Let us first review the simpler case of linear surfaces: Half space P = (a, b, c, d) contains point p = (x,y,z), if and only if ax + by + cz + d ≤ 0. This can also be expressed as the product:

a b c d

•

x y z 1

≤   0

or, in other words, L p^T ≤ 0. To find the plane that contains transformed points M•p^T, you need to replace L with L•M^-1.

Similarly, we can express quadratic equation

ax² + by² + cz² + dyz + ezx + fxy + gx + hy + iz + j ≤ 0

x y z 1

•

a f e g 0 b d h 0 0 c i 0 0 0 j

•

x y z 1

≤   0

or in other words, p Q p^T = 0, where Q denotes the 10 quadratic coefficients arranged into the above 4×4 matrix.

This means that if you want to find the three-variable quadratic equation that contains transformed points M•p^T, you need to replace Q by   (M^-1)^T Q M^-1   because that will give you:

(M p^T)^T     (M^-1)^T Q M^-1     (M p^T)     =

p^T M^T     (M^-1)^T Q M^-1     M p^T     =

p Q p^T

when you create the product of

(M^-1)^T P M^-1

x*x x*y x*z x*1
y*x y*y y*z y*1
z*x z*y z*z z*1
1*x 1*y 1*z 1*1

A E I M B F J N C G K O D H L P	→	A E+B I+C M+D 0 F J+G N+H 0 0 K O+L 0 0 0 P

Computing the surface normal:

Once you have computed your transformed values for [a,b,c,d,e,f,g,h,i,j], you can use high school differential calculus to compute its vector valued derivative:

f(x,y,z) = ax² + by² + cz² + dyz + ezx + fxy + gx + hy + iz + j

f'(x,y,z) = [  2ax + ez + fy + g  ,  2by + dz + fx + h  ,  2cz + dy + ex + i  ]

N = normalize(f'(x,y,z))

Light sources do not need to be at infinity. A local light source can add interesting chiarioscuro effects to your scene.

Unlike light sources at infinity, a local light source at some position (x,y,z) in your scene will more brightly illuminate surfaces that are near to it, and less brightly illuminate surface that are farther from it.

If a local light source were an infinitely small point, then intensity drop-off from that point would be 1/r², where r is the distance of the surface from (x,y,z). In reality, light sources, such as desk lamps, tend to have some non-zero extent, and so their intensity drops off with distance more slowly than 1/r².

A simple computational trick to model this slower dropoff is to define a power p, where 1.0 ≤ p ≤ 2.0, and define the dropoff function as 1/r^p.

Another visual trick you might use with local light sources is to give a very high ambient component to the surface material of one of your objects (eg: a sphere), and then place a local light source inside this object. The object will then appear to be a light source that illuminates the other objects in your scene.

Fog and atmosphere is very easy to implement and can make your ray traced scenes look a lot cooler.

In the presence of fog, objects gradually fade from view, as a function of their distance from your eye. Uniform fog (ie: fog with the same density everywhere) is quite easy to model and render. The key observation is that for any uniform fog there is some distance D that light will travel before half of the light has been scattered away by fog particles. The other half of the light continues along unobstructed.

Now consider what happens when the light which has made it through the fog travels further by another equal distance D. Half of the remaining light will now be scattered away, and half of the remaining light will make it through the second distance. In other words, the light that makes it through a distance 2D unobstructed is (1/2)². And in general, the amount of light that makes it through a distance kD is (1/2)^k.

To calculate uniform fog which scatters half of its light every multiple of D, you need to measure the distance d traveled by the ray (ie: the distance from the ray origin to the surface point that the ray hits). The proportion of this light that makes it through the fog unscattered will be t=(1/2)^d/D. The effect of the fog can be shown by mixing t(surfaceColor) + (1-t)(fogColor), where surfaceColor is the color that you would have calculated had there been no fog. Some uniform shade of light gray, such as (0.8,0.8,0.8), generally works well for the fog color.

In class, we briefly went over the general case of boolean modeling, but trying to implement the general case would be too much for one week.

So for now let's just talk about the simplest case (and a very useful one): intersection of convex shapes.

A convex shape is one that is not reentrant. That is, once a ray has entered the shape, the ray can then exit the shape at most once, and then never reenter that shape again. For example, a cube and a cylinder are convex, but a donut is not, since a single ray can enter and then exit the same donut twice.

Unlike the general case, the rule when raytracing for intersection of two convex shapes is very simple:

• Let A and B be two shapes.
  
• Let V+tW be a ray.
  
• Let the entry and exit roots of A be t_A0 and t_A1, respectively.
  
• Let the entry and exit roots of B be t_B0 and t_B1, respectively.

• The ray will enter A∩B at max(t_A0,t_B0)
  
• The ray will exit A∩ B at min(t_A1,t_B1)

Example

To ray trace to a unit cylinder (a tube with front and back end-caps):

• Start with the infinite unit tube:   x²+y²-1 ≤ 0
• Then intersect with half space:   x - 1 ≤ 0
• Then intersect with half space:   -x - 1 ≤ 0