Slides

C481 B581 Computer Graphics
Dana Vrajitoru

Hidden Surfaces

Determine which parts of the scene are seen by the viewer (camera) and which are not. Allows the user to create a mental 3D spatial representation of the 2D image.

For an object, determine the point of view and orientation

For several objects, determine their position with respect to each other:

Depth Comparison

If two points P₁(x₁, y₁, z₁) and P₂(x₂, y₂, z₂) result in the same projection, compare the distance between each of them and

the projection plane for the parallel projection
the projection center for the perspective projection.

If we project on the 0xy plane, we just have to compare z₁ and z₂: the point with smaller depth is visible.

Detection algorithms:

object-space methods
image-space methods.

General Idea

We consider that the user's eye is on the projection plane for the parallel projection.
For the perspective projection, we use the projection matrix to deform the scene first.
The projected image is then be obtained with a parallel projection.
Thus, we only need to solve the problem for the parallel projection.

Back-Face Detection

An object-space method.

Consider the normal to a surface N and the view vector V. If angle(V, N) < 90, then the face is a back face.

In the case where the projection is on the (0xy) plane, if the z component of the normal, Nz > 0, then the face is hidden.

Only works if the object is closed. Can be used to preprocess the scene.

Z-Buffer Algorithm

An image-space method.

For each pixel (x,y) in the projection plane, keep the value of the closest z value that has been drawn on it.

1. Initialize the projection area to a background color.
2. Initialize the z-buffer to a maximal number (back clipping plane).
3. Scan-convert one polygon (triangle) at a time. For each converted point, compare its z with the value stored for that pixel in the z-buffer. If the new value is smaller, draw the point.

A-Buffer Algorithm

Similar to the z-buffer method, only considers transparency.

For each pixel record all the scan-converted points, with depth and opacity parameter.

Compute the color of the pixel based on all this information.

color(P) = a2 * color(P2) + (1-a2) (a1*color(P1) + (1-a1) (a3*color(P3) + (1-a3)*background))
= 0.8 c2 + 0.2 (0.3 c1 + 0.7 (0.5 c3 + 0.5 bcgr)
= 0.8 c2 + 0.06 c1 + 0.065 c3 + 0.065 bcgr

Hidden Surfaces in OpenGL

Enable the z-buffer:
glEnable(GL_DEPTH_TEST);

Disable the z-buffer:
glDisable(GL_DEPTH_TEST);

Clear the z-buffer:
glClear(GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT);