area) isometric filling of the Riemannian circle. Topology Remarkably, it is possible to turn an ordinary sphere inside out in a three-dimensional space with possible self-intersections but without creating any creases, in a process called sphere eversion. The antipodal quotient of the sphere is the surface called the real projective plane, which can also be thought of as the Northern Hemisphere with antipodal points of the equator identified. Curves on a sphere Circles Circles on the sphere are, like circles in the plane, made up of all points a certain distance from a fixed point on the sphere. The intersection of a sphere and a plane is a circle, a point, or empty. Great circles are the intersection of the sphere with a plane passing through the center of a sphere: others are called small circles. More complicated surfaces may intersect a sphere in circles, too: the intersection of a sphere with a surface of revolution whose axis contains the center of the sphere (are coaxial) consists of circles and/or points if not empty. For example, the diagram to the right shows the intersection of a sphere and a cylinder, which consists of two circles. If the cylinder radius were that of the sphere, the intersection would be a single circle. If the cylinder radius were larger than that of the sphere, the intersection would be empty. Loxodrome In navigation, a rhumb line or loxodrome is an arc crossing all meridians of longitude at the same angle. Loxodromes are the same as straight lines in the Mercator projection. A rhumb line is not a spherical spiral. Except for some simple cases, the formula of a rhumb line is complicated. Clelia curves A Clelia curve is a curve on a sphere for which the longitude and the colatitude satisfy the equation . Special cases are: Viviani's curve () and spherical spirals () such as Seiffert's spiral. Clelia curves approximate the path of satellites in polar orbit. Spherical conics The analog of a conic section on the sphere is a spherical conic, a quartic curve which can be defined in several equivalent ways, including: as the intersection of a sphere with a quadratic cone whose vertex is the sphere center; as the intersection of a sphere with an elliptic or hyperbolic cylinder whose axis passes through the sphere center; as the locus of points whose sum or difference of great-circle distances from