User:Peter Mercator/Draft for cylindrical projections

Proposal for a new page on Cylindrical Projections
This is an outline of a proposed new page. I suspect a title of simply 'Cylindrical Projections' would suffice but any other suggestions would be welcome. The intention is that the page should cover the theoretical aspects of cylindrical projections, in all their aspects, in tangent and secant versions, and also on both sphere and ellipsoid. It will be more mathematical than the current page on projections (but it will not be 'modern' maths). Specific examples of projections will be used to illustrate the discussion but this page will not provide a comprehensive list (as expected of the new page 'List of projections'). The new page will duplicate some statements in Map projection, the proof of the projection formulae in Mercator projection and in scale (map), as well as derivation of scale and angle transformation formulae in scale (map). There would be consequent edits to these pages. Some of the figures in preparation can be viewed at ../Draft figures. The proposed contents suggested are as follows.

Classification of cylindrical projections

 * Sphere or ellipsoid.
 * Aspect: normal, transverse, oblique, space oblique.
 * Tangent or secant.

The basic cylindrical projection (CP)
The prototype is the normal, tangent projection of the sphere for which meridians and parallels project to straight lines of constant x and y respectively. I prefer to follow Snyder and take x=R*longitude so that the projection equator has the same length as the true equator. Scale unity on equator but varying elsewhere. Projection reduced to printed map by a constant RF (representation fraction). The singularities at the poles.

Angle transformations in the basic CP
The relation between an azimuth (alpha) on the sphere and the corresponding bearing (beta) on the projection (map). This distinction between azimuth and bearing is not universal but it helps to adopt a precise convention. The angle transformation formulae for an arbitrary y function.

Scale factors in the basic CP
Point scale defined. The expressions for the meridian scale (h) and the parallel scale (k). The scale along an arbitrary azimuth. General y function.

Special cases
For each of the following present the coordinate transformation (the y function), the scale factors h and k and in an arbitrary direction, the relation between and azimuth and bearing. Picture for each showing graticule.
 * Equidistant
 * Mercator. Derive form of y by imposing rhumb line condition. Equivalence with isotropy of scale.
 * Equal area: Derive by imposing h=k.
 * Central cylindrical. The only true perspective projection
 * Miller(?)

Scale variation and distortion
Graph showing superimposed meridian scale factors (h) for the examples above. Tissot's indicatrix for the examples.

Secant Projections
Equator as a standard parallel for the tangent projection. Two standard parallels for secant projections. Important method of bounding scale variation, particularly in a range of latitudes close to the equator. Graph of scale (h) variation close to the equator comparing tangent and secant versions of Mercator. Quantitative discussion for the Mercator (as in scale (map). Applications such as Gall and Behrmann.

Transverse projections
Rotating the graticule so that its pole lies on the equator. The formulae relating the angles on the normal graticule and the the rotated graticule. Illustration of geometry but refer to external source for the details of the spherical trigonometry involved. Meridians and parallels now complex curves. Concept of convergence. Illustrations for two examples only. In each case give explicit formulae for x, y, h, k Stress that these are not suitable for world maps and were developed for large scale maps over restricted longitude ranges. (eg Cassini for British maps in the nineteenth century. )
 * Cassini. The transverse aspect of the equidistant projection.
 * Transverse Mercator.

Ellipsoid forms
Not necessary for world maps since order e squared corrections only significant for large scale maps. Use for normal Mercator large scale maps of limited regions close to the equator and transverse Mercator for many countries. Use in the set 60 transverse Mercator projections for UTM. All these applications are secant forms. Explicit formulae and outline derivation for normal Mercator. No explicit formulae for Transverse Mercator on ellipsoid but these should be added to other pages (eg UTM) at a later stage.