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
(1)
at and assuming that the parametric
functions
for , 2, 3 are given by power
series which converge for small . If the curve is contained in no plane
for small ,
then a coordinate transformation puts the parametric
equations in the normal form
(2)
(3)
(4)
for integers ,
, , called the local numerical invariants.
do Carmo, M.; Fischer, G.; Pinkall, U.; and Reckziegel, H. "Singularities of Space Curves." §3.1 in Mathematical
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"On the Singularities of Curves of Double Curvature." Amer. J. Math.8,
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Modelle aus den Sammlungen von Universitäten und Museen, Bildband. Braunschweig,
Germany: Vieweg, pp. 58-59, 1986.Gray, A. "Curves in " 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.
and assuming that the parametric
functions 
for 
, 2, 3 are given by power
series which converge for small 
. If the curve is contained in no plane
for small 
,
then a coordinate transformation puts the parametric
equations in the normal form
,
, 
, called the local numerical invariants.
" and "Curves in Space."
§1.2 and Ch. 8 in