F Mixed cylindric map projections
Mixed cylindric map projections of the ellipsoid-of-revolution. Lambert projection and Sanson–Flamsteed
projection, generalized Lambert projection and generalized Sanson–Flamsteed projection. The mapping
equations and the conditions. Vertical coordinates and horizontal coordinates, horizontal weighted mean
and vertical weighted mean. Deformation analysis of vertically/horizontally averaged equiareal cylindric
mappings. Cauchy–Green deformation tensor and principal stretches.
The mixed spherical map projections of equiareal, cylindric type to be considered here are based
upon the Lambert projection and the sinusoidal Sanson–Flamsteed projection. These cylindric and
pseudo-cylindrical map projections of the sphere are generalized to the ellipsoid-of-revolution (biaxial
ellipsoid). They are used in consequence by two lemmas to generate a horizontal (a vertical) weighted
mean of equiareal cylindric map projections of the ellipsoid-of-revolution. Its left–right deformation
analysis via further results leads to the left–right principal stretches/eigenvalues and left–right eigenvectors/eigenspace as well as the maximal left–right angular distortion for these new mixed cylindric
map projections of ellipsoidal type. Detailed illustrations document the cartographic synergy of mixed
cylindric map projections.
Mixed cylindric map projections of the sphere are very popular projections, for instance, those
equiareal cylindric projections named after H. C. Foucaut (1862) or A. M. Nell (1890) and E. Hammer
(1900). The variants are the horizontal or vertical weighted means of the Lambert equiareal projections
and the equiareal pseudo-cylindrical mapping of sinusoidal type which is also known as Sanson–
Flamsteed projection of the sphere. In this chapter, let us exclusively aim at mixed cylindric map
projections of the ellipsoid-of-revolution.
Section F-1.
In Section F-1, we derive the conditions which must be fulfilled for a pseudo-cylindrical equiareal
map projection of a biaxial ellipsoid and the general form of ellipsoidal equiareal pseudo-cylindrical
mapping equations of Box F.2.
Section F-2.
In Section F-2, we start from the setup of the ellipsoidal generalized Lambert projection and the
ellipsoidal generalized Sanson–Flamsteed projection, which are both equiareal. By two lemmas, we
shall present the vertical–horizontal mean of the generalized Lambert projection and the generalized
Sanson–Flamsteed projection of the biaxial ellipsoid.
Section F-3.
The deformation analysis of vertically and horizontally averaged equiareal cylindric mappings is the
topic of Section F-3. We especially compute the left–right principal stretches, their corresponding
eigenvectors/eigenspace, and the maximal left angular distortion.
The following references are appropriate. The equiareal cylindric map projection of the sphere is
addressed to J. H. Lambert (1772), while the equiareal pseudo-cylindrical map projection of the sphere,
namely of sinusoidal type, to N. Sanson (1650), J. Cossin (1570), and J. Flamsteed (1692). Equally
weighted vertical and horizontal components of Lambert and Sanson–Flamsteed map projections of
the sphere have been presented by H. C. Foucaut (1862), A. M. Nell (1890), and E. Hammer (1900).
The theory of weighted means of general map projections of the sphere has been critically reviewed
by W. R. Tobler (1973).
Mixed cylindric map projections of the ellipsoid-of-revolution. Lambert projection and Sanson–Flamsteed
projection, generalized Lambert projection and generalized Sanson–Flamsteed projection. The mapping
equations and the conditions. Vertical coordinates and horizontal coordinates, horizontal weighted mean
and vertical weighted mean. Deformation analysis of vertically/horizontally averaged equiareal cylindric
mappings. Cauchy–Green deformation tensor and principal stretches.
The mixed spherical map projections of equiareal, cylindric type to be considered here are based
upon the Lambert projection and the sinusoidal Sanson–Flamsteed projection. These cylindric and
pseudo-cylindrical map projections of the sphere are generalized to the ellipsoid-of-revolution (biaxial
ellipsoid). They are used in consequence by two lemmas to generate a horizontal (a vertical) weighted
mean of equiareal cylindric map projections of the ellipsoid-of-revolution. Its left–right deformation
analysis via further results leads to the left–right principal stretches/eigenvalues and left–right eigenvectors/eigenspace as well as the maximal left–right angular distortion for these new mixed cylindric
map projections of ellipsoidal type. Detailed illustrations document the cartographic synergy of mixed
cylindric map projections.
Mixed cylindric map projections of the sphere are very popular projections, for instance, those
equiareal cylindric projections named after H. C. Foucaut (1862) or A. M. Nell (1890) and E. Hammer
(1900). The variants are the horizontal or vertical weighted means of the Lambert equiareal projections
and the equiareal pseudo-cylindrical mapping of sinusoidal type which is also known as Sanson–
Flamsteed projection of the sphere. In this chapter, let us exclusively aim at mixed cylindric map
projections of the ellipsoid-of-revolution.
Section F-1.
In Section F-1, we derive the conditions which must be fulfilled for a pseudo-cylindrical equiareal
map projection of a biaxial ellipsoid and the general form of ellipsoidal equiareal pseudo-cylindrical
mapping equations of Box F.2.
Section F-2.
In Section F-2, we start from the setup of the ellipsoidal generalized Lambert projection and the
ellipsoidal generalized Sanson–Flamsteed projection, which are both equiareal. By two lemmas, we
shall present the vertical–horizontal mean of the generalized Lambert projection and the generalized
Sanson–Flamsteed projection of the biaxial ellipsoid.
Section F-3.
The deformation analysis of vertically and horizontally averaged equiareal cylindric mappings is the
topic of Section F-3. We especially compute the left–right principal stretches, their corresponding
eigenvectors/eigenspace, and the maximal left angular distortion.
The following references are appropriate. The equiareal cylindric map projection of the sphere is
addressed to J. H. Lambert (1772), while the equiareal pseudo-cylindrical map projection of the sphere,
namely of sinusoidal type, to N. Sanson (1650), J. Cossin (1570), and J. Flamsteed (1692). Equally
weighted vertical and horizontal components of Lambert and Sanson–Flamsteed map projections of
the sphere have been presented by H. C. Foucaut (1862), A. M. Nell (1890), and E. Hammer (1900).
The theory of weighted means of general map projections of the sphere has been critically reviewed
by W. R. Tobler (1973).