Framed general helix and framed ζ₃-slant helix in ℝ⁴

In this paper, we focus on general and ζ₃-slant helices with any singular points in four-dimensional Euclidean space, which are called framed general and ζ₃-slant helices, respectively. Then, we state and prove the conditions of necessity and su ciency for any framed curves to be general helices or ζ₃-slant helices in ℝ⁴. Also, we give some characterizations for framed helices.