Normal curves in n-dimensional Euclidean space

Abstract

Abstract In this paper, we give a generalization of normal curves to n-dimensional Euclidean space. Then we obtain a necessary and sufficient condition for a curve to be a normal curve in the n-dimensional Euclidean space. We characterize the relationship between the curvatures for any unit speed curve to be congruent to a normal curve in the n-dimensional Euclidean space. Moreover, the differentiable function f(s) $f ( s ) $ is introduced by using the relationship between the curvatures of any unit speed curve in En $E^{n}$. Finally, the differential equation characterizing a normal curve can be solved explicitly to determine when the curve is congruent to a normal curve.

Keywords

• Normal curve
• Curvatures
• Explicit solutions
• Position vector