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Umphrey's McGee
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Umphrey's McGee
From left to right: Jake Cinninger, Kris Myers, Ryan Stasik and Brendan Bayliss performing in April 2007.
Background information
Origin
Notre Dame, Indiana / Chicago, Illinois, United States
Genres
Rock, progressive rock, jam, math rock, jazz fusion, funk rock
Years active
1997–present
Associated acts
Widespread Panic
Website
Official website
Members
Brendan Bayliss
Joel Cummins
Ryan Stasik
Andy Farag
Jake Cinninger
Kris Myers
Past members
Mike Mirro
Umphrey's McGee is an American jam band originally from Notre Dame, Indiana. The band's approach shares many elements with groups like Phish and the Grateful Dead such as varying set lists, improvisation, playing two sets per night, and allowing fans to tape concerts, they are much more influenced musically by progressive rock artists such as King Crimson, Yes, Pink Floyd, Frank Zappa, The Mahavishnu Orchestra, and early Genesis, as well as heavy metal bands such as Iron Maiden and Van Halen. The band also identifies The Police, The Beatles, and Led Zeppelin as primary influences.[1] UM experiments with many musical styles, including rock, metal, funk, jazz, blues, reggae, electronic, bluegrass, and folk.
Contents
[hide]
1 History
1.1 Early years (1997-2000)
1.2 Full line-up with Mike Mirro (2001-2002)
1.3 Current line-up (2003-present)
2 Members
3 Discography
3.1 Studio releases
3.2 Live releases
3.3 DVD releases
4 References
5 External links
History[edit]
Early years (1997-2000)[edit]
Umphrey's McGee at 2006 Bonnaroo Music Festival
Formed at the University of Notre Dame in December 1997 by guitarist/vocalist Brendan Bayliss, bassist Ryan Stasik, keyboardist Joel Cummins, and drummer Mike Mirro, Umphrey's McGee combined members of Tashi Station and Stomper Bob, two Notre Dame rock bands. Early concerts consisted of both originals and cover songs, including Guns N' Roses' "Patience" and Vince Guaraldi's Peanuts theme "Linus and Lucy" as well as songs by Phish, moe., and The Grateful Dead. According to Cummins, "The origins are of an actual person. Brendan Bayliss, our other lead singer and guitar player has a cousin... and his name is actually Humphrey Magee but it’s not possessive with the apostrophe s..."[2]
In mid-1998, after only 8 months together, the band released their debut album, Greatest Hits Vol. III. Having a released studio disc allowed them to more readily book live performances. Though long out of print, the album contains songs that remain staples of Umphrey's live sets, such as "Divisions", "Phil's Farm", "FF", and "All in Time".
Shortly after the release of the album, the band added a fifth member, percussionist Andy Farag. Farag's father became the band's agent, and a second pressing of Greatest Hits Vol. III featured Farag in the album's inner sleeve and credits. By the end of the year, Umphrey's McGee, along with peers Ali Baba's Tahini, were one of the most popular bands in the South Bend/Notre Dame area.[citation needed] They began performing outside of the area at colleges and house parties, allowing listeners to tape and trade their music freely. In 1998, the band released their first live album, Songs for Older Women.
Guitarist Jake Cinninger was added to the band in September 2000. Cinninger brought a heaviness to the Umphrey's sound, as well as a large repertoire of original music, much of which comprised the bulk of the catalog from his previous band, Ali Baba's Tahini. The band also adopted several songs written by Ali Baba's Tahini frontman Karl Engelmann (who now fronts Asheville-based rock band Mother Vinegar). Shortly after Cinninger's arrival, the band released another live album, One Fat Sucka, which contained live performances recorded in the summer and fall of 2000.
Full line-up with Mike Mirro (2001-2002)[edit]
In 2001, the band began practicing intense improvisational exercises. One of their first productive sessions took place in a hotel in the "Jimmy Stewart Ballroom", prompting the band to call their onstage improvisational excursions "Jimmy Stewart". This form of improvisation differs in approach from the methods utilized by many of their jam band peers.[3]
Jake Cinninger and Joel Cummins released solo albums in 2001 and 2002, respectively. At least two songs from each of those albums have made their way into Umphrey's concert repertoire.[citation needed]
June 2002 saw the release of the band's first proper recording, Local Band Does OK. That summer, the first annual Bonnaroo Music Festival took place in rural Manchester, Tennessee, and featured over 30 bands performing in front of nearly 100,000 people.[citation needed] Umphrey's played to their largest audience to date. The band sold more CDs than any other artist on a bill that included Widespread Panic, Trey Anastasio, and Norah Jones.[citation needed]
The band almost broke up in late 2002 when drummer Mike Mirro announced he was leaving the band to attend medical school.[citation needed] After hundreds of audition tapes were reviewed by the percussionist Andy Farag, the band settled on the first one they received, which was from Kick the Cat drummer Kris Myers.[citation needed]
Current line-up (2003-present)[edit]
2003 was a year of change for UM. New drummer Kris Myers had to learn their extensive repertoire of originals and covers. The band continued their grueling tour schedule, logging over 150 shows for the year. The band began experimenting with a new "Lego" style of songwriting which involved piecing together original sections and lifted pieces of "Jimmy Stewart" improvisations to create new songs, such as Ocean Billy, #5, and Wife Soup. That summer, UM entered the recording studio in Chicago for their first studio album with Myers. In the Fall, the UM Live program was started by "Sound Caresser" Kevin Browning. Each night's show would be offered on CD, for sale directly after the show. This soon translated into a partnership with Disc Logic for online distribution, and a few years later all of the band's shows would be moved to UMLive.net. In November, the band released their first DVD, Live from the Lake Coast, and later that month their first official release featuring Kris Myers, Local Band Does Oklahoma (Recorded on April 23 in Oklahoma City).
In 2004, the band released Anchor Drops, recorded in several different studios throughout the Chicago area. Shortly before the release of the album, the band scored a distribution deal for both Anchor Drops and Local Band Does OK, and for the first time, Umphrey's McGee music was available in stores all over the country. The album was said to be the "sound of a band reaching maturity".[4][specify] Once again, Umphrey's headed to Bonnaroo, playing a coveted late-night spot. The band performed to over 20,000 people, many of whom had never heard the band before. In July, the band was declared by Rolling Stone to "have become odds-on favorites in the next-Phish sweepstakes."[5]
Starting in 2005, the band began producing Umphrey's McGee Podcasts, an effort to provide fans with highlights of recent shows. Each podcast is around 75 minutes in length and is released about twice a month, and it has 20,000 listeners. "In the Kitchen" won the Jammy Award for Song of the Year. That evening, the band performed with Huey Lewis for the first time (they were also joined by Mavis Staples, Sinead O'Connor, and Jeff Coffin). That summer, the band participated in the Big Summer Classic tour, which was the first time UM played Red Rocks.
On April 4, 2006, Umphrey's released their fourth studio effort, Safety In Numbers. They appeared on Jimmy Kimmel Live! alongside Huey Lewis in support of the new album on May 18, 2006. The band was featured at many of the major U.S. festivals throughout the summer, playing 129 shows before a three-night stand in Chicago for New Year's Eve. The Bottom Half, a 2-disc album compiled from the Safety in Numbers sessions, was released a year later on April 3, 2007. Near the end of 2006, Umphrey's began a program to provide complimentary tickets to audience tapers in exchange for them distributing the recordings digitally online.
A live double-album titled Live at the Murat was released on October 16, 2007. The album was recorded over 2 nights at the Egyptian Room in the Murat Centre in Indianapolis, Indiana on April 6–7, 2007. It received a 4-star review from Rolling Stone. The album also won a Jammy for best live album.[6]
The band's 2008 Halloween show was the first to feature a mash-up theme. Riffing on the idea of mash-ups popularized by DJs, they arrange new composition from 2-3 songs (a mix of covers and originals). This tradition has continued on Halloween every year since.
Mantis was released on January 20, 2009. The album concentrated more on the progressive style that was associated with Anchor Drops. Unlike previous albums, most of the songs went unplayed until the album was released. Later in the year, Jefferson Waful took over duties as lighting director.
In 2010, the band debuted UMBowl, an interactive concert in 4 sets (or quarters), each with a different theme. Ticket purchasers are sent ballots to vote for the selections for two quarters, while the other two quarters are based upon interactive text submissions/votes. October 3 saw the debut of the Stew Art (S2) events. Each S2 event, limited to 50 fans, was an hour-long event that had the band improvising based upon text submissions from the audience. The S2 has since become a regular feature for UMBowl.
The band's seventh studio album Death By Stereo, was released September 13, 2011. The album features a mix of new songs as well as old songs that the band has used in their live rotation for years, such as "Hajimemashite" and "The Floor". With the album's pre-order came a set of packages for super fans, including fishing trips and bar crawls with the band. The Bill Graham For A Day option for a private show became a reality, and has since inspired several follow-up events.
In 2013, the band debuted "Headphones and Snowcones", a program available at their concerts which provides a soundboard feed delivered to a set of headphones via wireless receiver.
Original drummer Mike Mirro died on January 30, 2014, at age 36.[7]
On June 10, 2014, UM released their eighth studio album, Similar Skin, their first album released on their new music label Nothing Too Fancy Music.
On April 7, 2015, UM released their ninth studio album, The London Session, an album recorded in one day at Abbey Road Studios.[8]
On November 11, 2016, UM released their tenth studio album ZONKEY, an experimental album consisting entirely of mashups from artists such as Beastie Boys, Beck, ZZ Top, Radiohead, AC/DC, Gorillaz, Nirvana, MGMT, and Ted Nugent, among others. Some of the mashups have been performed live by the band for years while some are entirely new compositions for this record.
On December 29–31, 2016, UM returned to Chicago for a 3-show New Year's Eve run, their first such run in Chicago for the holiday since performing in the city for New Year's Eve each year from 2000-2010. The shows took place at the Riviera Theatre and Aragon Ballroom.
Members[edit]
Current members
Brendan Bayliss - guitar, vocals (1997–present)
Joel Cummins - keyboards, vocals (1997–present)
Ryan Stasik - bass guitar, no vocals (1997–present)
Andy Farag - percussion (1998–present)
Jake Cinninger - guitar, vocals (2000–present)
Kris Myers - drums, vocals (2003–present)
Former members
Mike Mirro - drums, vocals (1997-2002)
Discography[edit]
Main article: Umphrey's McGee discography
Studio releases[edit]
Greatest Hits Vol. III (1998)
Local Band Does OK (2002)
Anchor Drops (2004)
Safety In Numbers (2006)
The Bottom Half (2007)
Mantis (2009)
Death By Stereo (2011)
Similar Skin (2014)
The London Session (2015)
Zonkey (2016)
Live releases[edit]
Songs for Older Women (1999)
One Fat Sucka (2001)
Local Band Does OKlahoma (2003)
Live at the Murat (2007)
Jimmy Stewart 2007 (2008)
Hall of Fame Class of 2010 (2011)
Hall of Fame Class of 2011 (2012)
Hall of Fame Class of 2012 (2013)
Hall of Fame Class of 2013 (2014)
Hall of Fame Class of 2014 (2015)
Hall of Fame Class of 2015 (2016)
DVD releases[edit]
Live from the Lake Coast (2002)
Wrapped Around Chicago - New Year's Eve at The Riviera (2005)
Soundstage: Umphrey's McGee - Live (2009)
Live at Summer Camp 2011 (2011)
Live from The Pageant: NYE Weekend 2011 (2012)
Live at Red Rocks (2012)
Live at the Tabernacle (2012)
STUMP Tour (2013)
UM Halloween: Live from the Riverside (2013)
Rage.Riv.Repeat. (2014)
References[edit]
Jump up ^ "Umphrey's McGee - Band". Umphreys.com. Retrieved 2014-02-05.
Jump up ^ "Sound Waves Magazine". swaves.com. 2008-04-08. Retrieved 2017-07-07.
Jump up ^ "Jimmy Stewart writeup". Web.archive.org. Archived from the original on April 10, 2010. Retrieved 2014-02-05.
Jump up ^ "Anchor Drops - Umphrey's McGee". www.jambands.com. Retrieved 2016-03-30.
Jump up ^ "Music News: Latest and Breaking Music News". Rolling Stone. Retrieved 2014-02-05.
Jump up ^ "Album Reviews and Ratings". Rolling Stone. 2014-01-28. Retrieved 2014-02-05.
Jump up ^ "Ex-Umphrey's McGee drummer dies". South Bend Tribune. Schurz Communications. 1 February 2014. Retrieved 9 March 2015.
Jump up ^ http://umphreys.com/2015/02/london-session/
External links[edit]
Wikimedia Commons has media related to Umphrey's McGee.
Official website
Youtube Channel
Setlist Archive
Umphrey's McGee at Rolling Stone
Umphrey's McGee Live Recordings at Internet Archive
Interview with Joel Cummins of Umphrey's McGee Ace Weekly, February 1, 2011
Umphrey's McGee Live Concert Photos from 1.29.2011 at the Electric Factory in Philadelphia
Live at North Coast Music Festival, 2010
Authority control
MusicBrainz: 3826a6e0-9ea5-4007-941c-25b9dd943981
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College Park, Maryland
From Wikipedia, the free encyclopedia
Jump to navigation Jump to search
City in Maryland, United States
College Park, Maryland
City
City of College Park
Downtown College Park

Flag
Seal
College Park
Location within the state of Maryland
College Park
College Park (the US)
Show map of Maryland Show map of the US Show all
Coordinates: 38°59′48″N 76°55′39″W / 38.99667°N 76.92750°W / 38.99667; -76.92750Coordinates: 38°59′48″N 76°55′39″W / 38.99667°N 76.92750°W / 38.99667; -76.92750
Country
United States
State
Maryland
County
Prince George's
Founded
1856
Incorporated
1945
Government
• Mayor
Patrick L. Wojahn[1]
Area[2]
• Total
5.68 sq mi (14.71 km2)
• Land
5.64 sq mi (14.61 km2)
• Water
0.04 sq mi (0.10 km2)
Elevation
69 ft (21 m)
Population (2010)[3]
• Total
30,413
• Estimate (2016)[4]
32,275
• Density
5,400/sq mi (2,100/km2)
Time zone
EST (UTC−5)
• Summer (DST)
EDT (UTC−4)
ZIP Codes
20740–20742
Area code
301
FIPS code
24-18750
GNIS feature ID
2390578
Website
www.collegeparkmd.gov
The City of College Park is in Prince George's County, Maryland[5] and is about 4 miles (6.4 km) from the northeast border of Washington, D.C. The population was 30,413 at the 2010 United States Census. It is best known as the home of the University of Maryland, College Park, and since 1994 the city has also been home to the National Archives at College Park, a facility of the U.S. National Archives, as well as to the National Oceanic and Atmospheric Administration (NOAA) Center for Weather and Climate Prediction (NCWCP).[6]
College Park's United States Postal Service ZIP Codes are 20740, 20741 (Berwyn Heights; North College Park) and 20742 (University of Maryland).
Contents
1 History
1.1 Development
1.2 Later events
2 Demographics
2.1 2010 census
2.2 2000 census
3 Government
4 Geography
4.1 Climate
4.2 Neighborhoods
4.3 Bordering areas
5 Economy
6 Transportation
6.1 Airport
6.2 Major highways
6.3 Public transportation
7 Historic sites
8 Media
9 Education
9.1 Colleges and universities
9.2 Primary and secondary schools
9.2.1 Public schools
9.2.2 Private schools
10 Museums
11 City-student politics
12 References
13 Further reading
14 External links
History[edit]
Development[edit]
College Park was developed beginning in 1889 near the Maryland Agricultural College (later the University of Maryland) and the College Station stop of the Baltimore and Ohio Railroad. The suburb was incorporated in 1945 and included the subdivisions of College Park, Lakeland, Berwyn, Oak Spring, Branchville, Daniel's Park, and Hollywood.
The original College Park subdivision was first plotted in 1872 by Eugene Campbell. The area remained undeveloped and was re-platted in 1889 by John O. Johnson and Samuel Curriden, Washington real estate developers. The original 125-acre (0.51 km2) tract was divided into a grid-street pattern with long, narrow building lots, with a standard lot size of 50 feet (15 m) by 200 feet (61 m). College Park developed rapidly, catering to those who were seeking to escape the crowded Washington, D.C., as well as to a rapidly expanding staff of college faculty and employees.
College Park originally included single-family residences constructed in the Shingle, Queen Anne, and Stick styles, as well as modest vernacular dwellings. Commercial development increased in the 1920s, aided by the increased automobile traffic and the growing campus along Baltimore Avenue / Route 1.
By the late 1930s, most of the original subdivision had been partially developed. Several fraternities and sororities from the University of Maryland built houses in the neighborhood. After World War II, construction consisted mostly of infill of ranch and split-level houses. After incorporation in 1945, the city continued to grow, and a municipal center was built in 1959.[7]
The Lakeland neighborhood was developed beginning in 1892 around the Baltimore and Ohio Railroad, whose Branchville and Calvert Road depots were located approximately one mile to the north and south, respectively. Lakeland was created by Edwin Newman, who improved the original 238 acres (0.96 km2) located to the west of the railroad. He also built a number of the original homes, a small town hall, and a general store. The area was originally envisioned as a resort-type community. However, due to the flood-prone, low-lying topography, the neighborhood attracted a lower-income population and became an area for African-American settlement. Around 1900, the Baltimore Gold Fish Company built five artificial lakes in the area to spawn goldfish and rare species of fish. A one-room school was built in 1903 for the African-American population; a new school was built in 1925.[7]
The Berwyn neighborhood was developed beginning about 1885 adjacent to the Baltimore and Ohio Railroad. It was created by Francis Shannabrook, a Pennsylvanian who purchased a tract of land between Baltimore Avenue and the railroad tracks. Shannabrook established a small depot, built a general store, and erected approximately 15 homes in the area to attract moderate-income families looking to move out of Washington. The neighborhood began to grow after 1900 when the City and Suburban Electric Railway entered the area. By 1925 approximately 100 single-family homes existed, mostly two-story, wood-frame buildings. The community housing continued to develop in the 1930s and 1940s with one story bungalows, Cape Cods, and Victorians and, later, raised ranches and split level homes.[7]
The Daniels Park neighborhood was developed beginning in 1905 on the east and west sides of the City and Suburban Electric Railway in north College Park. Daniels Park was created by Edward Daniels on 47 acres (19 ha) of land. This small residential subdivision was improved with single-family houses arranged along a grid pattern of streets. The houses—built between 1905 and the 1930s—range in style from American Foursquares to bungalows.[7]
The Hollywood neighborhood was developed in the early 20th century along the City and Suburban Electric Railway. Edward Daniels, the developer of Daniels Park, planned the Hollywood subdivision as a northern extension of that earlier community. Development in Hollywood was slow until after World War II when Albert Turner acquired large tracts of the northern part of the neighborhood in the late 1940s. Turner was able to develop and market brick and frame three-bedroom bungalows beginning in 1950. By 1952, an elementary school had been built. Hollywood Neighborhood Park, a 21-acre (8.5 ha) facility along the Baltimore and Ohio Railroad line, is operated by the Maryland-National Capital Park and Planning Commission.[7]
Later events[edit]
In 1943, due to World War II efforts to conserve rail transport, the Washington Senators relocated their spring training camp in College Park. The location of 1943 Major League Baseball spring training camps was limited to an area east of the Mississippi River and north of the Ohio River.[8]
On September 24, 2001, a multiple-vortex F3 tornado hit the area. This storm moved at peak intensity through the University of Maryland College Park campus, and then moved north parallel to I-95 to the Laurel area, where F3 damage was also noted. The damage path from the storm was measured at 17.5 miles (28.2 km) in length. The tornado caused 2 deaths and 55 injuries and $101 million in property damage. The two deaths were sisters who died when their car was picked up and hurled over a building before being slammed to the ground. Both young women were University of Maryland students.[9] This tornado was part of the Maryland, Virginia, and Washington, D.C., tornado outbreak of 2001, one of the most dramatic recent tornado events to directly affect the Baltimore-Washington metropolitan area.
Image produced at the Student Design Charrette for a new College Park.
By the turn of the 21st century, College Park began experiencing significant development pressure. Both students and city residents acknowledged the city's lack of amenities and poor sense of place. In 2002, the city and county passed the Route 1 Sector Plan, which allowed and encouraged mixed use development along College Park's main roadway. Recent projects—like the East Campus Redevelopment Initiative, The University View, The Varsity, and Landmark student apartments and the Northgate Condos—give many in the community hope that the city, like other notable American college towns, might one day have a vibrant downtown and a diverse population.
The University of Maryland's Student Government Association sponsored a design charrette in April 2006 to envision the future of College Park. In July 2006, a group of students created Rethink College Park—a community group providing a website to share information about development and to encourage public dialogue.
Since 2009, other notable architectural additions to College Park have been: a parking garage (with The Ledo Restaurant on ground level) in downtown near the intersection of Route 1 and Knox Road; The University View and The Varsity student apartment towers with ground floor retail businesses; graduate school apartment towers adjacent to The View apartments; and The Hotel at the University of Maryland.
Demographics[edit]
Historical population
Census
Pop.
%±
1950 11,170 —
1960 18,482 65.5%
1970 26,156 41.5%
1980 23,614 −9.7%
1990 21,927 −7.1%
2000 24,657 12.5%
2010 30,413 23.3%
Est. 2016 32,275 [4] 6.1%
U.S. Decennial Census[10]
The median income for a household in the city was $50,168, and the median income for a family was $62,759 (these figures had risen to $66,953 and $82,295 respectively as of a 2007 estimate[11]). Males had a median income of $40,445 versus $31,631 for females. The per capita income for the city was $16,026. About 4.2% of families and 19.9% of the population were below the poverty line, including 6.9% of those under age 18 and 9.2% of those age 65 or over.
2010 census[edit]
As of the census[3] of 2010, there were 30,413 people, 6,757 households, and 2,852 families residing in the city. The population density was 5,392.4 inhabitants per square mile (2,082.0/km2). There were 8,212 housing units at an average density of 1,456.0 per square mile (562.2/km2). The racial makeup of the city was 63.0% White, 14.3% African American, 0.3% Native American, 12.7% Asian, 0.1% Pacific Islander, 6.0% from other races, and 3.5% from two or more races. Hispanic or Latino of any race were 11.9% of the population.
There were 6,757 households of which 18.4% had children under the age of 18 living with them, 30.6% were married couples living together, 7.9% had a female householder with no husband present, 3.7% had a male householder with no wife present, and 57.8% were non-families. 24.8% of all households were made up of individuals and 6.6% had someone living alone who was 65 years of age or older. The average household size was 2.79 and the average family size was 3.18.
The median age in the city was 21.3 years. 7.6% of residents were under the age of 18; 60.7% were between the ages of 18 and 24; 15.7% were from 25 to 44; 11% were from 45 to 64; and 5.1% were 65 years of age or older. The gender makeup of the city was 53.1% male and 46.9% female.
2000 census[edit]
As of the census[12] of 2000, there were 24,657 people, 6,030 households, and 3,039 families residing in the city. The population density was 4,537.5 people per square mile (1,753.2/km²). There were 6,245 housing units at an average density of 1,149.2 per square mile (444.1/km²). The racial makeup of the city was 68.82% White, 15.93% Black or African American, 0.33% Native American, 10.03% Asian, 0.01% Pacific Islander, 2.57% from other races, and 2.31% from two or more races. 5.54% of the population were Hispanic or Latino of any race.
There were 6,030 households out of which 19.8% had children under the age of 18 living with them, 38.6% were married couples living together, 8.3% had a female householder with no husband present, and 49.6% were non-families. 25.7% of all households were made up of individuals and 9.1% had someone living alone who was 65 years of age or older. The average household size was 2.65 and the average family size was 3.11.
In the city, the age distribution of the population shows 10.5% under the age of 18, 51.3% from 18 to 24, 19.8% from 25 to 44, 11.3% from 45 to 64, and 7.2% who were 65 years of age or older. The median age was 22 years. For every 100 females, there were 110.3 males. For every 100 females age 18 and over, there were 111.2 males.
Government[edit]
The Government of College Park is a Council-Manager form of government. The city manager is appointed by the city council and the mayor elected every two years. The council has eight members, representing four districts in the city. City Council meetings are held once a week at the College Park City Hall.
The current Mayor of College Park is Patrick L. Wojahn, who took office in 2015. Previous mayors were:[13]
William A. Duvall (1945–1951)
Charles R. Davis (1951–1963)
William W. Gullett (1963–1969)
William R. Reading (1969–1973)
Dervey A. Lomax (1973–1975)
St. Clair Reeves (1975–1981)
Alvin J. Kushner (1981–1987)
Anna Latta Owens (1987–1993)
Joseph E. Page (1993–1997)
Michael J. Jacobs (1997–2001)
Stephen A. Brayman (2001–2009)
Andrew M. Fellows (2009–2015)
College Park has six government departments:
Administration
Community Resources
Finance
Planning
Public Services
Public Works
As of September 2011, College Park belongs to Maryland's 5th congressional district.
Geography[edit]
College Park is located at 38°59′48″N 76°55′39″W / 38.99667°N 76.92750°W / 38.99667; -76.92750 (38.996560, -76.927509).[14]
According to the United States Census Bureau, the city has a total area of 5.68 square miles (14.71 km2), of which, 5.64 square miles (14.61 km2) is land and 0.04 square miles (0.10 km2) is water.[2]
Climate[edit]
The climate in this area is characterized by hot, humid summers and generally mild to cool winters. According to the Köppen climate classification system, College Park has a humid subtropical climate, abbreviated "Cfa" on climate maps.[15]
Neighborhoods[edit]
Autoville/Cherry Hill
Berwyn
Branchville
Calvert Hills
College Park Woods
Crystal Springs/Patricia Court
Daniels Park
Hollywood
Lakeland
North College Park
Old Town
Sunnyside
Yarrow
Bordering areas[edit]
Beltsville (North)
Berwyn Heights (East)
University Park (Southwest)
Riverdale Park (South)
Adelphi (West)
Hyattsville (Southwest)
Economy[edit]
According to the City's 2015 Comprehensive Annual Financial Report,[16] the top employers in the city are:
#
Employer
Employees
1 University of Maryland, College Park 15,720
2 University of Maryland University College 2,000
3 National Oceanic and Atmospheric Administration 825
4 Food and Drug Administration 800
5 National Archives and Records Administration 689
6 American Center for Physics 500
7 IKEA 450
Transportation[edit]
Airport[edit]
College Park Airport is one of the oldest continuously operating airports in the United States and is one of the oldest airports in the world, having been in continuous operation since 1909. Its future status is uncertain, as it lies just a few miles outside the restricted airspace of Washington, D.C. In 1977, the airport was added to the National Register of Historic Places.[17]
Area commercial airports include Baltimore-Washington International Airport, Reagan National Airport, and Washington Dulles International Airport.
Major highways[edit]
I-495 joins I-95 at the College Park Interchange, connecting with Alexandria and Baltimore. West of the interchange, I-495 continues west toward Silver Spring, Maryland and Northern Virginia.
US 1 is a major north-south roadway running through College Park, serving as the main street in the downtown area. It is a major route to Beltsville and Laurel to the north and Hyattsville and Washington, D.C. to the south.
Public transportation[edit]
College Park has a station on Washington Metro's Green Line.
College Park–University of Maryland Station on the Washington Metro's Green Line is in College Park; a large commuter parking garage was completed in 2004 adjacent to the Metro station. MARC trains run on CSX tracks adjacent to the Green Line and stop at a small station next to the College Park Metro station. The Metro station lies at what had been the historic junction of Calvert Road and the CSX tracks.
College Park had streetcar service from 1903 to 1962 along what is now Rhode Island Avenue and the College Park Trolley Trail.
Historic sites[edit]
The following is a list of historic sites in College Park identified by the Maryland-National Capital Park and Planning Commission.[18] Part of the city is located within the Calvert Hills Historic District; listed on the National Register of Historic Places in 2002.[19]
Site name
Image
Location
M-NCPPC Inventory
Number
Comment
1
Baker-Holliday House 5005 Huron Street 66-027-24 Located in Daniels Park.
2
Bowers-Sargent House 9312 Rhode Island Avenue 66-027-28 Located in Daniels Park.
3
College Park Airport 6709 Corporal Frank S. Scott Drive 66-004 Listed on the National Register of Historic Places, September 23, 1977
4
College Park Woman's Club 4711 Knox Road 66-021-09 Owned by the City of College Park.
5
Cory House 4710 College Avenue 66-021-08
6
Holbrook House 4618 College Avenue 66-021-31
7
Lake House (Presbyterian Parsonage) 8524 Potomac Avenue 66-018 Located in Berwyn.
8
LaValle House 5013 Huron Street 66-027-25 Located in Daniels Park.
9
McDonnell House 7400 Dartmouth Avenue 66-021-10
10
National Archives Archeological Site Address Restricted 66-036 Listed on the National Register of Historic Places, August 22, 1996
11
The Rossborough Inn Baltimore Avenue (US 1) 66-035-02 Located on the University of Maryland campus.
12
Taliaferro House 7406 Columbia Avenue 66-021-30
Media[edit]
See also: List of newspapers in Maryland, List of radio stations in Maryland, and List of television stations in Maryland
UMTV (University of Maryland)
WMUC broadcasts from the University of Maryland campus, with a range of two (2) miles (3 km) – roughly from the campus to the Beltway. It is also broadcast over the internet at www.wmucradio.com.[20]
The Diamondback, a student publication, is distributed five days a week on a limited basis downtown, including in city hall, and widely on the campus of the University of Maryland.
A College Park edition of The Gazette (a weekly publication distributed widely featuring community news) is available throughout the city and is distributed free.
The oldest operational Persian Podcast is called Radio College Park as it is produced by a group of Iranian graduate students at the University of Maryland, College Park.
The city is part of the Washington, D.C. television market (DMA #9). The city of College Park and the campus of the University of Maryland are the setting for the 2016 novel, "Journey" {Sunfall, book 1} by D. Gideon.
Education[edit]
University Hills Park pond, College Park, Maryland
HJ Patterson Hall, University of Maryland, College Park
McKeldin Library, University of Maryland, College Park
Colleges and universities[edit]
The University of Maryland, College Park, the flagship institution of the University System of Maryland, is located within the College Park city limits.
Primary and secondary schools[edit]
Public schools[edit]
College Park is served by Prince George's County Public Schools. The city is zoned to several different schools.[21]
Elementary school students are zoned to:[22]
Hollywood Elementary School (in College Park)
Paint Branch Elementary School (in College Park)
Berwyn Heights Elementary School (in Berwyn Heights)
University Park Elementary School (in University Park)
Cherokee Lane Elementary School (Adelphi CDP)
Middle school students are zoned to:[23]
Greenbelt Middle School (in Greenbelt)
Hyattsville Middle School (in Hyattsville)
Buck Lodge Middle School (Adelphi CDP)
High school students are zoned to:[24]
High Point High School (Beltsville CDP)
Northwestern High School (Hyattsville)
Parkdale High School (Riverdale Park)
Other area public high schools include: Eleanor Roosevelt High School (Greenbelt).[25]
Charter schools:
College Park Academy Public Charter School, 6–12
Private schools[edit]
Private schools include:[25]
Dar-us-Salaam/Al Huda School, K–12 (College Park)[26][27]
Berwyn Baptist School, PreK–8
Friends Community School, K–8
Holy Redeemer School, K–8
Saint Francis International School St. Mark Campus, K–8, Hyattsville[28] – formerly St. Mark the Evangelist School,[29] closed and merged into Saint Francis International, which opened in 2010.[30]
Museums[edit]
The Art Gallery at the University of Maryland
College Park Aviation Museum
National Museum of Language
City-student politics[edit]
Like many college towns, College Park has had its share of political controversy. Occasionally, University of Maryland students plan voter registration drives and seek to elect one of their own to the city council. City residents, including students living within the city are eligible[31] to run for city council if they are at least 18 years of age. Over the past twenty years there have been multiple attempts, none of which were successful until Marcus Afzali won a seat in 2009.
1993 – Dana L. Loewenstein & Michael J. Moore – Perhaps the most controversial of all student races was that of Loewenstein, a former president of the Panhellenic Association, the sorority umbrella organization at the university. A year after she had lost the election, she was charged with 16 counts of perjury, 16 counts of aiding and advising to falsely register voters and faced a maximum prison sentence of over 200+ years. Ms. Loewenstein's opponent in the council race, Michael Smith, joined former council member Chester Joy in filing a complaint with the Prince George's County Board of Elections days before the Nov. 2 election in an effort to intimidate students from voting. The complaint alleged that 16 of her sorority pledges lived in one district but registered in another. The complaint was turned over to the state's attorney, who filed criminal charges against Loewenstein a year after she lost the election, despite no student voting illegally. The complaint alleged that all of the pledges lived in on-campus dorms but used the house address as their residence. Loewenstein was found not guilty by the Circuit Court.
2001 – Mike Mann & Daniel Dorfman – In November 2001, Michael Mann[32] and Daniel Dorfman, sought the two District 3 seats on the College Park City Council. Campaigning against incumbent Eric Olson and for an open seat created by then-councilman Brayman's decision to run for mayor, the two campaigned heavily to inform students there was a council race going on that year, and registered over 700 students to vote in the municipal election. Despite their hard work and an almost year-long campaign, they were defeated.
2007 – Nick Aragon – In January 2007, Nick Aragon lost a special election for the city council. Two incumbents created a vacancy when they were elected to higher county offices. In turn, the city was forced to hold a special election after the November 2006 elections. The city chose an election date during the university's winter recess, a time when many students were away from the city. With some help from the Student Government Association (SGA)[33] and an endorsement by College Park Mayor Steve Brayman, the Aragon campaign encouraged students to use absentee ballots, although few actually did, and Aragon lost the election.
2009 – Marcus Afzali – Marcus Afzali, a 24-year-old doctoral student in the Department of Government and Politics at UMD, won a seat on the city council representing District 4 in November 2009. Afzali attributed displays of "energy"—exemplified by taking time to knock on doors and reach out to residents—as the cause behind his success. The 2009 election is notable not only for Afzali's performance at the polls, but for the fact that both District 4 incumbents lost.[34][35]
References[edit]
Jump up ^ "College Park, Prince George's County, Maryland". Maryland Manual On-Line. Maryland State Archives. January 31, 2017. Retrieved May 15, 2017.
^ Jump up to: a b "US Gazetteer files 2010". United States Census Bureau. Archived from the original on June 26, 2012. Retrieved January 25, 2013.
^ Jump up to: a b "American FactFinder". United States Census Bureau. Retrieved January 25, 2013.
^ Jump up to: a b "Population and Housing Unit Estimates". Retrieved June 9, 2017.
Jump up ^ U.S. Geological Survey Geographic Names Information System: College Park, Maryland
Jump up ^ "NOAA Center for Weather and Climate Prediction" (PDF). National Weather Service. National Oceanic and Atmospheric Administration. Retrieved 11 October 2016.
^ Jump up to: a b c d e "Community Summary Sheet, Prince George's County" (PDF). College Park, Maryland. Maryland State Highway Administration, 1999. May 10, 2008.
Jump up ^ Suehsdorf, A. D. (1978). The Great American Baseball Scrapbook, p. 103. Random House. ISBN 0-394-50253-1.
Jump up ^ "NWS Sterling, VA – Sept 24 tornado report". National Oceanic and Atmospheric Administration.
Jump up ^ "Census of Population and Housing". Census.gov. Archived from the original on May 12, 2015. Retrieved June 4, 2015.
Jump up ^ "College Park, MD Factsheet". United States Census Bureau. Retrieved 2009-09-15.
Jump up ^ "American FactFinder". United States Census Bureau. Archived from the original on September 11, 2013. Retrieved January 31, 2008.
Jump up ^ "College Park Mayors". Maryland Manual On-Line. Maryland State Archives. December 7, 2015. Retrieved May 15, 2017.
Jump up ^ "US Gazetteer files: 2010, 2000, and 1990". United States Census Bureau. February 12, 2011. Retrieved April 23, 2011.
Jump up ^ "College Park, Maryland Köppen Climate Classification (Weatherbase)". Weatherbase.
Jump up ^ "Comprehensive Annual Financial Report for the fiscal year ended June 30, 2015" (PDF). City of College Park, Maryland. November 20, 2015. p. 87. Retrieved October 26, 2016.
Jump up ^ National Park Service (2008-04-15). "National Register Information System". National Register of Historic Places. National Park Service.
Jump up ^ M-NCPPC Illustrated Inventory of Historic Sites: Prince George's County, Maryland (Prince George's County, Maryland), 2006 Archived 2008-07-25 at the Wayback Machine..
Jump up ^ "Calvert Hills Historic District". Maryland Historical Trust. Retrieved October 29, 2008.
Jump up ^ www.wmucradio.com.
Jump up ^ "District_BIG_WALL_MAP_2009d_36x48_July_2013.pdf." City of College Park. Retrieved on January 31, 2018. See also: City's listing of area schools, neighborhood map
Jump up ^ "NEIGHBORHOOD ELEMENTARY SCHOOLS AND BOUNDARIES SCHOOL YEAR 2017-2018." Prince George's County Public Schools. Retrieved on January 31, 2018.
Jump up ^ "NEIGHBORHOOD MIDDLE SCHOOLS AND BOUNDARIES SCHOOL YEAR 2017-2018." Prince George's County Public Schools. Retrieved on January 31, 2018.
Jump up ^ "NEIGHBORHOOD HIGH SCHOOLS AND BOUNDARIES SCHOOL YEAR 2017-2018." Prince George's County Public Schools. Retrieved on January 31, 2018.
^ Jump up to: a b "Local Schools." Prince George's County Public Schools. Retrieved on January 31, 2018.
Jump up ^ "Contact Us." Al Huda School. Retrieved on February 1, 2018. "5301 Edgewood Road, College Park, MD 20740"
Jump up ^ "About Al-Huda School." Al Huda School. Retrieved on February 1, 2018.
Jump up ^ "Contact Us." Saint Francis International School. Retrieved on January 31, 2018. "St. Mark Campus 7501 Adelphi Road Hyattsville, MD 20783"
Jump up ^ "St. Mark's School in Hyattsville holds reunion to marks its 50th year ." Catholic Standard', Roman Catholic Archdiocese of Washington. Wednesday, October 15, 2008. Retrieved on January 31, 2018. "St. Mark Campus 7501 Adelphi Road Hyattsville, MD 20783"
Jump up ^ Roberts, Tom. "Maryland Catholic school finds its footing amid demographic shifts." Catholic Standard', Roman Catholic Archdiocese of Washington. Wednesday, October 15, 2008. Retrieved on February 1, 2018.
Jump up ^ Maryland State Board of Elections. "Voter Registration Introduction".
Jump up ^ Michael Mann Archived 2007-09-27 at the Wayback Machine.
Jump up ^ "Student Government Association (SGA)".
Jump up ^ "- The Diamondback". Retrieved 27 July 2016.
Jump up ^ "Marcus D. Afzali (I)". The Washington Post. Archived from the original on September 30, 2012.
Further reading[edit]
A Guide to the City of College Park, from the College Park City Hall.
External links[edit]
Wikimedia Commons has media related to College Park, Maryland.
Wikivoyage has a travel guide for College Park (Maryland).
Official website
Geographic data related to College Park, Maryland at OpenStreetMap
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Solution of triangles
From Wikipedia, the free encyclopedia
Jump to: navigation, search
Solution of triangles (Latin: solutio triangulorum) is the main trigonometric problem of finding the characteristics of a triangle (angles and lengths of sides), when some of these are known. The triangle can be located on a plane or on a sphere. Applications requiring triangle solutions include geodesy, astronomy, construction, and navigation.
Contents
[hide]
1 Solving plane triangles
1.1 Trigonomic relations
1.1.1 Notes
1.2 Three sides given (SSS)
1.3 Two sides and the included angle given (SAS)
1.4 Two sides and non-included angle given (SSA)
1.5 A side and two adjacent angles given (ASA)
1.6 A side, one adjacent angle and the opposite angle given (AAS)
2 Solving spherical triangles
2.1 Three sides given (spherical SSS)
2.2 Two sides and the included angle given (spherical SAS)
2.3 Two sides and non-included angle given (spherical SSA)
2.4 A side and two adjacent angles given (spherical ASA)
2.5 A side, one adjacent angle and the opposite angle given (spherical AAS)
2.6 Three angles given (spherical AAA)
2.7 Solving right-angled spherical triangles
3 Some applications
3.1 Triangulation
3.2 The distance between two points on the globe
4 See also
5 References
6 External links
Solving plane triangles[edit]
Standard notation for a triangle
A general form triangle has six main characteristics (see picture): three linear (side lengths a, b, c) and three angular (α, β, γ). The classical plane trigonometry problem is to specify three of the six characteristics and determine the other three. A triangle can be uniquely determined in this sense when given any of the following:[1][2]
Three sides (SSS)
Two sides and the included angle (SAS)
Two sides and an angle not included between them (SSA), if the side length adjacent to the angle is shorter than the other side length.
A side and the two angles adjacent to it (ASA)
A side, the angle opposite to it and an angle adjacent to it (AAS).
Three angles (AAA) on the sphere (but not in the plane).
For all cases in the plane, at least one of the side lengths must be specified. If only the angles are given, the side lengths cannot be determined, because any similar triangle is a solution.
Trigonomic relations[edit]
Overview of particular steps and tools used when solving plane triangles
The standard method of solving the problem is to use fundamental relations.
Law of cosines
a 2 = b 2 + c 2 − 2 b c cos  α {\displaystyle a^{2}=b^{2}+c^{2}-2bc\cos \alpha }
b 2 = a 2 + c 2 − 2 a c cos  β {\displaystyle b^{2}=a^{2}+c^{2}-2ac\cos \beta }
c 2 = a 2 + b 2 − 2 a b cos  γ {\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos \gamma }
Law of sines
a sin  α = b sin  β = c sin  γ {\displaystyle {\frac {a}{\sin \alpha }}={\frac {b}{\sin \beta }}={\frac {c}{\sin \gamma }}}
Sum of angles
α + β + γ = 180 ∘ {\displaystyle \alpha +\beta +\gamma =180^{\circ }}
Law of tangents
a − b a + b = tan  [ 1 2 ( α − β ) ] tan  [ 1 2 ( α + β ) ] . {\displaystyle {\frac {a-b}{a+b}}={\frac {\tan[{\frac {1}{2}}(\alpha -\beta )]}{\tan[{\frac {1}{2}}(\alpha +\beta )]}}.}
There are other (sometimes practically useful) universal relations: the law of cotangents and Mollweide's formula.
Notes[edit]
To find an unknown angle, the law of cosines is safer than the law of sines. The reason is that the value of sine for the angle of the triangle does not uniquely determine this angle. For example, if sin β = 0.5, the angle β can equal either 30° or 150°. Using the law of cosines avoids this problem: within the interval from 0° to 180° the cosine value unambiguously determines its angle. On the other hand, if the angle is small (or close to 180°), then it is more robust numerically to determine it from its sine than its cosine because the arc-cosine function has a divergent derivative at 1 (or −1).
We assume that the relative position of specified characteristics is known. If not, the mirror reflection of the triangle will also be a solution. For example, three side lengths uniquely define either a triangle or its reflection.
Three sides given (SSS)[edit]
Three sides given
Let three side lengths a, b, c be specified. To find the angles α, β, the law of cosines can be used:[3]
α = arccos  b 2 + c 2 − a 2 2 b c β = arccos  a 2 + c 2 − b 2 2 a c . {\displaystyle {\begin{aligned}\alpha &=\arccos {\frac {b^{2}+c^{2}-a^{2}}{2bc}}\\[4pt]\beta &=\arccos {\frac {a^{2}+c^{2}-b^{2}}{2ac}}.\end{aligned}}}
Then angle γ = 180° − α − β.
Some sources recommend to find angle β from the law of sines but (as Note 1 above states) there is a risk of confusing an acute angle value with an obtuse one.
Another method of calculating the angles from known sides is to apply the law of cotangents.
Two sides and the included angle given (SAS)[edit]
Two sides and the included angle given
Here the lengths of sides a, b and the angle γ between these sides are known. The third side can be determined from the law of cosines:[4]
c = a 2 + b 2 − 2 a b cos  γ . {\displaystyle c={\sqrt {a^{2}+b^{2}-2ab\cos \gamma }}.}
Now we use law of cosines to find the second angle:
α = arccos  b 2 + c 2 − a 2 2 b c . {\displaystyle \alpha =\arccos {\frac {b^{2}+c^{2}-a^{2}}{2bc}}.}
Finally, β = 180° − α − γ.
Two sides and non-included angle given (SSA)[edit]
Two sides and a non-included angle given
Two solutions for the triangle
This case is not solvable in all cases; a solution is guaranteed to be unique only if the side length adjacent to the angle is shorter than the other side length. Assume that two sides b, c and the angle β are known. The equation for the angle γ can be implied from the law of sines:[5]
sin  γ = c b sin  β . {\displaystyle \sin \gamma ={\frac {c}{b}}\sin \beta .}
We denote further D = c/b sin β (the equation's right side). There are four possible cases:
If D > 1, no such triangle exists because the side b does not reach line BC. For the same reason a solution does not exist if the angle β ≥ 90° and b ≤ c.
If D = 1, a unique solution exists: γ = 90°, i.e., the triangle is right-angled.
If D < 1 two alternatives are possible.
If b ≥ c, then β ≥ γ (the larger side corresponds to a larger angle). Since no triangle can have two obtuse angles, γ is an acute angle and the solution γ = arcsin D is unique.
If b < c, the angle γ may be acute: γ = arcsin D or obtuse: γ′ = 180° - γ. The figure on right shows the point C, the side b and the angle γ as the first solution, and the point C′, side b′ and the angle γ′ as the second solution.
Once γ is obtained, the third angle α = 180° − β − γ.
The third side can then be found from the law of sines:
a = b sin  α sin  β {\displaystyle a=b\ {\frac {\sin \alpha }{\sin \beta }}}
or
a = c cos  β ± b 2 − c 2 sin 2  β {\displaystyle a=c\cos \beta \pm {\sqrt {b^{2}-c^{2}\sin ^{2}\beta }}}
A side and two adjacent angles given (ASA)[edit]
One side and two adjacent angles given
The known characteristics are the side c and the angles α, β. The third angle γ = 180° − α − β.
Two unknown side can be calculated from the law of sines:[6]
a = c sin  α sin  γ ; b = c sin  β sin  γ . {\displaystyle a=c\ {\frac {\sin \alpha }{\sin \gamma }};\quad b=c\ {\frac {\sin \beta }{\sin \gamma }}.}
or
a = c sin  α sin  α cos  β + sin  β cos  α {\displaystyle a=c{\frac {\sin \alpha }{\sin \alpha \cos \beta +\sin \beta \cos \alpha }}}
b = c sin  β sin  α cos  β + sin  β cos  α {\displaystyle b=c{\frac {\sin \beta }{\sin \alpha \cos \beta +\sin \beta \cos \alpha }}}
A side, one adjacent angle and the opposite angle given (AAS)[edit]
The procedure for solving an AAS triangle is same as that for an ASA triangle: First, find the third angle by using the angle sum property of a triangle, then find the other two sides using the law of sines.
Solving spherical triangles[edit]
Spherical triangle
The general spherical triangle is fully determined by three of its six characteristics (3 sides and 3 angles). Note that the sides a, b, c of a spherical triangle are measured by angular rather than linear units, based on the corresponding central angles.
The solution of triangles for non-Euclidean spherical geometry has some differences from the planar case. For example, the sum of the three angles α + β + γ depends on the triangle. In addition, there are no unequal similar triangles, and so the problem of constructing a triangle with specified three angles has a unique solution. The basic relations used to solve a problem are similar to those of the planar case: see Law of cosines (spherical) and Law of sines (spherical).
Among other relationships that may be useful are the half-side formula and Napier's analogies:[7]
tan  c 2 cos  α − β 2 = tan  a + b 2 cos  α + β 2 {\displaystyle \tan {\frac {c}{2}}\cos {\frac {\alpha -\beta }{2}}=\tan {\frac {a+b}{2}}\cos {\frac {\alpha +\beta }{2}}}
tan  c 2 sin  α − β 2 = tan  a − b 2 sin  α + β 2 {\displaystyle \tan {\frac {c}{2}}\sin {\frac {\alpha -\beta }{2}}=\tan {\frac {a-b}{2}}\sin {\frac {\alpha +\beta }{2}}}
cot  γ 2 cos  a − b 2 = tan  α + β 2 cos  a + b 2 {\displaystyle \cot {\frac {\gamma }{2}}\cos {\frac {a-b}{2}}=\tan {\frac {\alpha +\beta }{2}}\cos {\frac {a+b}{2}}}
cot  γ 2 sin  a − b 2 = tan  α − β 2 sin  a + b 2 . {\displaystyle \cot {\frac {\gamma }{2}}\sin {\frac {a-b}{2}}=\tan {\frac {\alpha -\beta }{2}}\sin {\frac {a+b}{2}}.}
Three sides given
Three sides given (spherical SSS)[edit]
Known: the sides a, b, c (in angular units). The triangle's angles are computed from the spherical law of cosines:
α = arccos  ( cos  a − cos  b cos  c sin  b sin  c ) , {\displaystyle \alpha =\arccos \left({\frac {\cos a-\cos b\ \cos c}{\sin b\ \sin c}}\right),}
β = arccos  ( cos  b − cos  c cos  a sin  c sin  a ) , {\displaystyle \beta =\arccos \left({\frac {\cos b-\cos c\ \cos a}{\sin c\ \sin a}}\right),}
γ = arccos  ( cos  c − cos  a cos  b sin  a sin  b ) . {\displaystyle \gamma =\arccos \left({\frac {\cos c-\cos a\ \cos b}{\sin a\ \sin b}}\right).}
Two sides and the included angle given
Two sides and the included angle given (spherical SAS)[edit]
Known: the sides a, b and the angle γ between them. The side c can be found from the law of cosines:
c = arccos  ( cos  a cos  b + sin  a sin  b cos  γ ) . {\displaystyle c=\arccos \left(\cos a\cos b+\sin a\sin b\cos \gamma \right).}
The angles α', β can be calculated as above, or by using Napier's analogies:
α = arctan  2 sin  a tan  ( γ 2 ) sin  ( b + a ) + cot  ( γ 2 ) sin  ( b − a ) , {\displaystyle \alpha =\arctan \ {\frac {2\sin a}{\tan({\frac {\gamma }{2}})\sin(b+a)+\cot({\frac {\gamma }{2}})\sin(b-a)}},}
β = arctan  2 sin  b tan  ( γ 2 ) sin  ( a + b ) + cot  ( γ 2 ) sin  ( a − b ) . {\displaystyle \beta =\arctan \ {\frac {2\sin b}{\tan({\frac {\gamma }{2}})\sin(a+b)+\cot({\frac {\gamma }{2}})\sin(a-b)}}.}
This problem arises in the navigation problem of finding the great circle between two points on the earth specified by their latitude and longitude; in this application, it is important to use formulas which are not susceptible to round-off errors. For this purpose, the following formulas (which may be derived using vector algebra) can be used:
c = arctan  ( sin  a cos  b − cos  a sin  b cos  γ ) 2 + ( sin  b sin  γ ) 2 cos  a cos  b + sin  a sin  b cos  γ , α = arctan  sin  a sin  γ sin  b cos  a − cos  b sin  a cos  γ , β = arctan  sin  b sin  γ sin  a cos  b − cos  a sin  b cos  γ , {\displaystyle {\begin{aligned}c&=\arctan {\frac {\sqrt {(\sin a\cos b-\cos a\sin b\cos \gamma )^{2}+(\sin b\sin \gamma )^{2}}}{\cos a\cos b+\sin a\sin b\cos \gamma }},\\\alpha &=\arctan {\frac {\sin a\sin \gamma }{\sin b\cos a-\cos b\sin a\cos \gamma }},\\\beta &=\arctan {\frac {\sin b\sin \gamma }{\sin a\cos b-\cos a\sin b\cos \gamma }},\end{aligned}}}
where the signs of the numerators and denominators in these expressions should be used to determine the quadrant of the arctangent.
Two sides and a non-included angle given
Two sides and non-included angle given (spherical SSA)[edit]
This problem is not solvable in all cases; a solution is guaranteed to be unique only if the side length adjacent to the angle is shorter than the other side length. Known: the sides b, c and the angle β not between them. A solution exists if the following condition holds:
b > arcsin  ( sin  c sin  β ) . {\displaystyle b>\arcsin(\sin c\,\sin \beta ).}
The angle γ can be found from the spherical law of sines:
γ = arcsin  ( sin  c sin  β sin  b ) . {\displaystyle \gamma =\arcsin \left({\frac {\sin c\,\sin \beta }{\sin b}}\right).}
As for the plane case, if b < c then there are two solutions: γ and 180° - γ.
We can find other characteristics by using Napier's analogies:
a = 2 arctan  [ tan  ( 1 2 ( b − c ) ) sin  ( 1 2 ( β + γ ) ) sin  ( 1 2 ( β − γ ) ) ] , α = 2 arccot  [ tan  ( 1 2 ( β − γ ) ) sin  ( 1 2 ( b + c ) ) sin  ( 1 2 ( b − c ) ) ] . {\displaystyle {\begin{aligned}a&=2\arctan \left[\tan \left({\tfrac {1}{2}}(b-c)\right){\frac {\sin \left({\tfrac {1}{2}}(\beta +\gamma )\right)}{\sin \left({\tfrac {1}{2}}(\beta -\gamma )\right)}}\right],\\[4pt]\alpha &=2\operatorname {arccot} \left[\tan \left({\tfrac {1}{2}}(\beta -\gamma )\right){\frac {\sin \left({\tfrac {1}{2}}(b+c)\right)}{\sin \left({\tfrac {1}{2}}(b-c)\right)}}\right].\end{aligned}}}
One side and two adjacent angles given
A side and two adjacent angles given (spherical ASA)[edit]
Known: the side c and the angles α, β. First we determine the angle γ using the law of cosines:
γ = arccos  ( sin  α sin  β cos  c − cos  α cos  β ) . {\displaystyle \gamma =\arccos(\sin \alpha \sin \beta \cos c-\cos \alpha \cos \beta ).\,}
We can find the two unknown sides from the law of cosines (using the calculated angle γ):
a = arccos  ( cos  α + cos  β cos  γ sin  β sin  γ ) , {\displaystyle a=\arccos \left({\frac {\cos \alpha +\cos \beta \cos \gamma }{\sin \beta \sin \gamma }}\right),}
b = arccos  ( cos  β + cos  γ cos  α sin  γ sin  α ) , {\displaystyle b=\arccos \left({\frac {\cos \beta +\cos \gamma \cos \alpha }{\sin \gamma \sin \alpha }}\right),}
or by using Napier's analogies:
a = arctan  [ 2 sin  α cot  ( c 2 ) sin  ( β + α ) + tan  ( c 2 ) sin  ( β − α ) ] , b = arctan  [ 2 sin  β cot  ( c 2 ) sin  ( α + β ) + tan  ( c 2 ) sin  ( α − β ) ] . {\displaystyle {\begin{aligned}a&=\arctan \left[{\frac {2\sin \alpha }{\cot({\frac {c}{2}})\sin(\beta +\alpha )+\tan({\frac {c}{2}})\sin(\beta -\alpha )}}\right],\\[4pt]b&=\arctan \left[{\frac {2\sin \beta }{\cot({\frac {c}{2}})\sin(\alpha +\beta )+\tan({\frac {c}{2}})\sin(\alpha -\beta )}}\right].\end{aligned}}}
One side, one adjacent angle and the opposite angle given
A side, one adjacent angle and the opposite angle given (spherical AAS)[edit]
Known: the side a and the angles α, β. The side b can be found from the law of sines:
b = arcsin  ( sin  a sin  β sin  α ) . {\displaystyle b=\arcsin \left({\frac {\sin a\,\sin \beta }{\sin \alpha }}\right).}
If the angle for the side a is acute and α > β, another solution exists:
b = π − arcsin  ( sin  a sin  β sin  α ) . {\displaystyle b=\pi -\arcsin \left({\frac {\sin a\,\sin \beta }{\sin \alpha }}\right).}
We can find other characteristics by using Napier's analogies:
c = 2 arctan  [ tan  ( 1 2 ( a − b ) ) sin  ( 1 2 ( α + β ) ) sin  ( 1 2 ( α − β ) ) ] , γ = 2 arccot  [ tan  ( 1 2 ( α − β ) ) sin  ( 1 2 ( a + b ) ) sin  ( 1 2 ( a − b ) ) ] . {\displaystyle {\begin{aligned}c&=2\arctan \left[\tan \left({\tfrac {1}{2}}(a-b)\right){\frac {\sin \left({\tfrac {1}{2}}(\alpha +\beta )\right)}{\sin \left({\frac {1}{2}}(\alpha -\beta )\right)}}\right],\\[4pt]\gamma &=2\operatorname {arccot} \left[\tan \left({\tfrac {1}{2}}(\alpha -\beta )\right){\frac {\sin \left({\tfrac {1}{2}}(a+b)\right)}{\sin \left({\frac {1}{2}}(a-b)\right)}}\right].\end{aligned}}}
Three angles given
Three angles given (spherical AAA)[edit]
Known: the angles α, β, γ. From the law of cosines we infer:
a = arccos  ( cos  α + cos  β cos  γ sin  β sin  γ ) , {\displaystyle a=\arccos \left({\frac {\cos \alpha +\cos \beta \cos \gamma }{\sin \beta \sin \gamma }}\right),}
b = arccos  ( cos  β + cos  γ cos  α sin  γ sin  α ) , {\displaystyle b=\arccos \left({\frac {\cos \beta +\cos \gamma \cos \alpha }{\sin \gamma \sin \alpha }}\right),}
c = arccos  ( cos  γ + cos  α cos  β sin  α sin  β ) . {\displaystyle c=\arccos \left({\frac {\cos \gamma +\cos \alpha \cos \beta }{\sin \alpha \sin \beta }}\right).}
Solving right-angled spherical triangles[edit]
The above algorithms become much simpler if one of the angles of a triangle (for example, the angle C) is the right angle. Such a spherical triangle is fully defined by its two elements, and the other three can be calculated using Napier's Pentagon or the following relations.
sin  a = sin  c ⋅ sin  A {\displaystyle \sin a=\sin c\cdot \sin A} (from the Law of sines (spherical))
tan  a = sin  b ⋅ tan  A {\displaystyle \tan a=\sin b\cdot \tan A}
cos  c = cos  a ⋅ cos  b {\displaystyle \cos c=\cos a\cdot \cos b} (from the law of cosines (spherical))
tan  b = tan  c ⋅ cos  A {\displaystyle \tan b=\tan c\cdot \cos A}
cos  A = cos  a ⋅ sin  B {\displaystyle \cos A=\cos a\cdot \sin B} (also from the law of cosines)
cos  c = cot  A ⋅ cot  B {\displaystyle \cos c=\cot A\cdot \cot B}
Some applications[edit]
Triangulation[edit]
Distance measurement by triangulation
Main article: Triangulation
If one wants to measure the distance d from shore to a remote ship via triangulation, one marks on the shore two points with known distance l between them (the baseline). Let α, β be the angles between the baseline and the direction to the ship.
From the formulae above (ASA case) one can compute the distance as the triangle height:
d = sin  α sin  β sin  ( α + β ) ℓ = tan  α tan  β tan  α + tan  β ℓ . {\displaystyle d={\frac {\sin \alpha \,\sin \beta }{\sin(\alpha +\beta )}}\ell ={\frac {\tan \alpha \,\tan \beta }{\tan \alpha +\tan \beta }}\ell .}
This method is used in cabotage. The angles α, β are defined by observation of familiar landmarks from the ship.
How to measure a mountain's height
As another example, if one wants to measure the height h of a mountain or a high building, the angles α, β from two ground points to the top are specified. Let ℓ be the distance between these points. From the same ASA case formulas we obtain:
h = sin  α sin  β sin  ( β − α ) ℓ = tan  α tan  β tan  β − tan  α ℓ . {\displaystyle h={\frac {\sin \alpha \,\sin \beta }{\sin(\beta -\alpha )}}\ell ={\frac {\tan \alpha \,\tan \beta }{\tan \beta -\tan \alpha }}\ell .}
The distance between two points on the globe[edit]
Main article: Great-circle distance
To calculate the distance between two points on the globe,
Point A: latitude λA, longitude LA, and
Point B: latitude λB, longitude LB
we consider the spherical triangle ABC, where C is the North Pole. Some characteristics are:
a = 90 o − λ B , {\displaystyle a=90^{\mathrm {o} }-\lambda _{\mathrm {B} },\,}
b = 90 o − λ A , {\displaystyle b=90^{\mathrm {o} }-\lambda _{\mathrm {A} },\,}
γ = L A − L B . {\displaystyle \gamma =L_{\mathrm {A} }-L_{\mathrm {B} }.\,}
If two sides and the included angle given, we obtain from the formulas
A B = R arccos  [ sin  λ A sin  λ B + cos  λ A cos  λ B cos  ( L A − L B ) ] . {\displaystyle \mathrm {AB} =R\arccos \left[\sin \lambda _{\mathrm {A} }\,\sin \lambda _{\mathrm {B} }+\cos \lambda _{\mathrm {A} }\,\cos \lambda _{\mathrm {B} }\,\cos \left(L_{\mathrm {A} }-L_{\mathrm {B} }\right)\right].}
Here R is the Earth's radius.
See also[edit]
Congruence
Hansen's problem
Hinge theorem
Snellius–Pothenot problem
Lénárt sphere
References[edit]
Jump up ^ "Solving Triangles". Maths is Fun. Retrieved 4 April 2012.
Jump up ^ "Solving Triangles". web.horacemann.org. Retrieved 4 April 2012.
Jump up ^ "Solving SSS Triangles". Maths is Fun. Retrieved 13 January 2015.
Jump up ^ "Solving SAS Triangles". Maths is Fun. Retrieved 13 January 2015.
Jump up ^ "Solving SSA Triangles". Maths is Fun. Retrieved 9 March 2013.
Jump up ^ "Solving ASA Triangles". Maths is Fun. Retrieved 13 January 2015.
Jump up ^ Napier's Analogies at MathWorld
Euclid (1956) [1925]. Sir Thomas Heath, ed. The Thirteen Books of the Elements. Volume I. Translated with introduction and commentary. Dover. ISBN 0-486-60088-2.
External links[edit]
Trigonometric Delights, by Eli Maor, Princeton University Press, 1998. Ebook version, in PDF format, full text presented.
Trigonometry by Alfred Monroe Kenyon and Louis Ingold, The Macmillan Company, 1914. In images, full text presented. Google book.
Spherical trigonometry on Math World.
Intro to Spherical Trig. Includes discussion of The Napier circle and Napier's rules
Spherical Trigonometry — for the use of colleges and schools by I. Todhunter, M.A., F.R.S. Historical Math Monograph posted by Cornell University Library.
Triangulator – Triangle solver. Solve any plane triangle problem with the minimum of input data. Drawing of the solved triangle.
TriSph – Free software to solve the spherical triangles, configurable to different practical applications and configured for gnomonic.
Spherical Triangle Calculator – Solves spherical triangles.
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