Cylindrical vs Conic vs Azimuthal Projections

Where the /learn/conformal-projections and /learn/equal-area-projections supports cover projection properties, this article covers projection shapes. The two are independent axes of classification: a projection has both a property family (what it preserves) and a shape family (how it is constructed). Together they form a 2D grid that locates every projection in the cartographic landscape. The /learn/what-is-a-map-projection pillar covers the impossibility theorem that makes the trade-offs necessary; this article covers the conceptual geometry of how each projection makes its trade-off.

Developable surfaces

A developable surface is a 3D surface that can be flattened onto a plane without stretching, tearing, or wrinkling. Three developable surfaces are used in cartographic projections:

• A cylinder can be unrolled into a rectangle. Wrap a sheet of paper around a can; cut it parallel to the cylinder's axis; unroll it flat.
• A cone can be unrolled into a sector of a circle (a pie-slice shape). Wrap a sheet of paper into a cone shape; cut from edge to apex; unroll it flat.
• A plane is already flat; no unrolling is needed.

The Earth — a sphere or oblate ellipsoid — is not a developable surface. Per Gauss's Theorema Egregium (covered in the /learn/what-is-a-map-projection pillar), the intrinsic Gaussian curvature of Earth cannot be flattened onto any of these three surfaces without distortion. Every projection must therefore choose which property to preserve and which to sacrifice.

The conceptual construction: wrap one of the three developable surfaces around Earth, transfer features from Earth's surface onto the developable surface following some specific rule, then unroll the developable surface flat. Different rules produce different projections within each shape family.

Cylindrical projections

A cylinder is wrapped around Earth, typically tangent to the equator (the cylinder axis aligned with Earth's rotation axis), and features are transferred from Earth onto the cylinder. After unrolling, the cylinder becomes a rectangular map where parallels are horizontal lines and meridians are vertical lines, both spaced according to the projection's rule.

Cylindrical projections work best for areas near the equator, where the cylinder is in contact with Earth's surface. They work badly toward the poles, where the cylinder is far from the surface and distortion grows rapidly.

Key cylindrical projections:

| Projection | Property | Notes | |---|---|---| | Plate carrée | Equidistant | Simplest possible — parallels and meridians equally spaced | | Mercator | Conformal | Parallel spacing increases by sec(φ); the navigator's projection | | Lambert cylindrical | Equal-area | Parallel spacing decreases by cos(φ) | | Gall-Peters | Equal-area | Lambert cylindrical with standard parallels at ±45° | | Behrmann | Equal-area | Lambert cylindrical with standard parallels at ±30° | | Web Mercator | Conformal-ish | Spherical Mercator used by every web tile map |

Cylindrical projections share the visual signature of straight parallels and straight meridians at right angles. They are mathematically simple, easy to compute, and well suited to rectangular display areas (computer screens, printed atlas pages).

Conic projections

A cone is set with its apex above Earth and its surface tangent to a parallel (or secant to two parallels). Features are transferred from Earth to the cone, then the cone is unrolled into a pie-slice shape. Parallels become arcs of concentric circles (centred at the unrolled cone's apex); meridians become straight lines radiating from that apex.

Conic projections work best for mid-latitude regions, especially ones that are wider east-west than north-south (most continental mid-latitude landmasses fit this pattern). They work badly near the equator (the cone is far from Earth's surface there) and at the poles (the cone's apex sits above the pole, far from the surface).

Key conic projections:

| Projection | Property | Notes | |---|---|---| | Lambert conformal conic | Conformal | Standard for ICAO aeronautical charts, FAA | | Albers conic equal-area | Equal-area | USGS standard for US-wide thematic maps | | Equidistant conic | Equidistant | Parallels equally spaced from the apex | | Bonne | Equal-area (pseudoconic) | Older but still used in national series |

Conic projections are typically used in secant form, with two standard parallels chosen for the region of interest. The two parallels bound a band of zero distortion; everywhere outside the band has growing distortion. For the contiguous US, the conventional Albers/Lambert standard parallels are 29.5°N and 45.5°N — the band covers most US population.

Azimuthal projections

A plane is set tangent to Earth at a single point (or secant to a small circle around the point). Features are transferred from Earth to the plane along straight lines from a chosen perspective point (the centre of Earth for orthographic projections, the antipode of the tangent point for stereographic, or infinity for orthographic variants).

Azimuthal projections work best for areas centred on the tangent point. They preserve azimuth (direction from the tangent point) by construction — straight lines from the tangent point on the map correspond to great circles from the tangent point on Earth — but distort everything outside the central area.

Key azimuthal projections:

| Projection | Property | Notes | |---|---|---| | Stereographic | Conformal | Oldest documented projection (Hipparchus ~150 BC); used for polar regions | | Lambert azimuthal | Equal-area | ETRS89-LAEA-Europe (EPSG:3035); continental thematic | | Azimuthal equidistant | Equidistant | Distance from tangent point preserved; used in UN logo | | Orthographic | None | View from infinite distance; produces a "globe view" | | Gnomonic | None | Straight lines are great circles; navigation route planning |

Azimuthal projections are especially natural for polar regions (north or south) and for hemispheric maps centred on a continent. The Lambert azimuthal equal-area centred on Europe (EPSG:3035) is the Inspire-recommended projection for thematic European mapping.

The "pseudo" variants

Each of the three primary shape families has a relaxed "pseudo" variant that modifies the rigid construction to improve visual balance:

• Pseudocylindrical: parallels are straight horizontal lines (like cylindrical projections), but meridians are curves. Examples: Sinusoidal, Mollweide, Eckert IV/VI, Robinson, Natural Earth, Equal Earth. The curved meridians distribute shape distortion more pleasantly than straight cylindrical meridians.
• Pseudoconic: parallels are circular arcs (like conic projections), but meridians are curves. The Bonne projection is the most common example. Less widely used than pseudocylindrical or pseudoazimuthal variants.
• Pseudoazimuthal: parallels and meridians are both curves, with one tangent point retaining special significance. Examples: Aitoff, Winkel Tripel, Hammer. The Winkel Tripel projection covered in its own support is the most prominent pseudoazimuthal projection in current use.

The pseudo variants are not less valid than their parent shapes — they are simply less rigidly constrained. Modern compromise projections almost all live in the pseudo families.

Tangent vs secant

A subtlety in cylindrical and conic projections: the developable surface can be either tangent to Earth (touching at one line — the equator for tangent cylindrical, one parallel for tangent conic) or secant (cutting through Earth at two lines).

Tangent forms have one line of zero distortion; secant forms have two. Most operational projections use secant forms because they distribute distortion more evenly:

• UTM uses a secant transverse Mercator with scale factor 0.9996 at the central meridian — making the cylinder cut Earth about 180 km on either side of the central meridian.
• Lambert conformal conic for aviation typically uses two standard parallels chosen for the latitude band of the chart.
• Albers conic equal-area for the US uses standard parallels at 29.5°N and 45.5°N — chosen specifically to bracket the contiguous states.

The secant variant trades a single line of perfect accuracy for two lines of perfect accuracy with smaller total distortion across the mapped region.

The 2D taxonomy

Combining the shape axis (this article) with the property axis (conformal vs equal-area vs compromise, covered in the respective supports) gives a 2D classification grid that locates every projection:

| | Conformal | Equal-area | Compromise | Equidistant | |---|---|---|---|---| | Cylindrical | Mercator | Lambert cyl., Peters | — | Plate carrée | | Conic | Lambert conf. conic | Albers conic | — | Equidistant conic | | Azimuthal | Stereographic | Lambert azim. | Orthographic | Azimuthal equidist. | | Pseudocylindrical | — | Mollweide, Eckert IV, Equal Earth | Robinson, Natural Earth | Sinusoidal | | Pseudoazimuthal | — | Hammer | Winkel Tripel, Aitoff | — | | Pseudoconic | — | Bonne | — | — |

Empty cells in the table mean no widely-used projection occupies the specific combination — for example, no pseudocylindrical conformal projection is in common use (the conformal requirement is restrictive enough that pseudo modifications usually break it). The Mercator, Lambert conformal conic, and stereographic are the standard conformal projections; everything else is a non-conformal projection in one of the other property families.

When choosing a projection for a specific task, identify the required property (conformal for navigation, equal-area for thematic data, compromise for general reference) and the appropriate shape (cylindrical for equatorial, conic for mid-latitudes, azimuthal for focal point). The 2D grid then narrows the choice to a handful of specific options.

A worked example: choosing a projection

Consider three concrete mapping tasks and the projection choice each implies:

Task 1: A US population-density choropleth. The region is mid-latitude (24°–49°N), east-west elongate (~57° longitude), and requires area preservation for the density data. Shape: conic (matches mid-latitude east-west elongate). Property: equal-area (thematic data is per unit area). Answer: Albers conic equal-area with standard parallels at 29.5°N and 45.5°N — the USGS standard.

Task 2: A world map of trans-Pacific shipping routes. Global coverage, requires angle preservation so that bearings can be read from the map. Shape: cylindrical (works across all longitudes). Property: conformal (bearing-preservation). Answer: Mercator — exactly the historical reason the projection was invented.

Task 3: An Antarctic research-station locator. Region centred on the South Pole, requires accurate distance and direction from the pole. Shape: azimuthal (centred on the pole). Property: equidistant or equal-area depending on what matters most. Answer: polar stereographic for conformal use, or Lambert azimuthal equal-area for thematic Antarctic data.

The shape taxonomy points to the right kind of projection; the property taxonomy points to the right specific projection within that kind. The combination is the cartographer's standard decision procedure.

Sources

• Snyder, Map Projections — A Working Manual (USGS Prof. Paper 1395) — the comprehensive reference.
• Snyder, Flattening the Earth (University of Chicago Press, 1993) — historical development of each projection family.
• Robinson, Snyder, Voxland, Album of Map Projections (USGS Bulletin 1453, 1989) — visual catalogue with sample maps for every major projection.

For the property axis of classification, see /learn/conformal-projections and /learn/equal-area-projections; for specific projections, see the individual supports linked from the 2D taxonomy table above.