A. Roncat:

"Visualizing Normal Equations in Least-Squares Adjustment";

Talk: 16th International Conference on Geometry and Graphics (ICGG 2014), Innsbruck, Austria; 2014-08-04 - 2014-08-08; in: "Proceedings of the 16th International Conference on Geometry and Graphics", H. Schröcker, M. Husty (ed.); Innsbruck University Press, Innsbruck (2014), ISBN: 978-3-902936-46-2.

The adjustment by least squares dates back to more than 200 years-commonly attributed to C.F. Gauss-and is widely applied in practically all disciplines where linear and non-linear regression is sought. These disciplines include natural sciences, engineering and also social sciences. German geodesist W. Niemeier entitled the adjustment by least squares as a brittle beauty ("spröde Schöne" in the original German text), highlighting both the possible rejection of a scholar at first sight and the attraction for a scientist dealing closer with this topic.

Besides its stochastic content, the setup of the least-squares adjustment approach by solving normal equations is not at last a geometric task. However, this aspect is seldom given great attention in relevant literature, especially in introductory textbooks on this topic.

However, special configurations allow even an examination from the viewpoint of descriptive geometry: This is the case if the parameter space does not exceed four dimensions and the space of observations can be embedded into it.

In order to reduce the brittle aspect mentioned above, this paper is intended to highlight the geometry of least-squares adjustment by visualizing it at the most possible level. It presents examples for the setup and solutions of such least-squares adjustment problems in two and three dimensions. The case of four-dimensional adjustments is discussed as well.

Least-Squares Adjustment, Descriptive Geometry, Normal Equations

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