1. **Problem Statement:** We need to prove the expression for the current pixel decision parameter in Bresenham's Line Drawing Algorithm: $$p_k = 2\Delta y x_k - 2\Delta x y_k + C$$ 2. **Background:** Bresenham's algorithm is used to draw a line on a pixel grid by deciding which pixel is closer to the theoretical line at each step. 3. **Line Equation:** The line between two points $(x_0, y_0)$ and $(x_1, y_1)$ can be written as: $$F(x,y) = \Delta y x - \Delta x y + C = 0$$ where: $$\Delta x = x_1 - x_0$$ $$\Delta y = y_1 - y_0$$ 4. **Constant $C$ Calculation:** Substitute point $(x_0, y_0)$ into $F(x,y)$: $$F(x_0,y_0) = \Delta y x_0 - \Delta x y_0 + C = 0$$ Solving for $C$: $$C = \Delta x y_0 - \Delta y x_0$$ 5. **Decision Parameter $p_k$:** At step $k$, the decision parameter is defined as: $$p_k = 2F(x_k + 1, y_k + 1/2)$$ This is scaled by 2 to avoid floating point operations. 6. **Substitute into $F$:** $$F(x_k + 1, y_k + 1/2) = \Delta y (x_k + 1) - \Delta x (y_k + 1/2) + C$$ Multiply by 2: $$p_k = 2\Delta y (x_k + 1) - 2\Delta x (y_k + 1/2) + 2C$$ 7. **Simplify $p_k$:** $$p_k = 2\Delta y x_k + 2\Delta y - 2\Delta x y_k - \Delta x + 2C$$ 8. **Rearranged form:** Grouping terms: $$p_k = 2\Delta y x_k - 2\Delta x y_k + (2\Delta y - \Delta x + 2C)$$ 9. **Conclusion:** The term $(2\Delta y - \Delta x + 2C)$ is a constant for the line, so we denote it as $C'$. Thus, $$p_k = 2\Delta y x_k - 2\Delta x y_k + C'$$ which proves the given expression for the current pixel decision parameter in Bresenham's algorithm.