Abstract
A rectifying curve is a twisted curve with the property that all of its rectifying planes pass through a fixed point. If this point is the origin of the Cartesian coordinate system, then the position vector of the rectifying curve always lies in the rectifying plane. A remarkable property of these curves is that the ratio between torsion and curvature is a nonconstant linear function of the arc-length parameter. In this paper, we give a new characterization of rectifying curves, namely, we prove that a curve is a rectifying curve if and only if it has a spherical involute. Consequently, rectifying curves can be constructed as evolutes of spherical twisted curves; we present an illustrative example of a rectifying curve obtained as the evolute of a spherical helix. We also express the curvature and the torsion of a rectifying spherical curve and give necessary and sufficient conditions for a curve and its involute to be both rectifying curves.
Highlights
The term rectifying curve was introduced by Chen in [1] to designate a curve whose position vector always lies in its rectifying plane
We prove that a curve is a rectifying curve if and only if it has a spherical involute
In this paper, we state and prove some new properties of rectifying curves based on the relationship with their involutes/evolutes
Summary
The term rectifying curve was introduced by Chen in [1] to designate a curve whose position vector always lies in its rectifying plane. We give a new characterization of rectifying curves based on the relationship with their evolutes/involutes. Allows us to develop a new method of constructing rectifying curves: as evolutes of spherical twisted curves.
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