190 5 “Sphere to tangential plane”: polar (normal) aspect
5-246 The orthographic projection
The orthographic projection, which is usually also called parallel projection or orthogonal projection,
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 → ∞ or R/D → ∞.
In Figs. 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 parameterized 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.
Technical
aside.
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 S
2
R are collected in Lemma 5.5.
Lemma 5.5 (Orthographic projection of the sphere to the polar tangential plane or the equatorial plane).
The orthographic projection of the sphere S
2
R to the tangential plane or to the equatorial plane is
parameterized by the two equations
x = R cos Φ cos Λ = X ,
y = R cos Φ sin Λ = Y ,
(5.90)
subject to the left Cauchy–Green eigenspace
left CG eigenspace =
E Λ , E Φ
1
sin Φ
.
(5.91)
End of Lemma.
Note that the northern orthographic projection covers all points of the northern hemisphere, while
the southern orthographic projection covers all points of the southern hemisphere. Accordingly, the
union of the two charts generated by a northern and a southern orthographic projection constitutes a
minimal atlas of the sphere. Additionally, let us here emphasize that the left maximal angular shear
of the gnomonic projection and the orthographic projection coincide.
Question.
Question: “What makes the orthographic projection (parallel projection, orthogonal projection) particularly useful in Geographic Information Systems?” Answer: “It is the property,
which is called concircular, that parallel circles of the sphere S
2
R are mapped onto circles
of T N S
2
R , T S S
2
R , or P
2
O . By means of r = R cos Φ, they are radius preserving – an essential
characteristic!”
5-246 The orthographic projection
The orthographic projection, which is usually also called parallel projection or orthogonal projection,
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 → ∞ or R/D → ∞.
In Figs. 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 parameterized 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.
Technical
aside.
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 S
2
R are collected in Lemma 5.5.
Lemma 5.5 (Orthographic projection of the sphere to the polar tangential plane or the equatorial plane).
The orthographic projection of the sphere S
2
R to the tangential plane or to the equatorial plane is
parameterized by the two equations
x = R cos Φ cos Λ = X ,
y = R cos Φ sin Λ = Y ,
(5.90)
subject to the left Cauchy–Green eigenspace
left CG eigenspace =
E Λ , E Φ
1
sin Φ
.
(5.91)
End of Lemma.
Note that the northern orthographic projection covers all points of the northern hemisphere, while
the southern orthographic projection covers all points of the southern hemisphere. Accordingly, the
union of the two charts generated by a northern and a southern orthographic projection constitutes a
minimal atlas of the sphere. Additionally, let us here emphasize that the left maximal angular shear
of the gnomonic projection and the orthographic projection coincide.
Question.
Question: “What makes the orthographic projection (parallel projection, orthogonal projection) particularly useful in Geographic Information Systems?” Answer: “It is the property,
which is called concircular, that parallel circles of the sphere S
2
R are mapped onto circles
of T N S
2
R , T S S
2
R , or P
2
O . By means of r = R cos Φ, they are radius preserving – an essential
characteristic!”