In the last module we looked at cylindrical coordinates -- a system of coordinates that is very useful when the important things about a three-dimensional point are its distance from the z-axis and its angle from the positive xz-plane. In this module we look at situations in which the important things about a point are its distance from the origin and, using terms from geography, its latitude and longitude. In this situation we use spherical coordinates.
This system of coordinates is very similar to the system -- longitude and latitude -- of coordinates used to describe points on the earth's surface. You may want to look at the module The Earth is Round -- Most Maps are Flat.
It is important to be aware of the units used with different coordinate systems. In Cartesian coordinates,
all three coordinates measure length and are in units of length. In spherical coordinates
only the first coordinate measures length and is in units of length. The other two coordinates measure angles and, thus, are dimensionless because angles measured in radians have no "units."
Converting Back-and-Forth from Spherical to Cartesian Coordinates
The pictures below are the key to converting from spherical to Cartesian coordinates and vice-versa.
Thus, we can convert from spherical to Cartesian coordinates by
We convert from Cartesian to spherical coordinates by
Warning: The notation for spherical coordinates is not standard. Sometimes they are written in a different order and sometimes the letters phi and theta are interchanged. In short, be careful when using spherical coordinates. You cannot assume that when someone else uses notation that looks like the notation here it is being used the same way.
Phong's Model -- Improved
We are now in a position to add another element of realism to our
model (Phong's Model) of specular lighting. Phong's model
says that the fraction of incident light that is reflected from a point on a surface
toward a viewer is
where phi is the angle between the line from the point to the viewer and the line from the point pointing in the direction of perfect mirrorlike reflection. The constant k depends on the intensity of the incident light and the physical make-up of the surface -- the fraction of the incident light that is reflected by the surface. When the constant n is very large the surface is very shiny and when the constant n is small it is more semi-glossy or even satin-like.
The graph below shows the functions
for n = 1,2, ... 10.
As n gets larger and larger, the functions tail off more
quickly for values of phi other than 0. This corresponds to
the fact that a shiny surface reflects most of the light very close to the
direction of perfect mirrorlike reflection. But this graph tells only half of
the story. All the functions in this graph have the same maximum value -- 1
at phi = 0 This agrees with our model so far because we have the same
constant k in Phong's formula
regardless of the value of n. This is unrealistic and it is this phenomenon that we can now address.
From now on we will write Phong's formula as
to indicate the fact that we need different constants for different values of n.
The reason we need different constants for different values of n is that when n is large most of the light is reflected very close the the direction of perfect mirrorlike reflection. Thus, the same total amount of outgoing light is more concentrated near the direction of perfect mirrorlike reflection.
We let T denote the total amount of light reflected in all directions from the given point. To make our computations simpler we will assume that T = 1 milliwatt. The actual value of the constant T depends on two factors -- the intensity of the incident light and the reflectivity or albedo of the surface (the fraction of incident light that is reflected).
The picture at the left shows a hemisphere of radius rho centimeters
centered at the point in question. The z-axis
in this picture is pointing in the direction of perfect mirrorlike reflection.
This hemisphere will capture all the light reflected by the point in question.
We can use this fact together with spherical coordinates to determine the values
of the constants kn.
Phong's model tells us the intensity of reflected light received by each point on this hemisphere
The factor 2 Pi rho2 in the denominator comes from the fact that the light intensity at a point is inversely proportional to the area of the hemisphere centered at the origin whose radius is the distance from the point to the origin.
since the total amount of reflected light is 1 milliwatt. But the intensity is not the same at each point on the hemisphere and simple multiplication must be replaced by integration.
Both sides of the equation above are measured in units like milliwatts. The left side is the total amount of light received by the hemisphere. The constant kn on the right side depends on the total amount of light that is incident on the point being studied and its albedo and is measured in milliwatts. The radius, rho, of the hemisphere is measured in units of length, and the symbol dA represents the area of a small section of the hemisphere and is measured in units of length2.
We can approximate this integral by breaking the hemisphere into small pieces as follows.
since the hemisphere has radius rho. But the widths are different. The sections near the north pole of the hemisphere are much thinner than those near its equator.
Notice that the units work out as expected. The only quantity in the formula above that has associated units is rho which is measured in units of length. Thus, as expected, area is measured in units of length2.
This leads to the estimates
The sums above are exactly the sums that would be used to estimate the integral
Since the total amount of light received by the hemisphere is 1 milliwatt we have
As usual it is worthwhile to take a close look at the units involved in these calculations. We are interested in an integral that can be written in various ways.
is used two different ways. In (1) it represents a small section of the hemisphere H and is measured in units of length2. In (2) it represents a section of the region P shown below and is dimensionless because the angles phi and theta are dimensionless.
|This constant is measured in units like milliwatts.|
|This is the intensity of the light measured at a point on the hemisphere. It is measured in units like milliwatts per length2.|
This is the area of a little section of the hemisphere and is measured in units
Notice that dA is a section of the region shown below
and is dimensionless.
|The integrand is measured in units of milliwatts and the final result is measured in the same units.|
to see the way that the glossiness of a surface affects the brightness of the highlights -- that is, to get a better feeling for Phong's model for specular lighting.