IC 418 spans 0.3 light-years across. IC 418 IC 418, also known as Spirograph Nebula, is a planetary nebula in the Milky Way Galaxy. The name derives from the intricate pattern of the nebula, which resembles a pattern which can be created using the Spirograph, a toy that produces geometric patterns (specifically, hypotrochoids and epitrochoids) on paper. Only a few million years ago, IC 418 was probably a common red giant star. Since running out of nuclear fuel, though, the outer envelope has begun expanding outward leaving a hot remnant core destined to become a white dwarf star. IC 418 lies about
Differential geometry of curves space curves are only distinguished by the way in which they bend and twist. Quantitatively, this is measured by the differential-geometric invariants called the "curvature" and the "torsion" of a curve. The fundamental theorem of curves asserts that the knowledge of these invariants completely determines the curve. Let (i) formula_1, (ii) formula_2, and (iii) formula_3 be a non-empty interval of real numbers. Then a vector-valued function of class formula_5 (i.e., the component-functions of formula_6 are formula_7-times continuously differentiable) is called a parametric formula_5-curve or a formula_5-parametrization. Note that formula_10 is called the image of the parametric curve. It is important