Conformal transformation has been applied for the adjustment of aerotriangulation in the past and recently adopted in the official specification for large scale mapping called “The National Large Scale Mapping Project”, in Japan, because its favourable results has been recognized from our experience.
The theoretical investigation on the conformal transformation together with the study on the characteristics of the error in aerotriangulation, is dealts in this paper to make clear the relation betwen them.
The conditions under which we are carrying out aerotriangulation in Japan, are assumed for the basic ground of the discusssions; that is; (i) requirements of higher accuracy on aerotriangulation for large scale mapping up to 1: 1, 000, (ii) relatively short strips, normally less than 15 modells, (iii) relatively dense controll networks; one trig, point in every 2 or 4 km (iv) rather steep terrain relief, hight relief more than 25 % of flight hight is not seldom, even up to 40 % are experienced smaller scale.
We concentrate to the accidental errors only for the moment, which are assumed to be caused mostly from the modell connection, and these deformation of modells can be considered as equivalent to the connective transformation mathematically.
Also This transformation must fulfil the Cauchy's conditions (3) at the same time, and it is nothing else but the conformal transformation, which can be considered as the combinations of rotation, dilatation or scale change and parallel displacement when infinitesimal parcells are taken into consideration.
The following mathematical features of the conformal trans formation shall be recognized as the advantages ; that is; (i) similarity of infinitesimal parcells and possibility of eliminating the residual errors after adjustment by rotation and scale change model by model at the time of plotting, (ii) orthogonality, which makes the least square computations very simple as given at (1) for the first order and (11), (12), (13) for the second order transformations, (iii) higer freedom caused from the least number of parameters, which gives the higher accuracy and less the number of neccessarry control points. The practical use of the conformal transformation is shown at fig 4.
It is sometime pointed ont that the second order conformal tansformation has the term y
2 and it is meaningless because it can be neglected compared with other terms; but we consider as follows.
The second order conformal tranformation are characterized by two groups of hyperbolla which are orthgonal to each other. The eccentricity of hyperbolla is aways greater than 1, and eccentricity of straight line can be considered as ∞ The curves, which were originally straight lines and deformed by the errors, shall conserve the characteristics of straight lines as possible. And as far as we consider the eccentricity of curves, the hyperbola is more favorable than parabola which has the eccentricity just equal to 1.
Many investigations on the errors of aerotriangulation gives the results equal to the conformal transformation, thoug they reem quite different apparently. Some of them has heen derived from the discussion along the strip axis, or infinitesimal narrow strips. And we can get the conformal transformation by extending these results outside of the axis applying the analytical connection on theory of complex functions.
The above discussions are based on some physical and mathematical considerations. Another approach may be possible starting from the Tailor's series, but it seems to be difficult to give any physical or practical meaning nor to prove the possibility of expanding the accidental errors into normal Tailor's series.
抄録全体を表示
