This document presents an optimization problem involving minimizing an energy function with respect to parameters p and q. It introduces notation for points P and Q, defines the energy function E in terms of distances between points, and describes taking its derivative and Hessian to solve for the minimum using Newton's method. The solution involves solving the linear system (A + λI)δx = a, where A is the Hessian and a is the negative gradient.

4. • 𝑖 𝑃𝑖 𝑖 = 1, … , 𝑚
𝑃𝑖 ≡ 𝐾𝑖 𝑅𝑖 𝒕𝑖 =
𝑓𝑢
0
0
𝛾
𝑓𝑣
0
𝑢0
𝑣0
1
𝑟11
𝑟21
𝑟31
𝑟12
𝑟22
𝑟23
𝑟13
𝑟23
𝑟33
𝑡1
𝑡2
𝑡3
𝐾𝑖 𝑅𝑖 𝒕𝑖
• 𝑗 𝒒 𝑗 j = 1, … , 𝑛
𝒒 𝑗 = 𝑋𝑗, 𝑌𝑗, 𝑍𝑗
T
𝒛𝑖𝑗 = 𝑢𝑖𝑗, 𝑣𝑖𝑗
T
𝒛𝑖𝑗
1
= 𝜆𝑃𝑖
𝒒𝑖
1

5. 𝑅𝑖, 𝒕𝑖 𝒑𝑖
𝒑1, … , 𝒑 𝑚 𝒒1, … 𝒒 𝑛
𝒑1, … , 𝒑 𝑚, 𝒒1, … 𝒒 𝑛
𝐸 𝒑1, … , 𝒑 𝑚, 𝒒1, … 𝒒 𝑛 =
1
2
෍
𝑖
෍
𝑗
𝑢𝑖𝑗 − 𝑢 𝒑𝑖, 𝒒 𝑗
2
+ 𝑣𝑖𝑗 − 𝑣 𝒑𝑖, 𝒒 𝑗
2

6. •
𝛿𝒙
𝐸(𝒙 + 𝛿𝒙)
𝐸 𝒙 + 𝛿𝒙 ≈ 𝐸 𝒙 +
𝑑𝐸 𝒙
𝑑𝒙
⋅ 𝛿𝒙 +
1
2
𝛿𝒙 ⋅
𝑑2 𝐸
𝑑𝒙2 𝛿𝒙
𝒈 ≡
𝑑𝐸 𝒙
𝑑𝒙
𝐻 ≡
𝑑2 𝐸
𝑑𝒙2 𝛿𝒙 𝐻𝛿𝒙 = −𝒈

7. •
𝐸 𝒙 =
1
2
𝒆 𝒙 2 𝐻
𝐻 ≈ 𝐴 ≡ 𝐽T
𝐽 𝐽 𝒆 𝒙
𝑎 ≡ −𝒈 = −𝐽 𝑇
𝒆(𝒙)
𝐴𝛿𝒙 = 𝒂
𝛿𝒙

8. •
𝐴 + 𝜆𝐼 𝛿𝒙 = 𝒂
𝜆

9. •