Objectives  Gauss-Markov model, the model with condition equations and Gauss-Helmert model are three classical adjustment models. Gauss-Helmert model can be seen as the combined form of the other two models. They are usually distinguished by the number of unknown parameters. However, this classification method cannot reflect the essential relationship between them. In addition, the connections between Gauss-Markov model and the conditional model has been intensively investigated, while that of Gauss-Helmert model has been ignored.

  Methods  Some equivalent objective functions of the adjustment of the least-square in the context of Gauss-Markov model and the conditional model are proposed based on the projection theory and the dual optimization technique. Based on the theory of matrix analysis, this paper realizes the conversion between Gauss-Markov model and Gauss-Helmert model according to partial orthogonality of coefficient matrices, thus completing the structure of these models and revealing the mathematical relationships.

  Results  The least-square adjustment within Gauss-Markov model and the model with condition equations are explained geometrically by employing the linear space and projector. The equivalence of the objective functions of these two models are also expounded. Combining the Pythagorean theorem and duality theory, we extend them to four equivalent objective functions and illustrate the construction principle. The transformation between Gauss-Markov model and Gauss-Helmert model are achieved by the so-called partial-orthogonality.

  Conclusions  The geometric reasoning of the least-squares adjustment is analyzed once more. Two objection functions are newly proposed, and the geometric relation and connotation between objective functions are revealed. Gauss-Helmert model is not only the combined model in terms of the model appearance, these three adjustment models can actually be mutually transformed. As a result, the understanding of the mathematical nature of adjustment models and the corresponding estimation in this paper has theoretical and practical value.