bundle block adjustment with 3d natural cubic splines

A.2 Derivation of arc-length parameterizationThe partial derivatives of symbolic representation of the arc-length parameterizationare introduced as follows.Du(t) = 2x ′ p(t)f w′w 2Du ′ (t) = −2x ′ p(t) f wDv(t) = 2y ′ p(t)f w′w 2Dv ′ (t) = −2y p(t) ′ f w dv′{Dw(t) = 2x ′ p(t) u′ w 2 − (u ′ w − uw ′ )2w− 2ywp(t) ′ v′ w 2 − (v ′ w − vw ′ })2w4 w 4Dw ′ (t) ={2x ′ p(t) u w + 2 2y′ p(t) v }w 2A 1 = t [2 − t 1 16 2 f(t 1) − 1 2 {−Du(t1 )r 11 − Dv(t 1 )r 21 − Dw(t 1 )r 31 }( ) t1 + t −1 {2 2+2f−Du( t 1 + t 2)r 11 − Dv( t 1 + t 2)r 21 − Dw( t 1 + t 22222+ 1 ]2 f(t 2) − 1 2 {−Du(t2 )r 11 − Dv(t 2 )r 21 − Dw(t 2 )r 31 }A 2 = t [2 − t 1 16 2 f(t 1) − 1 2 {−Du(t1 )r 12 − Dv(t 1 )r 22 − Dw(t 1 )r 32 }( ) t1 + t −1 {2 2+2f−Du( t 1 + t 2)r 12 − Dv( t 1 + t 2)r 22 − Dw( t 1 + t 22222+ 1 ]2 f(t 2) − 1 2 {−Du(t2 )r 12 − Dv(t 2 )r 22 − Dw(t 2 )r 32 }A 3 = t [2 − t 1 16 2 f(t 1) − 1 2 {−Du(t1 )r 13 − Dv(t 1 )r 23 − Dw(t 1 )r 33 }( ) t1 + t −1 {2 2+2f−Du( t 1 + t 2)r 13 − Dv( t 1 + t 2)r 23 − Dw( t 1 + t 22222+ 1 ]2 f(t 2) − 1 2 {−Du(t2 )r 13 − Dv(t 2 )r 23 − Dw(t 2 )r 33 }A 4 = t [2 − t 1 16 2 f(t 1) − 1 2 {Du(t1 ){r 12 (Z i (t 1 ) − Z C ) − r 13 (Y i (t 1 ) − Y C )}103})r 31 }})r 32 }})r 33 }