For the first issue, we can exploit our fourth component of our homogeneous coordinates to allow perspective transforms. Consider the matrix \(\mathbf{P}\) below

If we apply \(\mathbf{P}\) to the homogeneous point \((x_e,y_e,z_e,1)\), we get:

Note now we have a forth component \(w=-z_e\) of the transformed point which is neither 0 (a vector) nor 1 (a point). The openGL pipeline recognizes this situation a perspective transformation and will automatically apply perspective division, dividing all components of the point by \(w\) provided this value is not 0. Doing so yield the final transformed point of

Note that after division, we have a geometric point, but our points have been transformed such that \(x_e\) and \(y_e\) have been scaled by \(-1/z_e\), which is the non-linear part of what we first derived. The other components of the transform can be computed using linear means.

To convert to clip coordinates, we will map our projected points in the \(\hat{x}\) and \(\hat{y}\) directions to the \(-1,1\) range.

For example, we know that the clip coordinates in the \(\hat{x}\) direction change by two units in the same span that the projected units change by \(r-l\) units (right-left). So we can write the clip coordinate

for some coefficient \(\beta\) which we determine by using one of the know conversions, e.g., \(x_p=r \implies x_c=1\). A little bit of math later and we can determine:

Repeating the same exercise in the \(\hat{y}\) direction gives us the conversion there.

These expressions are still expressed in terms of the projected coordinates we computed earlier, not the original eye coordinates. Using \(x_p = \frac{-n x_e}{z_e}\) and \(y_p = \frac{-n y_e}{z_e}\) and getting everything over a common \(-z_e\) denominator results in:

Note that the inner terms are linear with respect to the eye coordinates. Combining with the projection portion of the matrix we derived early, we now have:

Note that to get the final clip coordinates derived above, OpenGL must apply the perspective division step after applying the matrix to the geometry.

For the final image display, the computation of the x and y positions are the most important, but for object occlusion, we cannot simply map all z positions to the near clipping plane. We must retain some depth information. The third row of our perspective matrix has two generic parameters \(A\) and \(B\) that we will now determine to help us retain depth information. If we apply the matrix \(\mathbf{M}\) to a point in eye space we get, after perspective division:

We haven’t talked too much about clip space or a closely related normalized device coordinates, but one slightly annoying feature is that the clipping volume is actually left handed, meaning the near clipping plane in the front maps to \(z=-1\), while the far clipping plane maps to \(z=+1\). You almost never need to worry about this, because there is not much that you actually do once the points have been converted to clip space. The only place where it becomes relevant is in constructing the projection matrices. If we map \(z_e=-n\) to \(z_c=-1\) and \(z_e=-f\) to \(z_c=1\) we can find:

For more details, see the Song Ho Ahn (안성호) Perspective Matrix page. This page also contains info on the much simpler Orthographic projection derivation.

To create realistic scenes, we must try to capture the interaction of light with surface materials. Consider a point \(p\) and a direction \(\vec{w_0}\) from \(p\) towards a viewer. The color or light intensity of observed at \(p\) in the \(\vec{w_0}\) is given by the rendering equation:

The rendering equation describes a conservation of light energy. If you knew all the light sources of a scene and how each of the light photons interacted with the materials, you could solve the rendering equation exactly and create an replica of a real scene. However, this is impractical in practice as we cannot trace the path of every photon computationally. One 800 lumen light bulb produces over \(10^{18}\) photons every second. Various approaches to estimating the rendering equation include radiosity, path tracing, and photon mapping. We will explore the concept of path tracing more during the midterm project, but this approach deviates a bit away from the pipeline approach we have been using so far. The basic idea is to trace paths from the viewer through the scene and see how the paths interact with surfaces and light sources and estimating \(L_o(p,\vec{w_0})\) by accumulating contributions along the path.

We will explore the simpler Blinn-Phong local illumination model first in the OpenGL pipeline before looking at more advanced global illumination models.

In this model we consider a simplified scene around a point \(p\), consisting of four unit vectors radiating out from \(p\).

Note that \(\hat{v}\) is \(\vec{w_0}\) in the rendering equation and \(\hat{l}\) is \(\vec{w_i}\) for some \(\vec{w_i}\).

The Blinn-Phong (James F. Blinn 1977, Bui Tuong Phong 1975) model approximates the scene interaction with light by describing three components to the illumination of a point:

The Blinn-Phong model evaluates each of these contributions separately and computes the final color as the sum of these contributions.

For each light in the scene, we will define diffuse and specular intensities, \(i_d\) and \(i_s\). These can be simply floating point values, or a three component vector describing separate intensities for the red, green, and blue channels. Since ambient light is not from a particular source, there is only one ambient light intensity \(i_a\) for the entire scene.