Family of Surfaces with a Common Special Polynomial Curves

Abstract

Polynomial curves play an important role in statistics, mathematics, engineering, computer
science, and many other fields. It is widely used to model and analyze data. By
using it to best simulate data points, it is possible to gain better insights and predictions
on data sets. In engineering applications, it is used to model material properties,
vibration analysis, temperature changes and similar factors. Computer graphics and
animations are frequently used to represent the surfaces of objects, especially in 3D
modeling and animation software for rendering surfaces and objects. In this paper, we
establish the necessary and sufficient conditions to parameterize a surface family on
which the polynomial curve of any given curve lies as an isogeodesic, isoasymptotic and
curvature line in E3. We also extend this idea on ruled surfaces. Finally, we present some
examples and picture the corresponding surfaces.