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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
Gott, Goldberg and Vanderbei’s double-sided disk map was designed to minimize all six types of map distortions. Not properly "a" map projection because it is on two surfaces instead of one, it consists of two hemispheric equidistant azimuthal projections back-to-back. [5] [6] [7] 1879 Peirce quincuncial: Other Conformal Charles Sanders Peirce
Like the orthographic projection and gnomonic projection, the stereographic projection is an azimuthal projection, and when on a sphere, also a perspective projection. On an ellipsoid , the perspective definition of the stereographic projection is not conformal, and adjustments must be made to preserve its azimuthal and conformal properties.
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 ...
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 ...
Using these properties one can show that any rotation can be represented by a unique angle φ in the range 0 ≤ φ ≤ π and a unit vector n such that n is arbitrary if φ = 0; n is unique if 0 < φ < π; n is unique up to a sign if φ = π (that is, the rotations R(π, ±n) are identical).
Stereographic projection of a complex number A onto a point α of the Riemann sphere. The Riemann sphere can be visualized as the unit sphere x 2 + y 2 + z 2 = 1 {\displaystyle x^{2}+y^{2}+z^{2}=1} in the three-dimensional real space R 3 {\displaystyle \mathbf {R} ^{3}} .
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.