Rotation Manipulation

suppose that :
,
then we have :

further, we can get the derivative :

Line Preservation

suppose that :
,
then we have :

further, we can get the derivative :

Shape Preservation

suppose that :
,
then we have :

further, we can get the derivative :

Boundary Preservation

further, we can get the derivative :

Optimization

Step 1 : Fix θ solve for V
Sparse Linear System :

V is the minimum point of total energy above, thus, let :

we can get the solution for V.

Step 2 : Fix V solve for θ
Part 1 : Fix Φ, update θ

suppose that :

we have :

then let :

by solving a sparse linear system, we can get the solution of θ :

Part 2 : Fix θ, update Φ

the solution of Φ can be easily approximately reached by enumeration based on iteration methods(increase β gradually).

The Leaning Tower of Pisa 5.5°

Palace Tower -6.1°

House Building -5.8°

The Oriental Pearl Tower 1.8°