In this thesis, we consider parametric surfaces with Pythagorean normals (PN surfaces), whose unit normals are determined by rational mappings. These rational mappings give PN surfaces interesting properties that are useful in solving various geometric problems in geometric modelling. In the introductory part of the thesis, we review and revisit the key concepts related to parametric surfaces in three-dimensional Euclidean space. This is followed by an in-depth treatment of the construction of polynomial PN surfaces using bilinear quaternion polynomials. The derived methods are then applied to solve an interpolation problem, where we search for a PN surface that interpolates given initial points and normal vectors. In the final part of the thesis, we focus on the construction of polynomial minimal PN surfaces using complex polynomials. Moreover, we present how the described method is applied to solve the Plateau problem, which deals with the existence of minimal surfaces with prescribed boundaries. The results presented in this work give an important step towards a better understanding and construction of polynomial surfaces in geometric modelling.
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