THIS appendix contains the Java 3D equations for fog, lighting, sound, and texture mapping. Many of the equations use the following symbols:

·	Multiplication
	Function operator for sound equations; Dot product for all other equations

E.1 Fog Equations

(Eq. E.1)

(Eq. E.2)

(Eq. E.3)

Fallbacks and Approximations

1. An implementation may approximate per-pixel fog by calculating the correct fogged color at each vertex and then linearly interpolating this color across the primitive.

2. An implementation may approximate exponential fog using linear fog by computing values of F and B that cause the resulting linear fog ramp to most closely match the effect of the specified exponential fog function.

3. An implementation will ideally perform the fog calculations in eye coordinates, which is an affine space. However, an implementation may approximate this by performing the fog calculations in a perspective space (such as device coordinates). As with other approximations, the implementation should match the specified function as closely as possible.

(Eq. E.4)

(Eq. E.5)

(Eq. E.6)

(Eq. E.7)

(Eq. E.8)

The parameters used in the lighting equation are

E	=	Eye vector
Ma	=	Material ambient color
Md	=	Material diffuse color
Me	=	Material emissive color
Ms	=	Material specular color
N	=	Vertex normal
shin	=	Material shininess

The per-light values are

di	=	Distance from vertex to light
Di	=	Spot light direction
expi	=	Spot light exponent
Kci	=	Constant attenuation
Kli	=	Linear attenuation
Kqi	=	Quadratic attenuation
Li	=	Direction from vertex to light
Lci	=	Light color
Si	=	Specular half-vector = || (Li + E) ||

Fallbacks and Approximations

1. An implementation may approximate the specular function using a different power function that produces a similar specular highlight. For example, the PHIGS+ lighting model specifies that the reflection vector (the light vector reflected about the vertex normal) is dotted with the eye vector and that this dot product is raised to the specular power. An implementation that uses such a model should map the shininess into an exponent that most closely matches the effect produced by the ideal equation.

2. Implementations that do not have a separate ambient and diffuse color may fall back to using an ambient intensity as a percentage of the diffuse color. This ambient intensity should be calculated using the following NTSC luminance equation:

I = 0.30 · Red + 0.59 · Green + 0.11 · Blue

(Eq. E.9)

E.3.1 Headphone Playback Equations

E.3.1.1 Interaural Time Difference (Delay)

1. The signal from the sound source to only one of the ears is direct. The ear farther from the sound is shadowed by the listener's head (); see Figure E-1.

(Eq. E.10)

where

Figure E-1 Signal to Only One Ear Is Direct

2. The signals from the sound source reach both ears by indirect paths around the head (); see Figure E-2.

(Eq. E.11)

where

The time from the sound source to the closer ear is , and the time from the sound source to the farther ear is , where S is the current AuralAttribute region's speed of sound.

If the sound is closer to the left ear, then

(Eq. E.12)

(Eq. E.13)

Figure E-2 Signals to Both Ears Are Indirect

The parameters used in the ITD equations are as follows:

	=	The smaller of the angles between Vh (or -Vh) and Va in radians
	=	Angle between Vh and radius to tangent point on Vt in radians
De	=	Distance between ears (interaural distance)
Dh	=	Distance from interaural center to sound source
Ec	=	Distance from sound source to ear closer to sound
Ef	=	Distance from sound source to ear farther from sound
P, P'	=	Arc path around the head that an indirect signal must travel to reach an ear
S	=	Speed of sound for the current AuralAttribute region
Va	=	Vector from center ear forward parallel to Z axis of head coordinates
Vc	=	Vector from sound source to ear closer to sound
Vh	=	Vector from center ear to sound source
Vt	=	Vector from sound source to tangent point on the listener's head

E.3.1.2 Interaural Intensity (Gain) Difference

(Eq. E.14)

(Eq. E.15)

(Eq. E.16)

Attenuation

For sound sources with a single distanceGain array defined, the intersection points of V_h (the vector from the sound source position through the listener's position) and the spheres (defined by the distanceGain array) are used to find the index k where d_k L d_k+1. See

For ConeSound sources with two distanceGain arrays defined, the intersection points of V_h and the ellipsi (defined by both the front and back distanceGain arrays) closest to the listener's position are used to determine the index k. See Figure E-4.

The equation for the distance gain is

(Eq. E.17)

Figure E-3 ConeSound with a Single Distance Gain Attenuation Array

Figure E-4 ConeSound with Two Distance Attenuation Arrays

Angular attenuation for both the spherical and elliptical cone sounds is identical. The angular distances in the attenuation array closest to are found and define the index k into the angular attenuation array elements. The equation for the angular gain is

(Eq. E.18)

(Eq. E.19)

(Eq. E.20)

Fallbacks and Approximations

1. If more than one lowpass filter is to be applied to the sound source (for example, both an angular filter and a distance filter are applied to a ConeSound source), it is necessary only to use a single filter, specifically the one that has the lowest cutoff frequency.

2. There is no requirement to support anything higher than very simple two-pole filtering. Any type of multipole lowpass filter can be used. If higher N-pole or compound filtering is available on the device on which sound rendering is being performed, use of these is encouraged, but not required.

E.3.1.3 Doppler Effect Equations

(Eq. E.21)

(Eq. E.22)

(Eq. E.23)

(Eq. E.24)

(Eq. E.25)

(Eq. E.26)

The feedback loop is repeated until AuralAttribute's reverberation feedback loop count is reached or Gr^j 0.000976 (effective zero amplitude, -60 dB, using the measure of -6 dB drop for every doubling of distance).

Fallbacks and Approximations

1. Reducing the number of feedback loops repeated while still maintaining the overall impression of the environment. For example, if -10 dB were used as the drop in gain for every doubling of distance, a scale factor of 0.015625 could be used as the effective zero amplitude, which can be reached in only 15 loop iterations (rather than the 25 needed to reach 0.000976).

2. Using preprogrammed "room" reverberation algorithms that allow selection of a fixed set of "reverberation types" (for example, large hall, small living room), which have implied reflection coefficients, delay times, and feedback loop durations.

E.3.2 Speaker Playback Equations

E.3.2.1 Monaural Speaker Output

E.3.2.2 Stereo Speaker Output

The equations for stereo speaker playback assume that the two speakers are placed symmetrically about the listener (at the same off-axis angle from the viewing axis at an equal distance from the center of the listener's head).

(Eq. E.27)

(Eq. E.28)

E.4 Texture Mapping Equations

E.4.1 Texture Lookup

(Eq. E.29)

(Eq. E.30)

(Eq. E.31)

If the texture boundary mode is CLAMP, then the s and t values are clamped to be in the range [0, 1] before being mapped into u and v values. Further, if s 1, then i is set to width - 1; if t 1, then j is set to height - 1.

The parameters in the point-sampled texture lookup equations are as follows:

width	=	Width, in pixels, of the texture image
height	=	Height, in pixels, of the texture image
s	=	Interpolated s coordinate at the pixel being textured
t	=	Interpolated t coordinate at the pixel being textured
u	=	u coordinate in texture image space
v	=	v coordinate in texture image space
i	=	Integer row address into texture image
j	=	Integer column address into texture image
T	=	Texture image

The above equations are used when the selected texture filter function-either the minification or the magnification filter function-is BASE_LEVEL_POINT. Java 3D selects the appropriate texture filter function based on whether the texture image is minified or magnified when it is applied to the polygon. If the texture is applied to the polygon such that more than one texel maps onto a single pixel, then the texture is said to be minified, and the minification filter function is selected. If the texture is applied to the polygon such that a single texel maps onto more than one pixel, then the texture is said to be magnified, and the magnification filter function is selected. The selected function is one of the following: BASE_LEVEL_POINT, BASE_LEVEL_LINEAR, MULTI_LEVEL_POINT, or MULTI_LEVEL_LINEAR. In the case of magnification, the filter will always be one of the two base level functions (BASE_LEVEL_POINT or BASE_LEVEL_LINEAR).

If the selected filter function is BASE_LEVEL_LINEAR, then a weighted average of the four texels that are closest to the sample point in the base level texture image is computed.

(Eq. E.32)

(Eq. E.33)

(Eq. E.34)

Mipmapping is the most common filtering technique for handling multiple levels of detail. If the implementation uses mipmapping, the equations for computing a texture color based on texture coordinates are simply those used by the underlying rendering API (such as OpenGL or PEX). Other filtering techniques are possible as well.

Fallbacks and Approximations

1. If the texture boundary mode is CLAMP, an implementation may use either the closest boundary pixel or the constant boundary color attribute for those values of s or t that are outside the range [0, 1].

2. An implementation can choose a technique other than mipmapping to perform the filtering of the texture image when the texture minification filter is MULTI_LEVEL_POINT or MULTI_LEVEL_LINEAR.

3. If mipmapping is chosen by an implementation as the method for filtering, it may approximate trilinear filtering with another filtering technique. For example, an OpenGL implementation may choose to use LINEAR_MIPMAP_NEAREST or NEAREST_MIPMAP_LINEAR in place of LINEAR_MIPMAP_LINEAR.

The equations for applying that color to the original pixel color are based on the texture mode, as follows:

REPLACE Texture Mode

(Eq. E.35)

(Eq. E.36)

(Eq. E.37)

BLEND Texture Mode

(Eq. E.38)

(Eq. E.39)

Note that C_rgb indicates the red, green, and blue channels of color C and that C indicates the alpha channel of color C. This convention applies to the other color variables as well.

If there is no alpha channel in the texture, a value of 1 is used for Ct in BLEND and DECAL modes.

When the texture mode is one of REPLACE, MODULATE, or BLEND, only certain of the red, green, blue, and alpha channels of the pixel color are modified, depending on the texture format, as described following:

• INTENSITY: All four channels of the pixel color are modified. The intensity value is used for each of Ct_r, Ct_g, Ct_b, and Ct in the texture application equations, and the alpha channel is treated as an ordinary color channel-the equation for C´rbg is also used for C´.

• LUMINANCE: Only the red, green, and blue channels of the pixel color are modified. The luminance value is used for each of Ct_r, Ct_g, and Ct_b in the texture application equations. The alpha channel of the pixel color is unmodified.

• ALPHA: Only the alpha channel of the pixel color is modified. The red, green, and blue channels are unmodified.

• LUMINANCE_ALPHA: All four channels of the pixel color are modified. The luminance value is used for each of Ct_r, Ct_g, and Ct_b in the texture application equations, and the alpha value is used for Ct.

• RGB: Only the red, green, and blue channels of the pixel color are modified. The alpha channel of the pixel color is unmodified.

• RGBA: All four channels of the pixel color are modified.