1. The Phong Illumination Model
𝐼= 𝑘 𝑎 𝐿 𝑎 + 𝑘 𝑑 𝐿 𝑑 𝑁 ∙ 𝐿 + 𝑘 𝑠 𝐿 𝑠 𝑉 ∙ 𝑅 𝑛
Problems: does not describe
Reflection
Refraction
Shadow
Generalize 𝐼 to describe the above 𝑹 𝑵 𝑳

2. Reflective/Transparency
Medium 1
Medium 2
θ 𝑖
θ 𝑜 𝑵
𝑽 𝒕 𝑽
𝑽 𝒓
P 𝑡 P
𝑽 : viewing direction
𝑽 𝒓 : mirror reflection direction
𝑽 𝒕 : transmission direction
𝑃: visible position
P 𝑟 : visible position along 𝑽 𝒓
P 𝑡 : visible position along 𝑽 𝒕
𝑵 : Normal vector at visible position P
ν: Reflectivity at visible position P
Material property: value between 0 to 1
τ: Transparency at visible position P
Note: for typical materials:
ν + τ < 1

3. Support for Reflection and Refraction
𝐼=𝐼 𝑃 × 1−τ−ν +𝐼 𝑃 𝑟 ×ν+𝐼 𝑃 𝑡 ×τ
𝐼 𝑃 - Phong model evaluated at position 𝑃
𝐼 𝑃 𝑟 - Phong model evaluated at Position 𝑃 𝑟
𝐼 𝑃 𝑡 - Phong model evaluated at Position 𝑃 𝑡
ν: Reflectivity at position 𝑃
τ : Transparency at position 𝑃
P 𝑟
θ 𝑖
θ 𝑜 𝑵
𝑽 𝒕 𝑽
𝑽 𝒓
P 𝑡 P

4. 𝑉 𝑡 : Refraction Direction
η: refraction index
When traveling from medium 1 to 2
η= sin θ 𝑖 sin θ 𝑜
𝑉 𝑡 =−[ 𝑉 𝜂 +( cos 𝜃 𝑜 − cos 𝜃 𝑖 𝜂 ) 𝑁
Medium 1
Medium 2
θ 𝑖
θ 𝑜 𝑵
𝑽 𝒕 𝑽

5. Reflection Direction 𝑽 𝒓 =2 cos 𝜃 𝑖 𝑁 − 𝑉 Or 𝑽 𝒓 =2( 𝑁 ∙ 𝑉 ) 𝑁 − 𝑉 𝑵 𝑽
𝑽 𝒓 =2( 𝑁 ∙ 𝑉 ) 𝑁 − 𝑉
θ 𝑖 𝑵 𝑽
𝑽 𝒓