M jrs M jgs M jbs Intensity of light reflected from surface j when illuminated by source i: I ijr

Transcription

1 Lighting Basics: ADS Model The ADS (ambient, diffuse, specular) model simulates shading in terms of these three components Light source i is modeled as emitting L ira L iga L iba L i = L ird L igd L ibd L irs L igs L ibs where we have three components for each of the primary colors Material j is modeled as reflecting these three components: M jra M jga M jba M j = M jrd M jgd M jbd M jrs M jgs M jbs Intensity of light reflected from surface j when illuminated by source i: I ijr = M jra L ira + M jrd L ird + M jrs L irs = I ijra + I ijrd + I ijrs Intensity of red light reflected from surface j when all sources considered: I jr = n i=1 (I ijra + I ijrd + I ijrs ) + I gr In simpler terms: I = L a M a + L d M d + L M s + I g = I a + I d + I s + I g where I g is the global ambient light Lighting is characterized by the vectors in the following diagram: 1

2 Lighting Basics: ADS Model - Calculations 1. Ambient Ambient reflection represented by factor k a 0 k a 1 Amount of ambient reflected light calculated as M a = k a 2. Diffuse Diffuse reflection represented by factor k d 0 k d 1 Lambert s Law: M d cosθ Amount of diffuse reflected light calculated as M d = k d cosθ I d = k d cosθ L d To perform lighting calculations, want to represent everything in terms of what is known: Locations of vertex, light source, eye Vectors from vertex to light source, vector from vertex to eye, vector normal to vertex 2

3 The reflection vector: Lighting Basics: ADS Model - Calculations (2) Given a light source and a vertex, want to compute the vector representing the reflected light in terms of the normal and the light source vector The vector from a to b is ˆncosθ (The projection of ˆl onto ˆn is ˆl ˆn = ˆl ˆn cosθ since ˆn is a unit vector Since ˆl = ˆn = 1, the length of this projection is cosθ Hence the vector from a to b is ˆncosθ ) s can be represented in terms of ˆl and ˆncosθ: s = ˆncosθ ˆl r can be similarly represented in terms of s and ˆncosθ: r = ˆncosθ + s = ˆncosθ + ˆncosθ ˆl = 2ˆncosθ ˆl Since cosθ ˆl ˆn: r = 2ˆn(ˆl ˆn) ˆl Given the above representation of cosθ in terms of ˆl and ˆn, we can represent I d as k d ˆl ˆn L d 3

4 Lighting Basics: ADS Model - Calculations (3) 3. Phong specular reflection Specular reflection represented by factor k s 0 k s 1 The relevant vectors: Amount of specular reflected light calculated as R s = k s cos α φ α is shininess coefficient α = mirror I s = k s cos α φl s As with reflection, we can replace cosφ with ˆn ˆv, resulting in I s = k s (ˆn ˆv) α L s 4

5 4. The halfway vector Lighting Basics: ADS Model - Calculations (4) The halfway vector is the vector that bisects the angle between ˆl and ˆv ĥ = ˆl + ˆv ˆl + ˆv ˆn ĥ is often used instead of ˆv ˆr in specular calculations It is sometimes referred to as the direction of maximum highlights This is because that - if the surface were rotated so that ˆn were to align with ˆv - the viewer would see the brightest specular highlight The advantage to using ĥ instead of ˆr is that computing r relies on n, which is variable for a non-planar surface, while if both the viewer and light source are at infinity, ĥ is constant The specular term in the Phong model would become k s (ˆn ĥ)α L s Note that while ψ and φ are not the same and thus the two formulations will give different results, neither is based on physical analysis, but rather on empirical observation 5. Intensity inversely proportional to distance This applies to both diffuse and specular light Distance factor: 1 a + bd + cd 2 5

6 6. Complete Phong lighting model: Lighting Basics: ADS Model - Calculations (5) I = OR (using the halfway vector) I = 1 a + bd + cd 2(k d(ˆn ˆl)L d + k s (ˆn ˆv) α L s ) + k a L a 1 a + bd + cd 2(k d(ˆn ˆl)L d + k s (ˆn ĥ)α L s ) + k a L a 6

7 Lighting Basics: Refraction As with reflection, we want to represent the refracted vector in terms of known values: ˆn, ˆl, θ, and φ The projection of ˆl onto ˆm is ˆncosθ ˆl The projection of ˆl onto ˆm is ˆmsinθ Solving for ˆm in terms of the above: The projection of ˆt onto ˆ n is ˆm = ˆ ncosφ The projection of ˆt onto ˆm is ˆmsinφ ˆncosθ ˆl sinθ ˆt can be represented in terms of the above two projections: (1) ˆt = ˆmsinφ ˆ ncosφ (2) 7

8 Substituting 1 in 2 for ˆm: Lighting Basics: Refraction (2) ˆt = sinφ sinθ (ˆncosθ ˆl) ˆ ncosφ (3) Snell s Law states that sinφ sinθ = η l η t where η t and η l are the indices of refraction of the respective media across which the light is transitioning Using Snell s Law and letting η = η l η t, equation 3 can be rewritten as ˆt = (ηcosθ cosφ)ˆn ˆlη (4) Using Snell s Law, we can represent cos φ in terms of η and cosθ: cosφ = 1 sin 2 φ = 1 η 2 sin 2 θ = 1 η 2 (1 cos 2 θ) (5) Substituting ˆn ˆl for cosθ in equation 5 gives cosφ = 1 η 2 (1 (ˆn ˆl) 2 ) (6) Finally, substituting ˆn ˆl for cosθ and equation 6 for cosφ in equation 4 gives ( ) ˆt = η(ˆn ˆl) ˆn 1 η 2 (1 (ˆn ˆl) 2 ) ˆlη (7) 8