The Mercator projection is a cylindrical map projection presented by the Flemish (Belgian) geographer and cartographer Gerardus Mercator, in 1569. It became the standard map projection for nautical purposes because of its ability to represent lines of constant course, known as rhumb lines or loxodromes, as straight segments. While the linear scale is constant in all directions around any point, thus preserving the angles and the shapes of small objects (which makes the projection conformal), the Mercator projection distorts the size and shape of large objects, as the scale increases from the Equator to the poles, where it becomes infinite.

Properties and historical details

Mercator's 1569 edition was a large planisphere measuring 202 by 124 cm, printed in eighteen separate sheets. As in all cylindrical projections, parallels and meridians are straight and perpendicular to each other. In accomplishing this, the unavoidable east-west stretching of the map, which increases as distance away from the equator increases, is accompanied by a corresponding north-south stretching, so that at every point location, the east-west scale is the same as the north-south scale, making the projection conformal. A Mercator map can never fully show the polar areas, since linear scale becomes infinitely high at the poles. Being a conformal projection, angles are preserved around all locations. However scale varies from place to place, distorting the size of geographical objects and conveying a distorted idea of the overall geometry of the planet. At latitudes greater than 70° north or south, the Mercator projection is practically unusable.

All lines of constant bearing (rhumb lines or loxodromes — those making constant angles with the meridians), are represented by straight segments on a Mercator map. This is precisely the type of route usually employed by ships at sea, where compasses are used to indicate geographical directions and to steer the ships. The two properties, conformality and straight rhumb lines, make this projection uniquely suited to marine navigation: courses and bearings are measured using wind roses or protractors, and the corresponding directions are easily transferred from point to point, on the map, with the help of a parallel ruler or a pair of navigational squares.

The name and explanations given by Mercator to his world map (Nova et Aucta Orbis Terrae Descriptio ad Usum Navigantium Emendata: "new and augmented description of Earth corrected for the use of sailors") show that it was expressly conceived for the use of marine navigation. Although the method of construction is not explained by the author, Mercator probably used a graphical method, transferring some rhumb lines previously plotted on a globe to a square graticule, and then adjusting the spacing between parallels so that those lines became straight, making the same angle with the meridians as in the globe.

The development of the Mercator projection represented a major breakthrough in the nautical cartography of the 16th century. However, it was much ahead of its time, since the old navigational and surveying techniques were not compatible with its use in navigation. Two main problems prevented its immediate application: the impossibility of determining the longitude at sea with adequate accuracy and the fact that magnetic directions, instead of geographical directions, were used in navigation. Only in the middle of the 18th century, after the marine chronometer was invented and the spatial distribution of magnetic declination was known, could the Mercator projection be fully adopted by navigators.

Several authors are associated with the development of Mercator projection:

German Erhard Etzlaub (c. 1460–1532), who had engraved miniature "compass maps" (about 10×8 cm) of Europe and parts of Africa, latitudes 67°–0°, to allow adjustment of his portable pocket-size sundials, was for decades declared to have designed "a projection identical to Mercator’s".

Portuguese mathematician and cosmographer Pedro Nunes (1502–1578), who first described the loxodrome and its use in marine navigation, and suggested the construction of several large-scale nautical charts in the cylindrical equidistant projection to represent the world with minimum angle distortion (1537).

English mathematician Edward Wright (c. 1558–1615), who formalized the mathematics of Mercator projection (1599), and published accurate tables for its construction (1599, 1610).

English mathematicians Thomas Harriot (1560–1621) and Henry Bond (c.1600–1678) who, independently (c. 1600 and 1645), associated the Mercator projection with its modern logarithmic formula, later deduced by calculus.

Mathematics of the projection

The following equations determine the x and y coordinates of a point on a Mercator map from its latitude φ and longitude λ (with λ₀ being the longitude in the center of map):

:

This is the inverse of the Gudermannian function:

:

The scale is proportional to the secant of the latitude φ, getting arbitrarily large near the poles, where φ = ±90°. Moreover, as seen from the formulas, the pole's y is plus or minus infinity.

The scaling factor for distances measured along lines of constant latitude φ is sec(φ) - this gives a scaling factor that is 1 at the equator (φ=0) and approaches infinity near the poles (φ = +/- 90 degrees). The vertical distance on the map between two points with the same longitude is more complex and depends on their respective latitudes - it is:

::::$\backslash ln\; \backslash left(\; \backslash frac\{\backslash tan(\backslash phi\_1)+\backslash sec(\backslash phi\_1)\}\{\backslash tan(\backslash phi\_2)+\backslash sec(\backslash phi\_2)\}\; \backslash right)$

Derivation of the projection

Assume a spherical Earth. (It is actually slightly flattened, but for small-scale maps the difference is immaterial. For more precision, interpose conformal latitude.) We seek a transform of longitude-latitude (λ, φ) to Cartesian (x, y) that is "a cylinder tangent to the equator" (i.e. x = λ) and conformal, so that:

:$\backslash frac\{\backslash partial\; x\}\{\backslash partial\; \backslash lambda\}\; =\; \backslash cos(\backslash varphi)\; \backslash frac\{\backslash partial\; y\}\{\backslash partial\; \backslash varphi\}$

:$\backslash frac\{\backslash partial\; y\}\{\backslash partial\; \backslash lambda\}\; =\; -\backslash cos(\backslash varphi)\; \backslash frac\{\backslash partial\; x\}\{\backslash partial\; \backslash varphi\}$

From x = λ we get

:$\backslash frac\{\backslash partial\; x\}\{\backslash partial\; \backslash lambda\}\; =\; 1$

:$\backslash frac\{\backslash partial\; x\}\{\backslash partial\; \backslash varphi\}\; =\; 0$

giving

:$1\; =\; \backslash cos(\backslash varphi)\; \backslash frac\{\backslash partial\; y\}\{\backslash partial\; \backslash varphi\}$

:$0\; =\; \backslash frac\{\backslash partial\; y\}\{\backslash partial\; \backslash lambda\}$

Thus y is a function only of φ with $y\text{'}=\backslash sec\backslash varphi$ from which a table of integrals gives

:$y\; =\; \backslash ln(|\backslash sec(\backslash varphi)\; +\; \backslash tan(\backslash varphi)|)\; +\; C.\backslash ,$

It is convenient to map φ = 0 to y = 0, so take C = 0.

Uses

As on all map projections, shapes or sizes are distortions of the true layout of the Earth's surface. The Mercator projection exaggerates areas far from the equator. For example:

Greenland takes as much area on the map as Africa, when in fact Africa's area is approximately 14 times greater than Greenland.

Alaska takes as much area on the map as Brazil, when Brazil's area is actually more than 5 times that of Alaska.

Finland appears with a greater north-south extent than India, although India's is greater.

Antarctica appears as the biggest continent, although it is actually the 5th in terms of area.

Although the Mercator projection is still used commonly for navigation, due to its unique properties, cartographers agree that it is not suited to general reference world maps due to its distortion of land area. Mercator himself used the equal-area sinusoidal projection to show relative areas. As a result of these criticisms, modern atlases no longer use the Mercator projection for world maps or for areas distant from the equator, preferring other cylindrical projections, or forms of equal-area projection. The Mercator projection is still commonly used for areas near the equator, however, where distortion is minimal.

Arno Peters stirred controversy when he proposed what is now usually called the Gall-Peters projection as the alternative to the Mercator. The projection is a specific parameterization of the cylindrical equal-area projection. A 1989 resolution by seven North American geographical groups decried the use of all rectangular-coordinate world maps, including the Mercator and Gall-Peters.

Google Maps uses a variant of the Mercator projection for its map images. Despite its obvious scale variation at small scales, the projection is well-suited as an interactive world map that can be zoomed seamlessly to large-scale (local) maps, where there is relatively little distortion due to the projection's near-conformality. (Google Satellite Maps, on the other hand, used a plate carrée projection until July 22, 2005.)

The Google Maps tiling system displays most of the world at zoom level 0 as a single 256 pixel-square image, excluding the polar regions. Since the Mercator coordinate x varies over 2π, the other coordinate is limited to -π ≤ y ≤ π. Because

$\backslash frac\{1\}\{2\}\; \backslash ln\backslash bigg(\backslash frac\{1+\backslash sin(\backslash varphi)\}\{1-\backslash sin(\backslash varphi)\}\backslash bigg)\; =\; \backslash pm\backslash pi\; \backslash Rightarrow\; \backslash varphi\; =\; \backslash pm\backslash arcsin\backslash bigg(\backslash frac\{\backslash mathrm\{e\}^\{2\; \backslash pi\}-1\}\{\backslash mathrm\{e\}^\{2\; \backslash pi\}+1\}\backslash bigg)$

the corresponding latitude extrema are φ = ±85.05113°. Latitude values outside this range are mapped using a different relationship that doesn't diverge at φ = ±90°.

See also

Cartography

Cassini projection

Dymaxion map

Equirectangular projection

Gall-Peters projection with resolution regarding the use of rectangular world maps

Gnomonic projection

Lambert conformal conic projection (used extensively in aviation)

Map projection

Mollweide projection

Nautical chart

Robinson projection

Reversed map

Tissot's Indicatrix

Transverse Mercator projection

Winkel Tripel projection

Notes

References

This paper can be downloaded from USGS pages.

Needham, Joseph (1986). Science and Civilization in China: Volume 3; Mathematics and the Sciences of the Heavens and the Earth. Taipei: Caves Books Ltd.

Needham, Joseph (1986). Science and Civilization in China: Volume 4, Physics and Physical Technology, Part 3, Civil Engineering and Nautics. Taipei: Caves Books Ltd.

Google Maps Coordinates

External links

Ad maiorem Gerardi Mercatoris gloriam - contains high resolution images of the 1569 world map by Mercator.

Table of examples and properties of all common projections, from radicalcartography.net.

An interactive Java Applet to study the metric deformations of the Mercator Projection.

Category:Cartographic projections Category:Conformal mapping

Name	Ryuichi
Background	solo_singer
Birth name
Born	May 20, 1970 Yamato, Kanagawa, Japan
Instrument	Vocal, acoustic guitar, guitar
Genre	Rock, pop
Occupation	Singer, composer, record producer, actor, author, race car driver
Years active	1989–present
Label	Columbia Music Entertainment, Avex Group
Associated acts	Luna Sea, Tourbillon
Url	kawamura-fc.com

is a Japanese singer-songwriter, musician, actor, and record producer. He is best known as the lead singer of the Japanese rock band Luna Sea. After the group disbanded in 2000, Ryuichi went on to a successful solo career. In 2005 he formed Tourbillon with Luna Sea guitarist Inoran and Hiroaki Hayama. He has recently rejoined Luna Sea, as they reunited in August 2010.

Ryuichi's debuting album Love topped the Japanese Oricon chart with sales of around 1,021,000 copies, making him the only male solo artist ever to have an album to sell over one million copies within its initial week in Oricon charts history. Also, that same album sold over 2,788,000 copies, therefore it holds the record for best selling male solo album. In 1997 alone, with his mini and full length albums and four singles (of which "Glass" sold over one million itself), he sold over six million records.

Career

1989–2000: Luna Sea, Solo Debut

Ryuichi Kawamura started his musical career as Luna Sea vocalist at the age of 19 while still in high school, when he was invited by Inoran to join the band. In 1992, hide of X Japan found them on the indie circuit, he eventually got them signed to X Japan co-founder Yoshiki Hayashi's Extasy Records. The band became commercially successful in the 90s. During Luna Sea's temporary break in 1997, Ryuichi started his solo career. Going by his full name, Ryuichi Kawamura, he changed his vocals from the powerful rock sound to more of a pop idol sound. That year, he released four singles, one mini-album and his first full album, Love. The album includes his cover version of the song "Namida Iro," which was formerly sung by Japanese female singer Noriko Sakai with his songwriting. Love topped the Japanese Oricon charts with the sales of around 1,021,000 copies, making him the only solo male artist ever to have an album to sell over one million copies within its initial one week in Oricon charts history.

2009–2010: Sora, Luna Sea Reunion, Debut in Cambodia

Following an overseas performance in Hong Kong on March 2009, Ryuichi released a new single "Heroine" (ヒロイン） on February 4, which was a collaborated his work with Tetsuro Oda. He then released the album Piano on April 1, which included the environmental-conservative themed single "Midori no Uta". Following the release of Piano, Ryuichi has currently been busy with several media promotional appearances, including a recent tour in Hong Kong which marked his first international tour as a solo artist.

In July, Ryuichi released a new photo exhibition book called "Japanesque". He went on to perform international tour in Taiwan in late August.

On September 30, 2009, Ryuichi released a new single "Brilliant Stars" including a tie-up song with the jewelery brand GemCeree. As with a new contract with Avex Group (after leaving Columbia Music Entertainment), Ryuichi performed at the "Hard na Yaon 2009" concert that same month along with many other artists from the company.

Following a single "Dakishimete (抱きしめて)" and the album Sora in February 2010, according to Ryuichi's official website, he would co-star in the second Broadway Musical Chicago in June. Also ,Ryuichi had held the tour "No Mic, No Speakers Concert #003" from August through September. On August 13 he performed at the Jack in the Box 2010 Summery festival. Also in August, he appeared with the other members of Luna Sea at a press conference in Hong Kong, where they officially announced their reunion and their "20th Anniversary World Tour Reboot -to the New Moon-".

On October 17, Ryuichi appeared in a television documentary, broadcast on BS Fuji, called Kawamura Ryuichi x Garyuu, where he traveled to the famous Angkor Wat. Awed by the structure and it's impressive echoes, he decided that he wants to perform there in the near future, playing acoustic guitar and piano but singing without the use of a microphone. It appears that Kawamura has already composed a new song inspired by the experience, though the lyrics have not yet been written. Naturally, he hopes to perform the song at Angkor Wat in 2011.

On a tidbit note ,Ryuichi also participated in Bobby Kim's (South Korea R&B; singer) album "My Soul" ,writing Japanese lyrics for the song "追憶の風".

2011

While reuniting with Luna Sea for the band's new album and DVD releases this year ,new projects for Ryuichi have already been planned. On December 23 ,Tokyograph reported that Kawamura and his fellow Luna Sea member Sugizo are collaborating with musical group m.o.v.e ,best known for making music for popular anime ,"Initial D" to provide a theme song for the upcoming movie「ガクドリ」(Gakudori) which is scheduled to open in theaters in Spring. Their song will be calked "oveRtaKerS," and will feature Kawamura on vocals and Sugizo on guitar.As of now ,the "oveRtaKerS" song has not been given a release date yet.

On January 7th ,Ryuichi announced he is aiming to outdo himself with a special solo concert at the Nippon Budokan on March 13, 2011. During the concert, which is titled "100の物語 〜終章（エピローグ)" (Genkai no Sono Saki e: 100 no Monogatari ~Epilogue~), he will perform a total of 100 songs over a span of more than 6 hours. Back in February 2008 ,Kawamura held a concert titled "70 no Monogatari" where he sang 71 songs from his original albums over a span of 4 hours at the same venue. This time, he will be increasing the setlist by nearly 50%. Tickets for the concert will be available on February 5.

In addition ,Ryuichi will co-star in the stage musical ,"Wuthering Heights" (known as "Arashigaoka") in Japan. Though several previous adaptions were based on Emily Bronte's classic novel but this one will be a new adaptation written by playwright Sanae Iijima and directed by Nobuhiro Nishikawa. Kawamura will take a lead role of the revenge-seeking Heathcliff and perform along with popular Japanese stars including voice actress Aya Hirano and singer Natsumi Abe. The musical will be performed at Tokyo's Akasaka ACT Theater on July 6-24, then at Osaka's Theater Drama City on July 25-31.

Personal life

Ryuichi was introduced to Kumi Sano by a mutual friend in early 2005, and several months later they began dating. Sano had been crowned Miss Japan in 2002. Although the pair stated that they had no plans for a wedding ceremony or a reception, a representative visited the local city office to register their marriage on January 23, 2006. Kumi gave birth to their first son on June 18, 2009.

He is good friends with Inoran. Together, they are avid car racers. Ryuichi also likes boxing and surfing, and says that he plans to open a surfing store once he retires from his singing career.

Discography

Albums

Cranberry Soda (June 21, 1997)

Love (November 22, 1997)

(December 19, 2001) (July 17, 2002)

Very Best of Songs... (September 26, 2002)

Dear... (March 24, 2004)

Vanilla (April 7, 2004)

Evergreen (May 24, 2006)

Orange (June 20, 2007)

Evergreen ~Anniversary Edition~ (December 5, 2007)

Piano (January 4, 2009)

Sora (February 24, 2010)

"Untitled self-covered album" (March 9,2011)

Singles

"I Love You/Cielo" (February 21, 1997)

"Glass" (April 23, 1997)

"Beat/Ki•se•ki" (July 18, 1997)

"Love is/Xyz" (October 15, 1997)

"Ne" (April 25, 2001)

(June 21, 2001) (August 22, 2001) (October 24, 2001) (November 21, 2001)

"Sugar Lady" (April 24, 2002)

"F114B" (March 3, 2004, official fan club members only)

"Spoon/Missing You" (March 3, 2004)

(April 18, 2007)

"Once Again" (October 31, 2007)

(2008, available only at Ryuichi's official website) (February 4, 2009)

"Brilliant Stars" (September 30, 2009)

(January 20, 2010)

DVD/VHS

Time of Legend 1997-2002

RK 10th Anniversary Action Final Yuki no Budokan ~70 Stories~ (May 2008)

Book

"VOICES" (autobiography - December 24,2010)

Achievement/Awards

Drama Academy Award - Best Newcomer (Spring 1997)

Best Japanese Male Singer (1997)

Filmography

Movies

Buraikan

Picaresque (2002)

TV dramas

Futari (1997)

Narita Rikon (1997)

Number One (2001)

Kowloon de Aimashou (2002)

References

External links

Official Website Official Shop

Category:1970 births Category:Living people Category:Luna Sea members Category:Japanese male singers Category:Japanese rock singers Category:Japanese singer-songwriters Category:Japanese actors Category:Japanese racecar drivers Category:People from Kanagawa Prefecture