Space Curve

A curve which may pass through any region of 3-D 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

$$\phi:I\to\Bbb{R}^3$$ (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

$$\phi_1(t) = t^{1+k_1}+\ldots$$ (2)

$$\phi_2(t) = t^{2+k_1+k_2}+\ldots$$ (3)

$$\phi_3(t) = t^{3+k_1+k_2+k_3}+\ldots$$ (4)

for integers $k_1$, $k_2$, $k_3\geq 0$, called the local numerical invariants.

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

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/Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, pp. 58-59, 1986.

Gray, A. ``Curves in $\Bbb{R}^n$'' and ``Curves in Space.'' §1.2 and Ch. 7 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 4-6 and 123-151, 1993.

Griffiths, P. and Harris, J. Principles of Algebraic Geometry. New York: Wiley, 1978.

Saurel, P. ``On the Singularities of Tortuous Curves.'' Ann. Math. 7, 3-9, 1905.

Staudt, C. von. Geometrie der Lage. Nürnberg, Germany, 1847.

Wiener, C. ``Die Abhängigkeit der Rückkehrelemente der Projektion einer unebenen Curve von deren der Curve selbst.'' Z. Math. & Phys. 25, 95-97, 1880.