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Space Curve


A curve which may pass through any region of three-dimensional space, as contrasted to a plane curve which must lie in a single plane. Von Staudt (1847) classified space curves geometrically by considering the curve

 phi:I->R^3
(1)

at t_0=0 and assuming that the parametric functions phi_i(t) for i=1, 2, 3 are given by power series which converge for small t. If the curve is contained in no plane for small t, then a coordinate transformation puts the parametric equations in the normal form

phi_1(t)=t^(1+k_1)+...
(2)
phi_2(t)=t^(2+k_1+k_2)+...
(3)
phi_3(t)=t^(3+k_1+k_2+k_3)+...
(4)

for integers k_1, k_2, k_3>=0, called the local numerical invariants.


See also

Curve, Cyclide, Fundamental Theorem of Space Curves, Helix, Plane Curve, Seiffert's Spherical Spiral, Skew Conic, Space-Filling Function, Spherical Curve, Spherical Spiral, Surface, Viviani's Curve

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References

do Carmo, M.; Fischer, G.; Pinkall, U.; and Reckziegel, H. "Singularities of Space Curves." §3.1 in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 24-25, 1986.Fine, H. B. "On the Singularities of Curves of Double Curvature." Amer. J. Math. 8, 156-177, 1886.Fischer, G. (Ed.). Plates 57-64 in Mathematische Modelle aus den Sammlungen von Universitäten und Museen, Bildband. Braunschweig, Germany: Vieweg, pp. 58-59, 1986.Gray, A. "Curves in R^n" and "Curves in Space." §1.2 and Ch. 8 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 5-7 and 181-206, 1997.Griffiths, P. and Harris, J. Principles of Algebraic Geometry. New York: Wiley, 1978.Kreyszig, E. Differential Geometry. New York: Dover, 1991.Saurel, P. "On the Singularities of Tortuous Curves." Ann. Math. 7, 3-9, 1905.Staudt, K. G. C. von. Geometrie der Lage. Nürnberg, Germany: Bauer und Raspe, 1847.Teixeira, F. G. Traité des courbes spéciales remarquables plane et gauches, 3 vols. Coimbra, Portugal, 1908-1915. Reprinted New York: Chelsea, 1971 and Paris: Gabay.Wiener, C. "Die Abhängigkeit der Rückkehrelemente der Projektion einer unebenen Curve von deren der Curve selbst." Z. Math. & Phys. 25, 95-97, 1880.

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Space Curve

Cite this as:

Weisstein, Eric W. "Space Curve." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SpaceCurve.html

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