∀𝑝𝑖 ∈ 𝑃, ∃𝑞𝑗 ∈ 𝑄 | 𝑇𝑝𝑖 − 𝑞𝑖 = 0

D 𝑃, 𝑄 = Ω

𝑇𝑝(𝑢, 𝑣) − 𝑞(𝑓 𝑢, 𝑣 , 𝑔(𝑢, 𝑣)) 2 𝑑𝑢𝑑𝑣 = 0

𝑝(𝑢, 𝑣) ∈ 𝑃, 𝑞(𝑢, 𝑣) ∈ 𝑄 𝑢, 𝑣 ∈ ℜ ×ℜ𝑓 𝑔

T =

𝑐𝛼𝑐𝛽𝑠𝛼𝑐𝛽−𝑠𝛽0

𝑐𝛼𝑠𝛽𝑠𝛾 − 𝑠𝛼𝑐𝛾𝑠𝛼𝑠𝛽𝑠𝛾 + 𝑐𝛼𝑐𝛾𝑐𝛽𝑠𝛾0

𝑐𝛼𝑠𝛽𝑐𝛾 + 𝑠𝛼𝑠𝛾𝑠𝛼𝑠𝛽𝑐𝛾 − 𝑐𝛼𝑠𝛾𝑐𝛽𝑐𝛾0

𝑡𝑥𝑡𝑦𝑡𝑧1

𝛼, 𝛽, 𝛾, 𝑡𝑥, 𝑡𝑦 , 𝑡𝑧

𝜕𝐷

𝜕𝛼= 0,

𝜕𝐷

𝜕𝑡𝑥= 0,

𝜕𝐷

𝜕𝛽= 0,

𝜕𝐷

𝜕𝑡𝑦= 0,

𝜕𝐷

𝜕𝛾= 0,

𝜕𝐷

𝜕𝑡𝑧= 0

𝑒 =

𝑖=1

𝑁

𝑇𝑝𝑖 − 𝑞𝑖2

𝑝𝑖 𝜖 𝑃 𝑞𝑖 𝜖 𝑄 𝑖 = 1…𝑁

𝑒 =

𝑖=1

𝑁

𝑇𝑝𝑖 − 𝑞𝑗2, with 𝑞𝑗 = |𝑞 min

𝑞𝜖𝑄𝑇𝑝𝑖 − 𝑞

𝑒𝑘 =

𝑖=1

𝑁

𝑇𝑘𝑝𝑖 − 𝑞𝑗𝑘 2 , with 𝑞𝑗

𝑘 = |𝑞 min𝑞𝜖𝑄𝑇𝑘−1𝑝𝑖 − 𝑞

𝑞𝑗𝑘

𝑒𝑘 =

𝑖=1

𝑁

𝑇𝑘𝑝𝑖 − 𝑞𝑗′𝑘 2 , with 𝑞𝑗

′𝑘 = 𝑇𝑘−1ℓ𝑖 ∩ 𝑄

where ℓ𝑖 = 𝑎| 𝑝𝑖 − 𝑎 × 𝑛𝑝𝑖 = 0

𝑇ℓ𝑖 ∩ 𝑄 ℓ𝑖 𝑄

𝑄

where 𝑆𝑗 is the tangent plane of 𝑄 at 𝑞𝑗

𝑒 =

𝑖=1

𝑁

𝑇𝑝𝑖 − 𝑞𝑗′ 2 , with 𝑞𝑗

′ = |𝑞 min𝑞𝜖𝑆𝑗𝑇𝑝𝑖 − 𝑞

𝑒 =

𝑖=1

𝑁

𝑇𝑝𝑖 − 𝑞𝑖2

𝑄

𝑒𝑘 =

𝑖=1

𝑁

𝑑𝑠2 𝑇𝑘𝑝𝑖 , 𝑆𝑗

𝑘

with 𝑆𝑗𝑘 = 𝑠| 𝑛𝑞𝑗

𝑘 ∙ 𝑞𝑗′𝑘 − 𝑠 = 0 , 𝑞𝑗

′𝑘 = 𝑇𝑘−1ℓ𝑖 ∩ 𝑄

𝑃 𝑄

𝒬

𝒫

𝑃 𝑄

𝒬

𝑇𝑘−1𝒫

𝑝𝑖 𝑑𝑠

𝑆𝑗𝑘

𝑛𝑖

𝑒𝑘 = 𝑖=1𝑁 𝑑𝑠

2 𝑇𝑘𝑝𝑖 , 𝑆𝑗𝑘

𝑝

𝒫

𝒬

𝑝 = (𝑥, 𝑦, 𝑃(𝑥, 𝑦))𝑃

- ℓ 𝑃 𝑝𝑝

𝑧

𝑥0 = 𝑥 𝑦0 = 𝑦

𝑝

𝒫

𝒬

𝑞0

𝑘𝑞𝑘 = 𝑥𝑘, 𝑦𝑘, 𝑧𝑘

ℓ𝑄 𝑞𝑘−1

𝒫

𝑞1

𝑞𝑘 − 𝑞𝑘−1 < 𝜖𝑑 𝜖𝑑 > 0

𝑒𝑘 = 𝑖=1𝑁 𝑑𝑠

2 𝑇 ∘ 𝑇𝑘−1𝑝𝑖 , 𝑆𝑗𝑘

𝛼, 𝛽, 𝛾 𝑇

where 𝑇 ∘ 𝑇𝑘−1 = 𝑇𝑘

𝑆𝑗𝑘 𝑄 𝑞𝑗

𝑘

𝑝𝑖 𝑇0 𝑇0𝑘

𝑝𝑖

𝑇𝑘−1 𝑝𝑖 𝑛𝑝𝑖 𝑝𝑖′ 𝑛𝑝𝑖

′

𝑞𝑖𝑘 𝑄

𝑆𝑖𝑘 𝑄 𝑞𝑖

′

𝑇 𝑒𝑘

𝑇𝑘 = 𝑇 ∘ 𝑇𝑘−1

𝛿 =𝑒𝑘−𝑒𝑘+1

𝑁′≤ 𝜖𝑒 , 𝜖𝑒 > 0

𝜖𝑒

𝑥 𝑢, 𝑣 , 𝑦 𝑢, 𝑣 𝑎𝑛𝑑 𝑧(𝑢, 𝑣)