Constant Ratio Quaternionic Curves in Euclidean Spaces

In this paper, we give some characterizations of quaternionic curves whose position vectors can be written as linear combinations of their Serret-Frenet vectors in Euclidean 3-space E3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{E}^{3}}$$\end{document} and Euclidean 4-space E4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{E}^{4}}$$\end{document}. We characterize such curves in terms of their curvature functions. Finally, we give the necessary and sufficient conditions for these types of curves to become constant ratio, T-constant, and N-constant.