# Conics

The following was graciously provided by Patty Ahmetaj. The source of the figures cited and much of this information is from Flattening the Earth: Two Thousand Years of Map Projections, by John Snyder. University of Chicago Press. 1993.

Graphics: Were gratiously provided by Paul S. Anderson.

In the Conical Projection the graticule is projected onto a cone tangent, or secant, to the globe along any small circle (usually a mid-latitude parallel). In the normal aspect (which is oblique for conic projections), parallels are projected as concentric arcs of circles, and meridians are projected as straight lines radiating at uniform angular intervals from the apex of the flattened cone. Conic projections are not widely used in mapping because of their relatively small zone of reasonable accuracy. The secant case, which produces two standard parallels, is more frequently used with conics. Even then, the scale of the map rapidly becomes distorted as distance from the correctly represented standard parallel increases. Because of this problem, conic projections are best suited for maps of mid-latitude regions, especially those elongated in an east- west direction. The United States meets these qualifications and therefore is frequently mapped on conic projections.

Variations in conic projection can take four forms:

• Varying the manner in which standard parallels or other constants are selected,
• spacing parallels to provide some arbitrary compromise of distortion,
• adapting the conic projection of the sphere to that of the ellipsoid, and
• developing pseudoconic projections such as the Bonne or other modifications that are not true conics.

## History & Use

Ptolemy (believed to have lived between A.D.90 and A.D.170), although he made no reference to a cone, introduced two projections with concentric, circular arcs for parallels of latitude (like conics) but with meridians that are broken straight lines or circular arcs. These projections although conic-like, were not conic.

Giovanni Matteo Contarini's (d. 1507) only known map, a world map of 1506, is a modification of Ptolemy's first conic-like projection (fig.1.8). Contarini doubled the span of the meridians from 180 degrees to the full 360 degrees; he extended latitudes to the north pole and he continued the meridians unbroken to about 35 degrees S without a bend at the Equator. Contarini had developed an equidistant conic projection having the same scale relationships north of the Equator as those of Ptolemy.

In 1507-08, Johannes Ruysch (d.1533) generally followed Contarini's modifications but placed the North Pole at the center of the circular arcs of latitude and equally spaced the parallels from this pole to the map limits at 38 degrees S (reconstructed in fig.1.21).

Wilhelm Schickard (1592-1635), an astronomer and mathematician of Tubingen, Germany, who designed and built a working model of the first modern mechanical calculator, was apparently the first of a half dozen 17th and 18th century mapmakers to use the conic projections for star maps.

### Equidistant or Simple Conic

Classified as:

• Conic
• Equally Spaced Parallels
• Neither Conformal nor Equal Area
• Equidistant Meridians converging at a common point
• This projection was developed by De L'isle or originating within the Coast Survey. It was used for field sheets and some charts of small areas until at least 1882.

It provided the base for maps of small and medium-sized countries throughout the century, as well as for several maps of the United States, Canada and Europe.

The simple conic projection was appropriate for small countries and regions regardless of shape, or for large countries or continents of predominant east-west extent. In one example, however, North America, a region more north-south in extent, was mapped on the simple conic projection in an atlas of 1896 (Rand McNally 1896, 40-41).

### Lambert Conformal Conic (Johanne Heinrich)

Classified as:

• Conic
• Conformal
• Developed by J.H. Lambert in 1772 (fig. 2.7). The projection was almost unknown as a Lambert Projection for over a century. Harding, Herschel and Boole had developed it independently in both spherical and ellipsoidal forms during the 19th century. World War I gave this projection new life, making it the standard projection for intermediate - and large-scale maps of regions in midlatitudes for which the transverse Mercator is not used.

Classified as:

• Conic
• Conformal

### Albers Equal-Area Conic (Heinrich Christian)

Classified as:

• Conic
• Equal Area
• The last of the basic conic projections (fig.3.16) to be developed with one of the three major properties of conformality, equivalence or equidistance along meridians was this equal-area presented by H.C. Albers (1805a), three months after Mollweide presented his elliptical world map in the same journal. Albers (1773-1833), a native and life long resident of Luneburg, Germany, derived the formulas for the projection of the sphere using two standard parallels.

Classified as:

• Conic
• Equal Area

### Perspective Conic

Classified as:

• Conic
• Perspective
• Neither Conformal nor Equal Area

### Polyconic

Classified as:

• Polyconic (this term is generically applied to any projection with circular arcs for parallels of latitude, whether or not they are concentric)
• Neither Conformal nor Equal Area
• The polyconic was applied as a specific projection in 1853 by Edward Bissell Hunt of the U.S. Coast Survey to one first proposed by Ferdinand Rudolph Hassler (Swiss-born, 1770-1843).

It was commonly, but not exclusively, used for coastal charts of the United States. When the U.S. Geologic Survey came into existence in 1879 and began issuing maps of land surveys, the polyconic was the only projection used for the agency's topographic quadrangles until the mid 20th century. This emphasis on usage by U.S. government agencies led to its use in several 19th century commercial atlases as well, for some maps of the United States, Canada, North America, Asia and Oceania.

The polyconic projection of Hassler (fig.3.18) is simultaneously universal for a given figure of the earth (sphere or ellipsoid), simply drawn, even for the ellipsoid, and employs useful scale characteristics. The projection is true to scale along the central meridian and along each parallel. It is neither conformal nor equal-area, and it is only free of distortion along the central meridian. Therefore, it should only be used for regions of predominant north-south extent.

### Rectangular Polyconic

Classified as:

• Polyconic
• Neither Conformal nor Equal Area
• The first reference to this projection was made in 1853. This projection was derived and used at that time by the U.S. Coastal Survey for portions of the United States exceeding about a square degree. It has since been used for topographic maps of the British War Office and thus has been called the War Office projection as well.

### Modified Polyconic - Int'l Map of the World Series

Classified as:

• Polyconic
• Neither Conformal nor Equal Area

### Bonne (Rigobert)

Classified as:

• Pseudoconic
• Equal Area
• The Bonne was the preferred projection for atlas maps of large countries and continents, giving the regions a uniform area scale and an appealing combination of curved meridians and curved parallels befitting a representation of the globe.

More than half the maps of the world's most populated continents were prepared on the Bonne projections. No maps of Africa were found on the Bonne. Australia was usually plotted on the Bonne projection if shown separately, but it was just as often part of a map of Oceania on a projection such as the Mercator. Other regions occasionally mapped on the Bonne included the Arabian Peninsula, China, Russia and the United States.

The Bonne projection played a significant role in 19th century larger-scale topographic mapping using ellipsoidal formulas. This began with its adoption in France after efforts by Bonne in 1802. After wartime delays, the Bonne projection was accepted by a special commission of the Depot de la Guerre (war office). It later received the name projection depot de la guerre in other European countries. These regions included Austria-Hungary (1:750,000 scale maps), Belgium (1:20,000 and reductions), Denmark (1:20,000), Italy (1:500,000), Netherlands (1:25,000), Russia (1:126,000), Spain (1:200,000), Switzerland (1:25,000 and 1:50,000), Scotland and Ireland (1:63,360 and smaller), as well as France (1:80,000 and 1:200,000) (Hinks 1912,65-66).

The Bonne may have been used for some of the U.S. Coast survey charts of the Delaware Bay in 1844-45 (Schott 1882, 293). One larger scale American use of the Bonne was for a nine-sheet map of the state of Virginia (then including West Virginia) prepared in 1825 by Herman Boye, a Richmond engineer of German ancestry, at a scale of five miles to the inch (1:316,800).

### Werner (Johannes)

Classified as:

• Pseudoconic
• Equal Area
• Cordiform (Heart Shaped)
• About 1500, Stabius invented a series of three attractive heart-shaped (cordiform) projections which were further publicized by Werner.

Werner was an ordained priest with a pastorate. His pastoral duties were limited, however, he spent much of his time studying astronomy, mathematics and geography (Folkert 1970-80). He related spherical trigonometry to the solution of various celestial and terrestrial problems, and designed several instruments to compute astronomical positions. His important treatises, prepared between 1505 and 1522, dealt especially with spherical triangles and conic sections.

After sporadic usage, the cordiform projection all but disappeared by the 18th century in favor of the Bonne projection. This more general adaptation of Werner's projection provides less angular distortion for a map of a continent.