Changing Direction of Light

If the direction of light does not need to be changed, Legendre polynomials is sufficient for representing the

luminous intensity distribution of the light source. However, in interior lighting design, it is often

necessary to change the direction of light. To change direction, spherical harmonic functions are introduced.

Thus, when changing the direction of light, the luminous intensity distribution expressed by Eq. 2 can be

replaced by using spherical harmonic functions. We express an arbitrary direction viewed from the position

of the light source with the polar coordinates (è, ö), where è is an angle from the z axis and ö is an angle

from the x axis as shown in the Fig. 2(a). Assuming that the direction of light is (è light, ö light), and the

direction (è, ö) has angle ã from the direction of light (see Fig. 2(b)), the luminous intensity distribution

expressed by Eq. 2 is rewritten by the following equation.

where and r, respectively, are unit vectors toward the direction of light and the direction whose angle is

ã from the direction of light, and (•) is the dot product.

By the addition theorem of spherical harmonic functions, the following equation is introduced [Hobson 55].

Where is a spherical harmonic function (see Appendix).

From Esq. 4 and 7, the luminous intensity distribution whose direction of light is is

expressed by the following equation.

N

l

,

Where

Equations 9 and 10 imply that

when changing the direction of light, the resulting image is generated by the

following procedure. First, basis luminances corresponding to each spherical harmonic function are

calculated. When the luminous intensity distribution of the light source is specified, weights, wl, for each

Legendre polynomial are calculated by using Eq. 3. Next, when the direction of light is specified, then

weights, Alm, for each spherical harmonic function are calculated by using Eq. 10. The luminance due to

luminous intensity distribution is obtained by multiplying basis luminances by the corresponding weights

and summing them. Note that from Eq. 9, (N +1)2 terms of spherical harmonic functions are required to

change the direction of light expressed by using Legendre polynomials up to degree N.

The color of the light source can also be handled using the proposed method. First, assuming the spectral

distribution of the light source is uniform, basis luminances for each spectral component, such as RGB, are

pre-computed. Luminances for each spectral component at a certain point are obtained by employing the

proposed method for each spectral component. Then, the luminances for each spectral component are

multiplied by the corresponding component ratios of the colored light source.

Дата добавления: 2015-11-16