“Sphere to Tangential Plane”: Polar (Normal) Aspect - Map Projections: Cartographic Information Systems - page 215

S 2 R onto the tangent space T N S 2 R : gnomonic projection, Tissot Fig. 5.21. Special perspective map of the sphere

ellipses, polar aspect, graticule 15 ◦ , shorelines

5-246 The Orthographic Projection

The orthographic projection , which is usually also called parallel projection or orthogonal projec-
tion , is generated as a parallel projection (orthogonal projection) of a point P ∈ S

2 R either on a

polar tangent plane of S

2 R or on a plane parallel to the polar tangent plane through the origin

O : compare with Figs. 5.22 and 5.23 . In the context of a general perspective mapping, we are

able to generate an orthographic projection by moving the perspective center O ∗ to infinity, i.e.
D

.InFigs. 5.22 and 5.23 , such a parallel projection (orthogonal projection)
is illustrated. In Box 5.16 , the characteristics of such a projection (namely, (i) the parameter-
ized mapping, (ii) the left principal stretches of the left Cauchy-Green eigenspace, (iii) the left
maximal angular shear, and (iv) the inverse parameterized mapping) are collected.

→∞ or R/D →∞

Note that the orthographic projection is used for charting
the Moon or the Earth, for example, on a TV screen.

The basic results of the orthographic projection (parallel projection, orthogonal projection) of
the sphere

2

S

R are collected in Lemma 5.5 .

Lemma 5.5 (Orthographic projection of the sphere to the polar tangential plane or the equatorial
plane).

2

The orthographic projection of the sphere S

R to the tangential plane or to the equatorial plane

is parameterized by the two equations