After you've experimented with the JAVA applet below you may want to go directly to one of the following sections of this module.
The Java applet below has the same sphere as the previous applet but now we have three different colored light sources. Experiment with this applet and describe the differences between this sphere and the sphere in the first two applets.
You may also want to go back to the three previous applets.
It might be particularly illuminating to compare the applet above with the second applet with the light source controls set as shown below.
All of the applets we used showed a somewhat unusual situation -- the sphere is in a dark room except for one white light source or three colored light sources. All of the light sources are bare bulbs. In practice, rooms usually have many light sources including light bouncing off the walls, the floor, and the ceiling. The first and third applet show a matte sphere with a rough surface. When light hits a rough surface it bounces in various directions. The applet above and the second applet show a more glossy but not perfectly mirror-like surface. When light hits a glossy surface it is more likely to bounce like a billiard ball. In this module we discuss the geometry of both matte and glossy surfaces. Visual artists often use the term specular lighting when they are dealing with glossy or semiglossy surfaces. We use a mathematical model called Lambertian reflection for matte surfaces and a mathematical model called Phong reflection for glossy and semiglossy surfaces.
The basic tool we need to work with the shading of a surface is normal vectors. At each point on a surface the normal vectors are the two unit vectors pointing straight out from the surface -- that is, perpendicular to the surface. For the sphere of radius R,
the outward pointing normal vector at a point (x, y, z) is the vector
The shading of a point on a surface -- its apparent brightness -- depends on two things
A little geometry shows that the amount of light received by a surface per square inch is proportional to cos theta where theta is the angle between the normal vector to the surface at the point being illuminated and the incoming light. What happens if cos theta is negative? Why?
|
In the Lambertian model for shading -- a matte surface illuminated by a
distant light source -- this is the only factor that changes the shading from one
point to another.
The book Multivariable Calculus, Linear Algebra, and Differential Equations in a Real and Complex World has a short discussion on Lambertian Reflection that is not repeated here. The sphere at the right is a matte sphere with the illumination coming from directly behind the viewer. The shading was done using Lambertian reflection and the picture was drawn by the first applet in this module. |
|
Look at your Mathematica or Maple window. It contains a "program" that draws a sphere shaded using Lambertian reflection with a single white light source directly behind the viewer. Look at this program and see how it works. Then modify it to show the sphere with the light source in several other positions. Then modify it to show the sphere illuminated by three light sources of different colors in different positions. Try to produce an interesting and attractive picture. One of my favorite works of art is a simple sphere illuminated in this way in an otherwise dark room.
If light hits a perfect mirror then it bounces so that the angle of coincidence is equal to the angle of incidence. Even though this is a three dimensional situation, a two dimensional picture captures eveything because all the action occurs in the plane determined by the incoming light ray and the normal vector to the surface at the illuminated point.
In a perfect mirror, the viewer sees a bright glare -- the reflection of the light source -- if the direction of the outgoing light ray is the same as the direction of the viewer.
In less than perfect specular reflection, the amount of light seen by the viewer looking at the point P coming from the light source S depends on the angle alpha between the outgoing light ray and the direction of the viewer. One commonly used model is called Phong's model and says that the light perceived by the viewer is proportional to
where n is a constant. Different values of the constant n model different kinds of surfaces -- surfaces that are more or less mirrorlike.
Suppose the sphere we've been looking at is floating above the plane