Explain the z-buffer algorithm for hidden surface removal with a suitable example.

The Z-buffer algorithm is a method used in computer graphics to determine which objects, or parts of objects, are visible and which are hidden behind other objects. It's a type of image-space method for hidden surface removal. The Z-buffer algorithm compares surface depths at each pixel position on a polygon.

Here's a step-by-step explanation of the Z-buffer algorithm:

1. Initialization: Start by initializing the Z-buffer and frame buffer so that for each pixel in the image, the depth is set to the maximum value (representing infinity) and the intensity is set to the background intensity.

2. Rendering: For each polygon in the scene, do the following steps:

   a. Projection: Project the polygon onto the view plane.

   b. Depth Calculation: For each pixel in the projected polygon, calculate the depth (z-value) of the polygon at this pixel.

   c. Comparison: Compare the depth of this polygon with the current depth stored in the Z-buffer at this pixel position. If the polygon's depth is less than the current depth, then update the Z-buffer with this new depth value, and update the frame buffer with the intensity of this polygon at this pixel. If the polygon's depth is greater than or equal to the current depth, then this pixel of the polygon is hidden and thus discarded.

3. Display: After all polygons have been processed, the frame buffer contains the intensities of the visible surfaces, and this image is displayed.

Example: Consider a simple 2D scene with two rectangles A and B, where B is partially covering A. When we apply the Z-buffer algorithm, we start with A. For each pixel of A, we calculate the depth and update the Z-buffer and frame buffer. Then we move to B. For each pixel of B, we calculate the depth and compare it with the depth in the Z-buffer. For the pixels where B is not covering A, the depth of B is greater than the depth in the Z-buffer, so these pixels are discarded. For the pixels where B is covering A, the depth of B is less than the depth in the Z-buffer, so we update the Z-buffer and frame buffer with the depth and intensity of B. The final image shows B partially covering A, which correctly represents the scene.