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Let = (,) be a graph (or directed graph) containing an edge = (,) with .Let be a function that maps every vertex in {,} to itself, and otherwise, maps it to a new vertex .The contraction of results in a new graph ′ = (′, ′), where ′ = ({,}) {}, ′ = {}, and for every , ′ = ′ is incident to an edge ′ ′ if and only if, the corresponding edge, is incident to in .
The straight-line distance between the central point on the map to any other point is the same as the straight-line 3D distance through the globe between the two points. c. 150 BC: Stereographic: Azimuthal Conformal Hipparchos* Map is infinite in extent with outer hemisphere inflating severely, so it is often used as two hemispheres.
Many large-scale maps use conformal projections because figures in large-scale maps can be regarded as small enough. The figures on the maps are nearly similar to their physical counterparts. A non-conformal projection can be used in a limited domain such that the projection is locally conformal. Glueing many maps together restores roundness.
The vertical lines PK and MQ are arcs of meridians of length Rδφ. [d] The horizontal lines PM and KQ are arcs of parallels of length R(cos φ)δλ. The corresponding points on the projection define a rectangle of width δx and height δy. For small elements, the angle PKQ is approximately a right angle and therefore
If maps were projected as in light shining through a globe onto a developable surface, then the spacing of parallels would follow a very limited set of possibilities. Such a cylindrical projection (for example) is one which: Is rectangular; Has straight vertical meridians, spaced evenly; Has straight parallels symmetrically placed about the ...
The vertical shear displaces points to the right of the y-axis up or down, depending on the sign of m. It leaves vertical lines invariant, but tilts all other lines about the point where they meet the y-axis. Horizontal lines, in particular, get tilted by the shear angle to become lines with slope m.
If a graph has vertices with high degree then it necessarily will have small angular resolution, but the angular resolution can be bounded below by a function of the degree. [12] The slope number of a graph is the minimum number of distinct edge slopes needed in a drawing with straight line segment edges (allowing crossings).
Topographic maps are also commonly called contour maps or topo maps. In the United States, where the primary national series is organized by a strict 7.5-minute grid, they are often called or quads or quadrangles. Topographic maps conventionally show topography, or land contours, by means of contour lines.