Equidistant Projection

Equidistant projection is a cartographic projection with the property of preserving scale along certain lines.

Content

1 Cylindrical Equidistant Projection
- 1.1 Mathematical Definition
2 Conical equidistant projection
- 2.1 Mathematical expression
3 See also
4 References

Cylindrical Equidistant Projection

The Blue Marble : Land Surface, Ocean Color and Sea Ice as an Example of Equidistant Projection

With this projection, both angles and area are distorted and the length scale in one of the main directions is preserved unchanged - a = const or b = const. Projection is used in modern geographic information systems , because geographic coordinates can be directly entered into a map. Today, along with the Mercator projection, the equidistance cylindrical projection is the de facto standard in computer applications.

Mathematical Definition

The following equations determine the x , y coordinates of a point with latitude φ and longitude λ for a projection with a fixed base point in (φ ₀ , λ ₀ ):

x=(\lambda -\lambda _{0}),

{\ displaystyle x = (\ lambda - \ lambda _ {0}),}

y=(\varphi -\varphi _{0})

{\ displaystyle y = (\ varphi - \ varphi _ {0})}

Plate-Carré - a variant of equidistant cylindrical projection with a base point (φ ₀ , λ ₀ ) = (0, 0)

Conical Equidistant Projection

In a conic equidistant projection, the scale is usually maintained along the meridians, as well as along some given parallel or pair of parallels.

Math expression

R _cp = 6371007 m. - the average radius of the Earth (WGS-84);

W - map width (in meters or pixels);

H - map height (in meters or pixels);

B - latitude;

L is the geographical longitude;

M - map scale (m / m or pix / m, usually M << 1), for the map of Russia it is recommended that M = H / 5,000,000 pixels / m;

L _c - middle meridian

L _m - meridian passing through the lower left corner of the map

B _m - latitude at the intersection of the central meridian with the lower edge of the map

Direct conversion:

y_{c}=MR_{cp}({\frac {\pi }{2}}-B_{m})

{\ displaystyle y_ {c} = MR_ {cp} ({\ frac {\ pi} {2}} - B_ {m})}

\alpha =\mathop {\rm {arctg}} {\frac {W}{2y_{c}}}

{\ displaystyle \ alpha = \ mathop {\ rm {arctg}} {\ frac {W} {2y_ {c}}}}

\beta =\alpha {\frac {(L-L_{c})}{(L_{c}-L_{m})}}

{\ displaystyle \ beta = \ alpha {\ frac {(L-L_ {c})} {(L_ {c} -L_ {m})}}}

R=y_{c}{\frac {({\frac {\pi }{2}}-B)}{({\frac {\pi }{2}}-B_{m})}}

{\ displaystyle R = y_ {c} {\ frac {({\ frac {\ pi} {2}} - B)} {({\ frac {\ pi} {2}} - B_ {m})}} }

{\begin{matrix}x=W/2+R\sin \beta \end{matrix}}

{\ displaystyle {\ begin {matrix} x = W / 2 + R \ sin \ beta \ end {matrix}}}

{\begin{matrix}y=y_{c}-R\cos \beta \end{matrix}}

{\ displaystyle {\ begin {matrix} y = y_ {c} -R \ cos \ beta \ end {matrix}}}

for computer graphics:

{\begin{matrix}y=H-y_{c}+R\cos \beta \end{matrix}}

{\ displaystyle {\ begin {matrix} y = H-y_ {c} + R \ cos \ beta \ end {matrix}}}

Equidistant Projection

Content

Cylindrical Equidistant Projection

Mathematical Definition

Conical Equidistant Projection

Math expression

See also

Links

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