Learning geometry through surface creation from the hypocycloid curves expansion with derivative operators for ornaments

Abstract

Geometry is one of the particular problems for students. Therefore, several methods have been developed to attract students to learn geometry. For undergraduate students, learning geometry through surface visualization is introduced. One topic is studying parametric curves called the hypocycloid curve. This paper presents the generalization of the hypocycloid curve. The curve is known in calculus and usually is not studied further. Therefore, the research's novelty is introducing the spherical coordinate to the equation to obtain new surfaces. Initially, two parameters are indicating the radius of 2 circles governing the curves in the hypocycloid equations. The generalization idea here means that the physical meaning of parameters is not considered allowing any real numbers, including negative values. Hence, many new curves are observed infinitely. After implementing the spherical coordinates to the equations and varying the parameters, various surfaces had been obtained. Additionally, the differential operator was also implemented to have several other new curves and surfaces. The obtained surfaces are useful for learning by creating ornaments. Some examples of ornaments are presented in this paper.