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Stereographic projection is conformal, meaning that it preserves the angles at which curves cross each other (see figures). On the other hand, stereographic projection does not preserve area; in general, the area of a region of the sphere does not equal the area of its projection onto the plane. The area element is given in (X, Y) coordinates by
In normal aspect, pseudoconical projections represent the central meridian as a straight line, other meridians as complex curves, and parallels as circular arcs. Azimuthal In standard presentation, azimuthal projections map meridians as straight lines and parallels as complete, concentric circles. They are radially symmetrical.
It is a generalization of near-sided perspective projection, allowing tilt. The stereographic projection, which is conformal, can be constructed by using the tangent point's antipode as the point of perspective. r(d) = c tan d / 2R ; the scale is c/(2R cos 2 d / 2R ). [36] Can display nearly the entire sphere's surface on a finite ...
Stereographic projection of a 3-sphere (again removing the north pole) maps to three-space in the same manner. (Notice that, since stereographic projection is conformal, round spheres are sent to round spheres or to planes.) A somewhat different way to think of the one-point compactification is via the exponential map. Returning to our picture ...
Stereographic projection as an inversion of a sphere. A stereographic projection usually projects a sphere from a point (north pole) of the sphere onto the tangent plane at the opposite point (south pole). This mapping can be performed by an inversion of the sphere onto its tangent plane.
It is possible to choose any projection plane parallel to the equator (except the South pole): the figures will be proportional (property of similar triangles). It is usual to place the projection plane at the North pole. Definition The pole figure is the stereographic projection of the poles used to represent the orientation of an object in space.
The stereographic projection, also known as the planisphere projection or the azimuthal conformal projection, is a conformal map projection whose use dates back to antiquity. Like the orthographic projection and gnomonic projection, the stereographic projection is an azimuthal projection, and when on a sphere, also a perspective projection.
Gall stereographic projection of the world. 15° graticule. Gall stereographic projection with 1,000 km indicatrices of distortion. The Gall stereographic projection, presented by James Gall in 1855, is a cylindrical projection. It is neither equal-area nor conformal but instead tries to balance the distortion inherent in any projection.