A toric section is a curve obtained by slicing a torus (generally a horn torus) with a plane. A spiric section is a special case of a toric section in which the slicing plane is perpendicular to both the midplane of the torus and to the plane .Consider a torus with tube radius . For a cutting plane parallel to the -plane, the toric section is either a single circle (for ) or two concentric circles (for ). For planes containing the z-axis, the section is two equal circles.Toric sections at oblique angles can be more complicated, passing from a crescent shape, through a U-shape, and into two disconnected kidney-shaped curves.
A cylindric section is the intersection of a plane with a right circular cylinder. It is a circle (if the plane is at a right angle to the axis), an ellipse, or, if the plane is parallel to the axis, a single line (if the plane is tangent to the cylinder), pair of parallel lines bounding an infinite rectangle (if the plane cuts the cylinder), or no intersection at all (if the plane misses the cylinder entirely; Hilbert and Cohn-Vossen 1999, pp. 7-8).
A curve on a surface whose tangents are always in the direction of principalcurvature. The equation of the lines of curvature can be writtenwhere and are the coefficients of the first and second fundamental forms.
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.
A space curve consisting of a spiral wound arounda helix. It has parametric equations(1)(2)(3)
The spherical curve obtained when moving along the surface of a sphere with constant speed, while maintaining a constant angular velocity with respect to a fixed diameter (Erdős 2000). This curve is given in cylindrical coordinates by the parametric equations(1)(2)(3)where is a positive constant and and are Jacobi elliptic functions (Whittaker and Watson 1990, pp. 527-528).Erdős (2000) provides a derivation of the equations of this curve, as well as an analysis of its properties, including conditions for obtaining periodic orbits.
As defined by Gray (1997, p. 201), Viviani's curve, sometimes also called Viviani's window, is the space curve giving the intersection of the cylinder of radius and center (1)and the sphere(2)with center and radius . This curve was studied by Viviani in 1692 (Teixeira 1908-1915, pp. 311-320; Struik 1988, pp. 10-11; Gray 1997, p. 201).Solving directly for and as a function of gives(3)(4)This curve is given by the parametric equations(5)(6)(7)for (Gray 1997, p. 201).From the parametric equations, it can be immediately seen that views of the curve from the front, top, and left are given by a lemniscate-like curve, circle, and parabolic segment, respectively. The lemniscate-like figure has parametric equations(8)(9)which can be written in Cartesian coordinatesas the quartic curve(10)Viviani's curve has arc length(11)where is a complete elliptic integral of the second kind.The arc length function,..
The conical spiral with angular frequency on a cone of height and radius is a space curve given by the parametric equations(1)(2)(3)The general form has parametric equations(4)(5)(6)This curve has arc length function, curvature,and torsion given by(7)(8)(9)
A Steinmetz curve is a curve of intersection of two perpendicularly placed cylinders of radii and comprising a Steinmetz solid. If the vertical cylinder has radius and the horizontal cylinder radius , then the Steinmetz curves are given by the parametric equations(1)(2)(3)(Gray 1997, p. 204).
The spherical curve taken by a ship which travels from the south pole to the north pole of a sphere while keeping a fixed (but not right) angle with respect to the meridians. The curve has an infinite number of loops since the separation of consecutive revolutions gets smaller and smaller near the poles.It is given by the parametric equations(1)(2)(3)where(4)and is a constant. Plugging in therefore gives(5)(6)(7)It is a special case of a loxodrome.The arc length, curvature,and torsion are all slightly complicated expressions.A series of spherical spirals are illustrated in Escher's woodcuts "Sphere Surface with Fish" (Bool et al. 1982, pp. 96 and 318) and "Sphere Spirals" (Bool et al. 1982, p. 319; Forty 2003, Plate 67).
The tangent indicatrix of a curve of constant precession is a spherical helix. The equation of a spherical helix on a sphere with radius making an angle with the z-axis is(1)(2)(3)The projection on the -plane is an epicycloid with radii(4)(5)
A helix, sometimes also called a coil, is a curve for which the tangent makes a constant angle with a fixed line. The shortest path between two points on a cylinder (one not directly above the other) is a fractional turn of a helix, as can be seen by cutting the cylinder along one of its sides, flattening it out, and noting that a straight line connecting the points becomes helical upon re-wrapping (Steinhaus 1999, p. 229). It is for this reason that squirrels chasing one another up and around tree trunks follow helical paths.Helices come in enantiomorphous left- (coils counterclockwise as it "goes away") and right-handed forms (coils clockwise). Standard screws, nuts, and bolts are all right-handed, as are both the helices in a double-stranded molecule of DNA (Gardner 1984, pp. 2-3). Large helical structures in animals (such as horns) usually appear in both mirror-image forms, although the teeth of a male narwhal, usually..
If two single-valued continuous functions (curvature) and (torsion) are given for , then there exists exactly one space curve, determined except for orientation and position in space (i.e., up to a Euclidean motion), where is the arc length, is the curvature, and is the torsion.