Framed curves in the Euclidean space
TLDR
In this paper, the curvature of a framed curve is defined, similarly to the curvatures of a regular curve and of a Legendre curve in the unit tangent bundle.Abstract:
Abstract A framed curve in the Euclidean space is a curve with a moving frame. It is a generalization not only of regular curves with linear independent condition, but also of Legendre curves in the unit tangent bundle. We define smooth functions for a framed curve, called the curvature of the framed curve, similarly to the curvature of a regular curve and of a Legendre curve. Framed curves may have singularities. The curvature of the framed curve is quite useful to analyse the framed curves and their singularities. In fact, we give the existence and the uniqueness for the framed curves by using their curvature. As applications, we consider a contact between framed curves, and give a relationship between projections of framed space curves and Legendre curves.read more
Citations
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Journal ArticleDOI
Curves and singularities, by J. W. Bruce and P. J. Giblin. Pp 222. 1984. ISBN 0-521-24945-7 (Hardback) £25, 27091-X (Paperback) £8·95 (Cambridge University Press)
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Tangent developables and Darboux developables of framed curves
Yanlin Li,Siyao Liu,Zhigang Wang +2 more
TL;DR: In this paper, the authors generalize the tangent developables of regular curves with linear independent condition to the Tangent Developables of framed curves and investigate the singularities of tangent developedables in Euclidean 3-space.
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Framed Surfaces in the Euclidean Space
TL;DR: In this paper, the basic invariants and curvatures of a smooth surface with a moving frame are introduced, and the existence and uniqueness of the fundamental invariants of the framed surfaces are established.
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The developable surfaces with pointwise 1-type Gauss map of Frenet type framed base curves in Euclidean 3-space
TL;DR: In this paper , the ruled developable surfaces with pointwise 1-type Gauss map of Frenet-type framed base (Ftfb) curve are introduced in Euclidean 3-space.
References
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Journal ArticleDOI
There is More than One Way to Frame a Curve
TL;DR: More than one way to frame a curve as mentioned in this paper is a well-known way to define a curve, and it has been used in many applications in computer science and computer engineering.
Book
Modern Differential Geometry of Curves and Surfaces with Mathematica
TL;DR: Modern Differential Geometry of Curves and Surfaces with Mathematica explains how to define and compute standard geometric functions, for example the curvature of curves, and presents a dialect ofMathematica for constructing new curves and surfaces from old.