In geometry, a polyhedron is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices.
A sphere is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. Formally, a sphere is the set of points that are all at the same distance r from a given point in three-dimensional space. That given point is the center of the sphere, and r is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians.
In cartography, a map projection is any of a broad set of transformations employed to represent the curved two-dimensional surface of a globe on a plane. In a map projection, coordinates, often expressed as latitude and longitude, of locations from the surface of the globe are transformed to coordinates on a plane. Projection is a necessary step in creating a two-dimensional map and is one of the essential elements of cartography.
A circle of latitude or line of latitude on Earth is an abstract east–west small circle connecting all locations around Earth at a given latitude coordinate line.
In computer graphics and computational geometry, a bounding volume for a set of objects is a closed region that completely contains the union of the objects in the set. Bounding volumes are used to improve the efficiency of geometrical operations, such as by using simple regions, having simpler ways to test for overlap.
In geometry, a zonohedron is a convex polyhedron that is centrally symmetric, every face of which is a polygon that is centrally symmetric. Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments in three-dimensional space, or as a three-dimensional projection of a hypercube. Zonohedra were originally defined and studied by E. S. Fedorove, a Russian crystallographer. More generally, in any dimension, the Minkowski sum of line segments forms a polytope known as a zonotope.
In geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's vertices. The word circumsphere is sometimes used to mean the same thing, by analogy with the term circumcircle. As in the case of two-dimensional circumscribed circles (circumcircles), the radius of a sphere circumscribed around a polyhedron P is called the circumradius of P, and the center point of this sphere is called the circumcenter of P.
The Mollweide projection is an equal-area, pseudocylindrical map projection generally used for maps of the world or celestial sphere. It is also known as the Babinet projection, homalographic projection, homolographic projection, and elliptical projection. The projection trades accuracy of angle and shape for accuracy of proportions in area, and as such is used where that property is needed, such as maps depicting global distributions.
A gnomonic projection, also known as a central projection or rectilinear projection, is a perspective projection of a sphere, with center of projection at the sphere's center, onto any plane not passing through the center, most commonly a tangent plane. Under gnomonic projection every great circle on the sphere is projected to a straight line in the plane. More generally, a gnomonic projection can be taken of any n-dimensional hypersphere onto a hyperplane.
The scale of a map is the ratio of a distance on the map to the corresponding distance on the ground. This simple concept is complicated by the curvature of the Earth's surface, which forces scale to vary across a map. Because of this variation, the concept of scale becomes meaningful in two distinct ways.
The sinusoidal projection is a pseudocylindrical equal-area map projection, sometimes called the Sanson–Flamsteed or the Mercator equal-area projection. Jean Cossin of Dieppe was one of the first mapmakers to use the sinusoidal, using it in a world map in 1570.
In geometry, a truncated 24-cell is a uniform 4-polytope formed as the truncation of the regular 24-cell.
Bernard Joseph Stanislaus Cahill, American cartographer and architect, was the inventor of the octahedral "Butterfly Map". An early proponent of the San Francisco Civic Center, he also designed hotels, factories and mausoleums like the Columbarium of San Francisco.
In geometry, the Waterman polyhedra are a family of polyhedra discovered around 1990 by the mathematician Steve Waterman. A Waterman polyhedron is created by packing spheres according to the cubic close(st) packing (CCP), also known as the face-centered cubic (fcc) packing, then sweeping away the spheres that are farther from the center than a defined radius, then creating the convex hull of the sphere centers.
In cartography, the normal cylindrical equal-area projection is a family of normal cylindrical, equal-area map projections.
The Cahill–Keyes projection is a polyhedral compromise map projection first proposed by Gene Keyes in 1975. The projection is a refinement of an earlier 1909 projection by Bernard Cahill. The projection was designed to achieve a number of desirable characteristics, namely symmetry of component maps (octants), scalability allowing the map to continue to work well even at high resolution, uniformity of geocells, metric-based joining edges, minimized distortion compared to a globe, and an easily understood orientation to enhance general usability and teachability.
The octant projection or octants projection, is a type of map projection proposed the first time, in 1508, by Leonardo da Vinci in his Codex Atlanticus. Leonardo's authorship would be demonstrated by Christopher Tyler, who stated "For those projections dated later than 1508, his drawings should be effectively considered the original precursors."
Leonardo's world map is the name assigned to a unique world map drawn using the "octant projection" and found loosely inserted among a Codex of Leonardo da Vinci preserved in Windsor. It features an early use of the toponym America and incorporates information from the travels of Amerigo Vespucci, published in 1503 and 1505. Additionally, the map depicts the Arctic as an ocean and Antarctica as a continent of about the correct size.
A polyhedral map projection is a map projection based on a spherical polyhedron. Typically, the polyhedron is overlaid on the globe, and each face of the polyhedron is transformed to a polygon or other shape in the plane. The best-known polyhedral map projection is Buckminster Fuller's Dymaxion map. When the spherical polyhedron faces are transformed to the faces of an ordinary polyhedron instead of laid flat in a plane, the result is a polyhedral globe.
