arrived at the answer. Formally the algorithm can be described as: gcd ( a , 0 ) = a {\displaystyle \gcd(a,0)=a} gcd ( a , b ) = gcd ( b , a m ...
Geometry # Given two points p (x1, y1) and q (x2, y2), calculate the number of integral points lying on the line joining them. Solution 1. If the edge formed by joining p and q is parallel to the X-axis, then the number of integral points between the vertices is : abs( y2 - y1 ) - 1 2. Similarly if edge is parallel to the Y-axis, then the number of integral points in between is : abs( x2 - x1 ) - 1 3. Else, we can find the integral points between the vertices using below formula: GCD(abs( x2 - x1 ) , abs( y2 - y1 )) - 1
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