In computational geometry, one common problem is to efficiently count the number of integer points (or lattice points) inside a triangle. This is not only a theoretical problem but also has practical applications in computer graphics, numerical simulations, and more. Here, we explore methods to achieve this task with optimal efficiency and accuracy.

Problem Definition

Given a triangle defined by three vertices with integer coordinates, the goal is to count the number of integer points that lie inside or on the boundary of the triangle. Let these vertices be $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$.

Exclusion-Inclusion Principle

An effective technique to solve this problem involves the use of Pick's Theorem and the Exclusion-Inclusion Principle. These methods utilize the geometry and discrete mathematics of the triangle.

Pick's Theorem

Pick's Theorem relates the area of the lattice polygon and its internal and boundary lattice points:

$$A = I + \frac{B}{2} - 1$$

Where: • $A$ is the area of the polygon. • $I$ is the number of interior lattice points. • $B$ is the number of boundary lattice points.

By rearranging, the number of interior lattice points $I$ can be found:

$$I = A + 1 - \frac{B}{2}$$

Boundary Points Calculation

The number of boundary points $B$ can be calculated using the GCD (Greatest Common Divisor) method for each edge of the triangle:

• For edge 1-2: $gcd(x_2 - x_1, y_2 - y_1)$ • For edge 2-3: $gcd(x_3 - x_2, y_3 - y_2)$ • For edge 3-1: $gcd(x_1 - x_3, y_1 - y_3)$

Adding these, but subtracting 3 for the vertices counted twice (once for each incident edge) gives us $B$.

Fast Computation Method

1. Area Calculation: Use the shoelace formula to get the area $A$: $$A = \frac{1}{2} \left| x\_1(y\_2 - y\_3) + x\_2(y\_3 - y\_1) + x\_3(y\_1 - y\_2) \right|$$
2. Interior and Boundary Points: Substitute the area $A$ and the computed boundary points $B$ into Pick's Theorem to find $I$ and consequently, the total number of lattice points: $I + B$.

Example

To illustrate, consider a triangle with vertices at $(0, 0)$, $(4, 0)$, and $(0, 3)$. Follow these steps:

1. Area Calculation: $$A = \frac{1}{2} \times \left|0 \times (0 - 3) + 4 \times (3 - 0) + 0 \times (0 - 0) \right| = 6$$
2. Boundary Points: • Edge 1-2: $gcd(4-0, 0-0) = 4$ • Edge 2-3: $gcd(0-4, 3-0) = 1$ • Edge 3-1: $gcd(0-0, 0-3) = 3$ • Total boundary points $B = 4 + 1 + 3 - 3 = 5$
3. Interior Points: • Using Pick's Theorem, $I = A + 1 - B/2 = 6 + 1 - 5/2 = 4$

Therefore, the total number of lattice points inside or on the boundary is $9$ (since $I + B = 4 + 5$).

Conclusion

Counting integer points inside a triangle efficiently involves leveraging Pick's Theorem along with systematic calculation of boundary points. The problem requires well-understood computational geometry techniques, such as the use of the GCD function for boundary determination and the application of the shoelace formula for area computation.

This approach encourages leveraging the power of classical mathematical tools in computational scenarios to increase efficiency and construct robust solutions.