text:
List of common physics notations
From Wikipedia, the free encyclopedia
Jump to: navigation, search
This is a list of common physical constants and variables, and their notations. Note that bold text indicates that the quantity is a vector.
Contents
[hide]
1 Latin characters
2 Greek characters
3 Other characters
4 See also
5 References
Latin characters[edit]
Symbol
Meaning
SI unit of measure
A {\displaystyle A} area meter squared (m2)
magnetic vector potential
amplitude meter
a {\displaystyle \mathbf {a} } acceleration meters per second squared (m/s2)
B {\displaystyle \mathbf {B} } magnetic flux density
also called the magnetic field density or magnetic induction tesla (T), or equivalently,
weber per square meter (Wb/m2)
C {\displaystyle C} capacitance farad (F)
heat capacity joule per kelvin (J K−1), or equivalently, joule per degree Celsius (J °C−1)
constant of integration varied depending on context
c {\displaystyle c} speed of light (in vacuum) 299,792,458 meter per second (m/s)
speed of sound 340.29 meter per second (m/s)
specific heat capacity joule per kilogram per kelvin (J kg−1 K−1), or equivalently, joule per kilogram per degree Celsius (J kg−1 °C−1)
viscous damping coefficient kilogram per second (kg/s)
D {\displaystyle \mathbf {D} } electric displacement field
also called the electric flux density coulomb per square meter (C/m2)
D {\displaystyle D} density kilograms per cubic meter (kg/m3)
d {\displaystyle d} distance meter (m)
direction unitless
impact parameter meter (m)
diameter meter (m)
differential (e.g. d x {\displaystyle dx} )
d A {\displaystyle d\mathbf {A} } differential vector element of surface area A, with infinitesimally
small magnitude and direction normal to surface S
square meter (m2)
d V {\displaystyle dV} differential element of volume V enclosed by surface S cubic meter (m3)
E {\displaystyle \mathbf {E} } electric field newton per coulomb (N C−1), or equivalently, volt per meter (V m−1)
E {\displaystyle E} energy joule (J)
Young's Modulus Pascal (Pa) or Newton per square meter (N/m2) or kilogram per meter per second squared (kg·m−1·s−2)
e {\displaystyle e} eccentricity unitless
2.71828... (base of the natural logarithm), electron, elementary charge
F {\displaystyle \mathbf {F} } force newton (N)
f {\displaystyle f} frequency hertz (Hz)
function
friction newton (N)
G {\displaystyle G} the gravitational constant newton meter squared per kilogram squared (N m2/kg2)
g {\displaystyle g} acceleration due to gravity meter per second squared (m/s2), or equivalently, newton per kilogramme (N/kg)
H {\displaystyle \mathbf {H} } magnetic field strength
also called just magnetic field ampere per meter (A/m)
H {\displaystyle H} Hamiltonian joule (J)
h {\displaystyle h} height meter (m)
Planck's constant joule second (J s)
ℏ {\displaystyle \hbar } reduced Planck's constant ( h 2 π ) {\displaystyle \textstyle \left({\frac {h}{2\pi }}\right)} joule second (J s)
I {\displaystyle I} action joule second (J s)
intensity watt per square meter (W/m2)
sound intensity watt per square meter (W/m2)
electric current ampere (A)
moment of inertia kilogram meter squared (kg m2)
i {\displaystyle i} intensity watt per square meter (W/m2)
imaginary unit
i ^ {\displaystyle \mathbf {\hat {i}} } Cartesian x-axis basis unit vector unitless
J {\displaystyle \mathbf {J} } free current density,
not including polarization or magnetization currents bound in a material ampere per square meter (A/m2)
impulse kilogram meter per second (kg m/s)
j ^ {\displaystyle \mathbf {\hat {j}} } Cartesian y-axis basis unit vector unitless
K {\displaystyle K} kinetic energy
2π/λ
joule (J)
k {\displaystyle k} Boltzmann constant joule per kelvin (J/K)
wavenumber radians per meter (m−1)
k ^ {\displaystyle \mathbf {\hat {k}} } Cartesian z-axis basis unit vector unitless
L {\displaystyle L} inductance henry (H)
luminosity watt (W)
angular momentum newton meter second (N m s or kg m2 s−1)
Lagrangian joule (J)
l {\displaystyle l} length meter (m)
M {\displaystyle M} magnetization ampere per meter (A/m)
moment of force
often simply called moment or torque newton meter (N m)
m {\displaystyle m} mass kilogram (kg)
N {\displaystyle N} normal vector unit varies depending on context
atomic number unitless
n {\displaystyle n} refractive index unitless
principal quantum number unitless
P {\displaystyle P} power watt (W)
p {\displaystyle \mathbf {p} } momentum kilogram meter per second (kg m/s)
pressure pascal (Pa)
Q {\displaystyle Q} electric charge coulomb (C)
Heat joule (J)
q {\displaystyle q} electric charge coulomb (C)
R {\displaystyle R} electrical resistance ohm (Ω)
Ricci tensor unitless
radiancy
gas constant joule per kilogramme kelvin (J/kgK)
r {\displaystyle \mathbf {r} } radius vector (position) meter (m)
r {\displaystyle r} radius of rotation or distance between two things such as the masses in Newton's law of universal gravitation meter (m)
S {\displaystyle S} surface area m2
entropy joule per kelvin (J/K)
action
s {\displaystyle s} arc length meter (m)
displacement
T {\displaystyle T} period second (s)
thermodynamic temperature
also called absolute temperature kelvin (K)
t {\displaystyle t} time second (s)
U {\displaystyle \mathbf {U} } four-velocity meter per second (m/s)
U {\displaystyle U} potential energy joule (J)
internal energy joule (J)
u {\displaystyle u} relativistic mass kilogram (kg)
energy density joule per cubic meter (J/m3) or joule per kilogram (J/kg) depending on the context
V {\displaystyle V} voltage
also called electric potential difference volt (V)
volume cubic meter (m3)
shear force
v {\displaystyle \mathbf {v} } velocity meter per second (m/s)
W {\displaystyle W} mechanical work joule (J)
w {\displaystyle w} width meter (m)
x {\displaystyle x} a generic unknown varied depending on context
displacement meter (m)
Z {\displaystyle Z} electrical impedance ohm (Ω)
Greek characters[edit]
Symbol
Name
Meaning
SI unit of measure
α {\displaystyle \alpha } alpha angular acceleration radian per second squared (rad/s2)
β {\displaystyle \beta } beta velocity in terms of the speed of light c unitless
γ {\displaystyle \gamma } gamma Lorentz factor unitless
photon
gamma ray
shear strain
Heat capacity ratio unitless
Δ {\displaystyle \Delta } delta a change in a variable (e.g. Δ x {\displaystyle \Delta x} ) unitless
Laplace operator
δ {\displaystyle \delta } displacement (usually small)
ϵ {\displaystyle \epsilon } epsilon permittivity farad per meter (F/m)
strain unitless
ε 0 {\displaystyle \varepsilon _{0}} epsilon
nought
The vacuum permittivity constant
ζ {\displaystyle \zeta } zeta damping ratio unitless
η {\displaystyle \eta } eta energy efficiency unitless
coefficient of viscosity
also called simply viscosity pascal second (Pa s)
θ {\displaystyle \theta } theta angular displacement radian (rad)
K {\displaystyle \mathrm {K} } kappa torsion coefficient
also called torsion constant newton meter per radian (N m/rad)
Λ {\displaystyle \Lambda } lambda cosmological constant per second squared (s−2)
λ {\displaystyle \lambda } wavelength meter (m)
λ {\displaystyle \lambda } eigenvalue non-zero vector
μ {\displaystyle \mathbf {\mu } } mu magnetic moment ampere square meter (A m2)
coefficient of friction unitless
dynamic viscosity Pascal second (Pa s)
permeability (electromagnetism) Henry per meter (H/m)
reduced mass kilogram (kg)
gravitational parameter
ν {\displaystyle \nu } nu frequency hertz (Hz)
kinematic viscosity meters squared per second (m2/s)
π {\displaystyle \pi } pi 3.14159... (irrational number)
ρ {\displaystyle \rho } rho mass density
usually simply called density kilogram per cubic meter (kg/m3)
free electric charge density,
not including dipole charges bound in a material coulomb per cubic meter (C/m3)
resistivity Ohm meter ( Ω {\displaystyle \Omega } m)
Σ {\displaystyle \Sigma } sigma summation operator
σ {\displaystyle \sigma } sigma electrical conductivity Siemens per meter (S/m)
normal stress
τ {\displaystyle \tau } tau torque newton meter (N m)
shear stress
time constant second (s)
6.28318... (2π)
Φ {\displaystyle \Phi } phi field strength unit varies depending on context
Magnetic Flux Weber (Wb)
ϕ {\displaystyle \phi } phi Electric Potential
Higgs Field Work Function
Ψ {\displaystyle \Psi } psi Wave function unitless
ω {\displaystyle \omega } omega Angular frequency
2πf
radian per second (rad/s)
Ω {\displaystyle \Omega } omega electric resistance ohm
Other characters[edit]
Symbol
Name
Meaning
SI unit of measure
∇ ⋅ {\displaystyle \nabla \cdot } nabla dot the divergence operator
often pronounced "del dot" per meter (m−1)
∇ × {\displaystyle \nabla \times } nabla cross the curl operator
often pronounced "del cross" per meter (m−1)
∇ {\displaystyle \nabla } nabla del (differential operator)
Δ {\displaystyle \Delta } Delta the Laplace operator per square meter (m−2)
∂ {\displaystyle \partial } "der", "dow", "die", "partial" or simply "d" partial derivative (e.g. ∂ y / ∂ x {\displaystyle \partial y/\partial x} )
◻ {\displaystyle \Box } D'Alembert operator ∇ 2 − ∂ t 2 {\displaystyle \nabla ^{2}-\partial _{t}^{2}}
See also[edit]
List of letters used in mathematics and science
Roman letters used in mathematics
Greek letters used in mathematics, science, and engineering
Physical constant
Physical quantity
International System of Units
ISO 31
Generally used symbols
References[edit]
Retrieved from "https://en.wikipedia.org/w/index.php?title=List_of_common_physics_notations&oldid=799325768"
Categories:
Physics-related lists
Navigation menu
Personal tools
Not logged in
Talk
Contributions
Create account
Log in
Namespaces
Article
Talk
Variants
Views
Read
Edit
View history
More
Search
Navigation
Main page
Contents
Featured content
Current events
Random article
Donate to Wikipedia
Wikipedia store
Interaction
Help
About Wikipedia
Community portal
Recent changes
Contact page
Tools
What links here
Related changes
Upload file
Special pages
Permanent link
Page information
Wikidata item
Cite this page
Print/export
Create a book
Download as PDF
Printable version
Languages
العربية
অসমীয়া
Русский
Українська
中文
Edit links
This page was last edited on 7 September 2017, at 00:50.
Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.
Privacy policy
About Wikipedia
Disclaimers
Contact Wikipedia
Developers
Cookie statement
Mobile view

key: html

type: local

value: div

attrs: id class

values: mw-page-base noprint

values: mw-head-base noprint

attrs: id class style

value: a

attrs: id class lang

values: firstHeading firstHeading en

values: siteSub noprint

values: jump-to-nav mw-jump

value: b

attrs: href class title

values:
/wiki/Vector_(geometry)
mw-redirect
Vector (geometry)

value: p

attrs: role tabindex class

values: button 0 togglelink

value: li

values: vertical-align: -0.338ex; width:1.754ex; height:2.343ex;

values:
/wiki/Magnetic_vector_potential
mw-redirect
Magnetic vector potential

values: vertical-align: -0.338ex; width:1.31ex; height:1.676ex;

values: vertical-align: -0.338ex; width:1.911ex; height:2.176ex;

values:
/wiki/Magnetic_field_density
mw-redirect
Magnetic field density

values:
/wiki/Square_meter
mw-redirect
Square meter

values: vertical-align: -0.338ex; width:1.777ex; height:2.176ex;

values: vertical-align: -0.338ex; width:1.017ex; height:1.676ex;

values:
/wiki/Specific_heat_capacity
mw-redirect
Specific heat capacity

values: vertical-align: -0.338ex; width:2.06ex; height:2.176ex;

values: vertical-align: -0.338ex; width:1.935ex; height:2.176ex;

values:
/wiki/Cubic_meter
mw-redirect
Cubic meter

values: vertical-align: -0.338ex; width:1.226ex; height:2.176ex;

values: vertical-align: -0.338ex; width:2.566ex; height:2.176ex;

values: vertical-align: -0.338ex; width:3.256ex; height:2.176ex;

values:
/wiki/Surface_normal
mw-redirect
Surface normal

values: vertical-align: -0.338ex; width:3.024ex; height:2.176ex;

values: vertical-align: -0.338ex; width:1.767ex; height:2.176ex;

values: vertical-align: -0.338ex; width:1.786ex; height:2.176ex;

values: vertical-align: -0.338ex; width:1.094ex; height:1.676ex;

values:
/wiki/Eccentricity_(orbit)
mw-redirect
Eccentricity (orbit)

values: vertical-align: -0.338ex; width:1.693ex; height:2.176ex;

values: vertical-align: -0.671ex; width:1.289ex; height:2.509ex;

values: vertical-align: -0.338ex; width:1.837ex; height:2.176ex;

values: vertical-align: -0.671ex; width:1.126ex; height:2.009ex;

values: vertical-align: -0.338ex; width:2.102ex; height:2.176ex;

values: vertical-align: -0.338ex; width:2.074ex; height:2.176ex;

values: vertical-align: -0.338ex; width:1.349ex; height:2.176ex;

values:
/wiki/Planck%27s_constant
mw-redirect
Planck's constant

values: vertical-align: -0.338ex; width:1.317ex; height:2.176ex;

values: vertical-align: -1.838ex; width:5.411ex; height:4.843ex;

values: \textstyle \left ( {\frac h {2\pi}}\right)

values: vertical-align: -0.338ex; width:1.182ex; height:2.176ex;

values: vertical-align: -0.338ex; width:0.813ex; height:2.176ex;

values: vertical-align: -0.338ex; width:1.347ex; height:3.009ex;

values: vertical-align: -0.338ex; width:1.391ex; height:2.176ex;

values: vertical-align: -0.671ex; width:1.347ex; height:3.343ex;

values: vertical-align: -0.338ex; width:2.076ex; height:2.176ex;

values: vertical-align: -0.338ex; width:1.222ex; height:2.176ex;

values: vertical-align: -0.338ex; width:1.421ex; height:3.009ex;

values: vertical-align: -0.338ex; width:1.593ex; height:2.176ex;

values:
/wiki/Lagrangian_Mechanics
mw-redirect
Lagrangian Mechanics

values: vertical-align: -0.338ex; width:0.704ex; height:2.176ex;

values: vertical-align: -0.338ex; width:2.453ex; height:2.176ex;

values: vertical-align: -0.338ex; width:2.051ex; height:1.676ex;

values: vertical-align: -0.338ex; width:1.405ex; height:1.676ex;

values: vertical-align: -0.338ex; width:1.756ex; height:2.176ex;

values: vertical-align: -0.671ex; width:1.496ex; height:2.009ex;

values: vertical-align: -0.671ex; width:1.849ex; height:2.509ex;

values: vertical-align: -0.671ex; width:1.08ex; height:2.009ex;

values: vertical-align: -0.338ex; width:1.774ex; height:2.176ex;

values:
/wiki/Electrical_resistance
mw-redirect
Electrical resistance

values:
/wiki/Ricci_tensor
mw-redirect
Ricci tensor

values: vertical-align: -0.338ex; width:1.113ex; height:1.676ex;

values:
/wiki/Position_vector
mw-redirect
Position vector

values: vertical-align: -0.338ex; width:1.059ex; height:1.676ex;

values: vertical-align: -0.338ex; width:1.51ex; height:2.176ex;

values: vertical-align: -0.338ex; width:1.101ex; height:1.676ex;

values: vertical-align: -0.338ex; width:1.647ex; height:2.176ex;

values:
/wiki/Period_(physics)
mw-redirect
Period (physics)

values: vertical-align: -0.338ex; width:0.85ex; height:2.009ex;

values: vertical-align: -0.338ex; width:2.067ex; height:2.176ex;

values: vertical-align: -0.338ex; width:1.793ex; height:2.176ex;

values: vertical-align: -0.338ex; width:1.34ex; height:1.676ex;

values:
/wiki/Relativistic_mass
mw-redirect
Relativistic mass

values: vertical-align: -0.338ex; width:1.798ex; height:2.176ex;

values: vertical-align: -0.338ex; width:1.421ex; height:1.676ex;

values: vertical-align: -0.338ex; width:2.446ex; height:2.176ex;

values:
/wiki/Mechanical_work
mw-redirect
Mechanical work

values: vertical-align: -0.338ex; width:1.675ex; height:1.676ex;

values: vertical-align: -0.338ex; width:1.691ex; height:2.176ex;

values: vertical-align: -0.338ex; width:1.498ex; height:1.676ex;

values:
/wiki/Alpha_(letter)
mw-redirect
Alpha (letter)

values: vertical-align: -0.671ex; width:1.342ex; height:2.509ex;

values:
/wiki/Beta_(letter)
mw-redirect
Beta (letter)

values: vertical-align: -0.838ex; width:1.273ex; height:2.176ex;

values:
/wiki/Shear_strain
mw-redirect
Shear strain

values: vertical-align: -0.338ex; width:1.946ex; height:2.343ex;

values: vertical-align: -0.338ex; width:3.286ex; height:2.343ex;

values: vertical-align: -0.338ex; width:1.059ex; height:2.343ex;

values: vertical-align: -0.338ex; width:0.955ex; height:1.676ex;

values:
/wiki/Epsilon_(letter)
mw-redirect
Epsilon (letter)

values:
/wiki/Strain_(materials_science)
mw-redirect
Strain (materials science)

values: vertical-align: -0.671ex; width:2.156ex; height:2.009ex;

values: vertical-align: -0.671ex; width:1.106ex; height:2.509ex;

values: vertical-align: -0.838ex; width:1.18ex; height:2.176ex;

values:
/wiki/Eta_(letter)
mw-redirect
Eta (letter)

values: vertical-align: -0.338ex; width:1.101ex; height:2.176ex;

values:
/wiki/Theta_(letter)
mw-redirect
Theta (letter)

values: vertical-align: -0.338ex; width:1.819ex; height:2.176ex;

values:
/wiki/Kappa_(letter)
mw-redirect
Kappa (letter)

values:
/wiki/Torsion_coefficient
mw-redirect
Torsion coefficient

values: vertical-align: -0.338ex; width:1.623ex; height:2.343ex;

values: vertical-align: -0.338ex; width:1.366ex; height:2.176ex;

values: vertical-align: -0.838ex; width:1.412ex; height:2.176ex;

values:
/wiki/Coefficient_of_friction
mw-redirect
Coefficient of friction

values:
/wiki/Dynamic_viscosity
mw-redirect
Dynamic viscosity

values: vertical-align: -0.338ex; width:1.243ex; height:1.676ex;

values:
/wiki/Kinematic_viscosity
mw-redirect
Kinematic viscosity

values: vertical-align: -0.338ex; width:1.342ex; height:1.676ex;

values: vertical-align: -0.838ex; width:1.212ex; height:2.176ex;

values:
/wiki/Rho_(letter)
mw-redirect
Rho (letter)

values: vertical-align: -0.338ex; width:1.689ex; height:2.176ex;

values:
/wiki/Electrical_conductivity
mw-redirect
Electrical conductivity

values:
/wiki/Stress_(physics)
mw-redirect
Stress (physics)

values: vertical-align: -0.338ex; width:1.212ex; height:1.676ex;

values:
/wiki/Newton_meter
mw-redirect
Newton meter

values:
/wiki/Phi_(letter)
mw-redirect
Phi (letter)

values:
/wiki/Magnetic_Flux
mw-redirect
Magnetic Flux

values: vertical-align: -0.671ex; width:1.396ex; height:2.509ex;

values:
/wiki/Electric_Potential
mw-redirect
Electric Potential

values:
/wiki/Higgs_field
mw-redirect
Higgs field

values: vertical-align: -0.338ex; width:1.456ex; height:1.676ex;

values: vertical-align: -0.338ex; width:3.12ex; height:2.176ex;

values: vertical-align: -0.338ex; width:4.281ex; height:2.176ex;

values: vertical-align: -0.338ex; width:1.946ex; height:2.176ex;

values: vertical-align: -0.338ex; width:1.329ex; height:2.343ex;

values: vertical-align: -0.838ex; width:6.336ex; height:2.843ex;

values: vertical-align: -1.005ex; width:8.274ex; height:3.343ex;

values: List of letters used in mathematics and science

values:
/wiki/Roman_letters_used_in_mathematics
mw-redirect
Roman letters used in mathematics

values:
catlinks mw-normal-catlinks
catlinks mw-normal-catlinks
interface

values: Discussion about edits from this IP address [ctrl-alt-n]

values: A list of edits made from this IP address [ctrl-alt-y]

values: You are encouraged to create an account and log in; however, it is not mandatory

values: You're encouraged to log in; however, it's not mandatory. [ctrl-alt-o]

values:
/wiki/List_of_common_physics_notations
View the content page [ctrl-alt-c]
c

values: Discussion about the content page [ctrl-alt-t]

values: p-namespaces navigation vectorTabs p-namespaces-label

values:
p-variants
navigation
vectorMenu emptyPortlet
p-variants-label

values: Past revisions of this page [ctrl-alt-h]

values: p-views navigation vectorTabs p-views-label

values:
p-cactions
navigation
vectorMenu emptyPortlet
p-cactions-label

values:
mw-wiki-logo
/wiki/Main_Page
Visit the main page

values:
/wiki/Main_Page
Visit the main page [ctrl-alt-z]
z

values: Featured content – the best of Wikipedia

values: Find background information on current events

values:
/wiki/Special:Random
Load a random article [ctrl-alt-x]
x

values: portal navigation p-navigation p-navigation-label

values: Guidance on how to use and edit Wikipedia

values: About the project, what you can do, where to find things

values: A list of recent changes in the wiki [ctrl-alt-r]

values: portal navigation p-interaction p-interaction-label

values: List of all English Wikipedia pages containing links to this page [ctrl-alt-j]

values: Recent changes in pages linked from this page [ctrl-alt-k]

values:
/wiki/Wikipedia:File_Upload_Wizard
Upload files [ctrl-alt-u]
u

values: A list of all special pages [ctrl-alt-q]

values: Permanent link to this revision of the page

values: Link to connected data repository item [ctrl-alt-g]

values: Printable version of this page [ctrl-alt-p]

values:
uls-settings-trigger
Language settings
0
button
true

values: قائمة الرموز المستخدمة في الفيزياء – Arabic

values: পদাৰ্থ বিজ্ঞানৰ সাধাৰণ সাংকেতিক চিহ্নসমূহৰ তালিকা – Assamese

values: Список обозначений в физике – Russian

values: Список позначень у фізиці – Ukrainian

values:
suggestions
display: none; font-size: 13px;

values: mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject page-List_of_common_physics_notations rootpage-List_of_common_physics_notations vector-nav-directionality skin-vector action-view


text:
United States twenty-dollar bill
From Wikipedia, the free encyclopedia
Jump to: navigation, search
Twenty dollars
(United States)
Value
$20
Width
156 mm
Height
66.3 mm
Weight
Approx 1 g
Paper type
75% cotton
25% linen
Years of printing
1861–present
Obverse
Design
Andrew Jackson
Design date
2003
Reverse
Design
White House
Design date
2003
The United States twenty-dollar bill ($20) is a denomination of U.S. currency. The seventh U.S. President (1829–37), Andrew Jackson has been featured on the front side of the bill since 1928, while the White House is featured on the reverse side.
As of December 2013, the average circulation life of a $20 bill is 7.9 years before it is replaced due to wear.[1] Approximately 11% of all notes printed in 2009 were $20 bills.[2] Twenty-dollar bills are delivered by Federal Reserve Banks in violet straps.
Contents
[hide]
1 Pre-Federal Reserve history
2 Federal Reserve history
3 Proposal for a woman's portrait
4 See also
5 References
6 External links
Pre-Federal Reserve history[edit]
This section does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed. (July 2008) (Learn how and when to remove this template message)
1863 $20 Legal Tender note
1880 $20 Legal Tender depicting Alexander Hamilton
Series 1905 $20 gold certificate
1861: A demand note with Lady Liberty holding a sword and shield on the front, and an abstract design on the back. The back is printed green.
1862: A note that is very similar, the first $20 United States note. The back is different, with several small variations extant.
1863: A gold certificate $20 note with an Eagle vignette on the face. The reverse has a $20 gold coin and various abstract elements. The back is orange.
1865: A national bank note with "The Battle of Lexington" and of Pocahontas' marriage to John Rolfe in black, and a green border.
1869: A new United States note design with Alexander Hamilton on the left side of the front and Victory holding a shield and sword. The back design is green.
1875: As above, except with a different reverse.
1878: A silver certificate $20 note with a portrait of Stephen Decatur on the right side of the face. The back design is black.
1882: A new gold certificate with a portrait of James Garfield on the right of the face. The back is orange and features an eagle.
1882: A new national bank note. The front is similar, but the back is different and printed in brown.
1886: A new silver certificate $20 note with Daniel Manning on the center of the face.
1890: A treasury (coin) note with John Marshall on the left of the face. Two different backs exist: both with abstract designs.
1902: A new national bank note. The front design features Hugh McCulloch, and the back has a vignette of an allegorical America.
1905: A new gold certificate $20 note with George Washington on the center of the face. The back design is orange.
Federal Reserve history[edit]
1914 $20 Federal Reserve Note.
1928 $20 small-size Federal Reserve Note.
Series 1929 $20 National Currency note issued by the Cleveland Federal Reserve Bank.
Series 1995 $20 Federal Reserve Note.
Series 1996 $20 Federal Reserve Note.
The security strip in a twenty-dollar bill glows green under a blacklight.
Andrew Jackson first appeared on the $20 bill in 1928. Although 1928 coincides with the 100th anniversary of Jackson's election as president, it is not clear why the portrait on the bill was switched from Grover Cleveland to Jackson. (Cleveland's portrait was moved to the new $1000 bill the same year). According to the U.S. Treasury, "Treasury Department records do not reveal the reason that portraits of these particular statesmen were chosen in preference to those of other persons of equal importance and prominence."[3]
The placement of Jackson on the $20 bill may be a historical irony; as president, he vehemently opposed both the National Bank and paper money and made the goal of his administration the destruction of the National Bank.[4][5] In his farewell address to the nation, he cautioned the public about paper money.[6]
1914: Began as a large-sized note, a portrait of Grover Cleveland on the face, and, on the back, a steam locomotive and an automobile approaching from the left, and a steamship approaching from the right.
1918: A federal reserve bank note with Grover Cleveland on the front, and a back design similar to the 1914 Federal Reserve Note.
1928: Switched to a small-sized note with a portrait of Andrew Jackson on the face and the south view of the White House on the reverse. The banknote is redeemable in gold or silver (at the bearer's discretion) at any Federal Reserve Bank.
1933: With the U.S. having abandoned the gold standard, the bill is no longer redeemable in gold, but rather in "lawful money", meaning silver.
1942: A special emergency series, with brown serial numbers and "HAWAII" overprinted on both the front and the back, is issued. These notes are designed to circulate on the islands, and be deemed invalid in the event of a Japanese invasion.
1948: The White House picture was updated to reflect renovations to the building itself, including the addition of the Truman Balcony, as well as the passage of time. Most notably, the trees are larger.
1950: Design elements like the serial numbers are reduced in size and moved around subtly, presumably for aesthetic reasons.
1963: "Will Pay To The Bearer On Demand" is removed from the front of the bill below the portrait, and the legal tender designation is shortened to "This note is legal tender for all debts, public and private" (eliminating "and is redeemable in lawful money at the United States Treasury, or at any Federal Reserve Bank.") Also, "In God We Trust" is added above the White House on the reverse. These two acts (one taking U.S. currency off silver backing, and the other authorizing the national motto) are coincidental, even if their combined result is implemented in one redesign. Also, several design elements are rearranged, less perceptibly than the change in 1950, mostly to make room for the slightly rearranged obligations.
1969: The new treasury seal appears on all denominations, including the $20.
1977: A new type of serial-number press results in a slightly different font. The old presses are gradually retired, and old-style serial numbers appear as late as 1981 for this denomination.
1992: Anti-counterfeiting features are added: microprinting around the portrait, and a plastic strip embedded in the paper. Even though the bills read Series 1990, the first bills were printed in April 1992.[7]
September 24, 1998: Received a completely new appearance to further deter counterfeiting; the picture of the White House was changed to the north side view. A larger, off-center portrait of Jackson was used on front, and several anti-counterfeiting features were added, including color-shifting ink, microprinting, and a watermark. The plastic strip now reads "USA 20" and glows green under a black light.[8] The bills were first printed in June 1998.[9]
October 9, 2003:[10] The current series of 20 dollar bills is released with light background shading in green and yellow, and no oval around Andrew Jackson's portrait (background images of eagles, etc. were also added to the front); the back is the same view of the White House, but without the oval around it. Ninety faint "20"s are scattered on the back in yellow as a "EURion constellation" to prevent photocopying. The first issue's series date is 2004 with Marin-Snow signatures. The bills were first printed in April 2003.[11]
Proposal for a woman's portrait[edit]
In a campaign called "Women on 20s", selected voters were asked to choose three of 15 female candidates to have a portrait on the $20 bill. The goal was to have a woman on the $20 bill by 2020, the centennial of the 19th Amendment which gave women the right to vote.[12] Among the candidates on the petition were Harriet Tubman, Eleanor Roosevelt, Rosa Parks, and Wilma Mankiller, the first female chief of the Cherokee Nation.[13][14][15]
On May 12, 2015, Tubman was announced as the winning candidate of that "grassroots" poll with more than 600,000 people surveyed and more than 118,000 choosing Tubman, followed by Roosevelt, Parks and Mankiller.[16]
On June 17, 2015, then-Treasury Secretary Jack Lew announced that a woman's portrait would be featured on a redesigned $10 bill by 2020, replacing Alexander Hamilton.[17] However, that decision was reversed, at least in part due to Hamilton's surging popularity following the hit Broadway musical Hamilton.[18]
On April 20, 2016, Lew officially announced that Alexander Hamilton would remain on the $10 bill, while Andrew Jackson would be replaced by Tubman on the front of the $20 bill, with Jackson appearing on the reverse.[19][20][21] Lew simultaneously announced that the five- and ten-dollar bills would also be redesigned in the coming years and put into production in the next decade.[19][21] Steven Mnuchin, the current U.S. Treasury Secretary, said that he will not commit to putting Tubman on the twenty-dollar bill, explaining "People have been on the bills for a long period of time. This is something we’ll consider; right now we have a lot more important issues to focus on."[22]
See also[edit]
Twenty Bucks, a 1993 movie that follows the travels of a $20 bill
References[edit]
Jump up ^ "How long is the life span of U.S. paper money?". Federal Reserve. Retrieved April 16, 2015.
Jump up ^ "Money Facts". Bureau of Engraving and Printing.
Jump up ^ "U.S. Currency FAQs". U.S. Bureau of Engraving and Printing. Archived from the original on May 5, 2015. Retrieved 13 May 2015.
Jump up ^ "Jackson as President". CliffsNotes. Retrieved November 20, 2007.
Jump up ^ "Jackson Vetoes Bank Bill — July 10, 1832". Miller Center of Public Affairs, University of Virginia. Retrieved November 20, 2007.
Jump up ^ Wikisource:Andrew Jackson's Farewell Address
Jump up ^ "USPaperMoney.Info: Series 1990 $20".
Jump up ^ "New $20 Bill Debuts September 24th - 09/21/98".
Jump up ^ "USPaperMoney.Info: Series 1996 $20".
Jump up ^ "Anti-Counterfeiting". Bureau of Engraving and Printing. United States Treasury. 2007. Retrieved May 13, 2015.
Jump up ^ "USPaperMoney.Info: Series 2004 $20".
Jump up ^ "Why the $20?". Women On 20s. Retrieved May 13, 2015.
Jump up ^ Tan, Avianne (April 8, 2015). "'Women on 20s' to Ask President Obama to Put One of These 4 Women on $20 Bill". ABC News. Retrieved April 9, 2015.
Jump up ^ Which country has the least sexist banknotes? BBC. April 13, 2015. Retrieved on April 14, 2015.
Jump up ^ "Final Round Candidates". Women On 20s. Retrieved May 13, 2015.
Jump up ^ "Harriet Tubman wins poll to replace Andrew Jackson on $20 bill". New York Post. Reuters. May 13, 2015. Retrieved July 27, 2015.
Jump up ^ "Andrew Jackson To Be Taken Off The $20 Bill". Huffington Post. April 17, 2016. Retrieved April 19, 2016.
Jump up ^ Nguyen, Tina. ""Hamilton" Fans, Rejoice: Founding Father to Stay on the $10 Bill". Vanity Fair.
^ Jump up to: a b "Treasury Secretary Lew Announces Front of New $20 to Feature Harriet Tubman, Lays Out Plans for New $20, $10 and $5" (Press release). United States Department of the Treasury. April 20, 2016. Retrieved April 27, 2016.
Jump up ^ White, Ben; McCaskill, Nolan D. "Treasury's Lew to announce Hamilton to stay on $10 bill". Politico. Retrieved April 20, 2016.
^ Jump up to: a b Korte, Gregory (April 21, 2016). "Anti-slavery activist Harriet Tubman to replace Jackson on the front of the $20 bill". USA Today. Retrieved August 28, 2017.
Jump up ^ Temple-West, Patrick (August 31, 2017). "Mnuchin dismisses question about putting Harriet Tubman on $20 bill". Politico. Retrieved 6 September 2017.
External links[edit]
Wikimedia Commons has media related to 20 United States dollar banknotes.
[hide]
v
t
e
United States currency and coinage
Topics
Federal Reserve System
Federal Reserve Note
U.S. dollar
U.S. Mint
Philadelphia
San Francisco
Denver
West Point
Bureau of Engraving and Printing
Mutilated currency
Current coinage
Penny (1¢)
Nickel (5¢)
Dime (10¢)
Quarter (25¢)
Half dollar (50¢)
Dollar ($1)
Bullion coinage
America the Beautiful Silver Bullion Coins
American Buffalo
American Gold Eagle
American Platinum Eagle
American Silver Eagle
American Palladium Eagle
Current paper money
$1
$2
$5
$10
$20
$50
$100
See also
Bicentennial coinage
Commemoratives
Early Commemoratives
Modern Commemoratives
Confederate dollar
Large denominations
Obsolete denominations
Promotional fake denominations
Counterfeit United States currency
Mill
Coin production
In God We Trust
E pluribus unum
Annuit cœptis
Nicknames
Replacement banknote
Sales tax token
Retrieved from "https://en.wikipedia.org/w/index.php?title=United_States_twenty-dollar_bill&oldid=805519077"
Categories:
1861 establishments in the United States
Currency introduced in 1861
Andrew Jackson
Banknotes of the United States
Twenty-base-unit banknotes
Hidden categories:
Use mdy dates from April 2015
Articles needing additional references from July 2008
All articles needing additional references
Navigation menu
Personal tools
Not logged in
Talk
Contributions
Create account
Log in
Namespaces
Article
Talk
Variants
Views
Read
Edit
View history
More
Search
Navigation
Main page
Contents
Featured content
Current events
Random article
Donate to Wikipedia
Wikipedia store
Interaction
Help
About Wikipedia
Community portal
Recent changes
Contact page
Tools
What links here
Related changes
Upload file
Special pages
Permanent link
Page information
Wikidata item
Cite this page
Print/export
Create a book
Download as PDF
Printable version
In other projects
Wikimedia Commons
Languages
Español
Français
Русский
Українська
中文
Edit links
This page was last edited on 15 October 2017, at 22:44.
Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.
Privacy policy
About Wikipedia
Disclaimers
Contact Wikipedia
Developers
Cookie statement
Mobile view
Enable previews

values: mw-page-base noprint

values: mw-head-base noprint

values: firstHeading firstHeading en

values: siteSub noprint

values: jump-to-nav mw-jump

attrs: href title

value: caption

value: th

attrs: colspan style

value: tr

value: td

attrs: alt src width height srcset data-file-width data-file-height

values: Edit section: Pre-Federal Reserve history

values: Help:Introduction to referencing with Wiki Markup/1

values: plainlinks metadata ambox ambox-content ambox-Unreferenced

attrs: alt src width height class srcset data-file-width data-file-height

values:
/wiki/Demand_note
mw-redirect
Demand note

values:
/wiki/United_States_Notes
mw-redirect
United States Notes

values:
/wiki/James_Garfield
mw-redirect
James Garfield

values:
/wiki/Small-sized_note
mw-redirect
Small-sized note

values:
/wiki/HAWAII_Overprint_Note
mw-redirect
HAWAII Overprint Note

values:
/wiki/Black_light
mw-redirect
Black light

values:
/wiki/Oval_(geometry)
mw-redirect
Oval (geometry)

values: Edit section: Proposal for a woman's portrait

values: Nineteenth Amendment to the United States Constitution

values: wikisource:Andrew Jackson's Farewell Address

values:
nofollow
external text
http://www.womenon20s.org/why_the_20

values:
nofollow
external text
http://www.womenon20s.org/candidates

values: -moz-column-width: 30em; -webkit-column-width: 30em; column-width: 30em; list-style-type: decimal;

values: commons:Category:20 United States dollar banknotes

values: Modern United States commemorative coins

values: Large denominations of United States currency

values: Obsolete denominations of United States currency

values: nowraplinks collapsible autocollapse navbox-inner

values: Category:1861 establishments in the United States

values: Category:Articles needing additional references from July 2008

values: Category:All articles needing additional references

values: Discussion about edits from this IP address [ctrl-alt-n]

values: A list of edits made from this IP address [ctrl-alt-y]

values: You are encouraged to create an account and log in; however, it is not mandatory

values: You're encouraged to log in; however, it's not mandatory. [ctrl-alt-o]

values:
/wiki/United_States_twenty-dollar_bill
View the content page [ctrl-alt-c]
c

values: Discussion about the content page [ctrl-alt-t]

values: p-namespaces navigation vectorTabs p-namespaces-label

values:
p-variants
navigation
vectorMenu emptyPortlet
p-variants-label

values: Past revisions of this page [ctrl-alt-h]

values: p-views navigation vectorTabs p-views-label

values:
p-cactions
navigation
vectorMenu emptyPortlet
p-cactions-label

values:
mw-wiki-logo
/wiki/Main_Page
Visit the main page

values:
/wiki/Main_Page
Visit the main page [ctrl-alt-z]
z

values: Featured content – the best of Wikipedia

values: Find background information on current events

values:
/wiki/Special:Random
Load a random article [ctrl-alt-x]
x

values: portal navigation p-navigation p-navigation-label

values: portal navigation p-wikibase-otherprojects p-wikibase-otherprojects-label

values: Billete de veinte dólares estadounidenses – Spanish

values: Billet de 20 dollars américains – French

values: Двадцать долларов США (банкнота) – Russian

values: Двадцять доларів США – Ukrainian

values: /static/images/wikimedia-button-1.5x.png 1.5x, /static/images/wikimedia-button-2x.png 2x

values: /static/images/poweredby_mediawiki_132x47.png 1.5x, /static/images/poweredby_mediawiki_176x62.png 2x

values: mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject page-United_States_twenty-dollar_bill rootpage-United_States_twenty-dollar_bill vector-experimental-print-styles vector-nav-directionality skin-vector action-view


text:
Mercator projection
From Wikipedia, the free encyclopedia
Jump to: navigation, search
Mercator projection of the world between 82°S and 82°N.
The Mercator projection with Tissot's indicatrix of deformation.
Mercator 1569 world map (Nova et Aucta Orbis Terrae Descriptio ad Usum Navigantium Emendate Accommodata) showing latitudes 66°S to 80°N.
The Mercator projection is a cylindrical map projection presented by the Flemish 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 that conserve the angles with the meridians. Although the linear scale is equal 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 of objects as the latitude increases from the Equator to the poles, where the scale becomes infinite. So, for example, landmasses such as Greenland and Antarctica appear much larger than they actually are relative to land masses near the equator, such as Central Africa.
Contents
[hide]
1 Properties and historical details
2 Uses
2.1 Web Mercator
3 Mathematics of the Mercator projection
3.1 The spherical model
3.2 Cylindrical projections
3.2.1 Small element geometry
3.3 Derivation of the Mercator projection
3.3.1 Inverse transformations
3.3.2 Alternative expressions
3.3.3 Truncation and aspect ratio
3.4 Scale factor
3.4.1 Area scale
3.5 Distortion
3.6 Accuracy
3.7 Secant projection
3.8 Generalization to the ellipsoid
3.9 Formulae for distance
3.9.1 On the equator
3.9.2 On other parallels
3.9.3 On a meridian
3.9.4 On a rhumb
4 See also
5 Notes
6 References
7 External links
Properties and historical details[edit]
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 in the Mercator projection 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. Being a conformal projection, angles are preserved around all locations.
Because the linear scale of a Mercator map increases with latitude, it distorts the size of geographical objects far from the equator and conveys a distorted perception of the overall geometry of the planet. At latitudes greater than 70° north or south the Mercator projection is practically unusable, since the linear scale becomes infinitely high at the poles. A Mercator map can therefore never fully show the polar areas (as long as the projection is based on a cylinder centered on the Earth’s rotation axis; see the Transverse Mercator projection for another application).
All lines of constant bearing (rhumbs or loxodromes—those making constant angles with the meridians) are represented by straight segments on a Mercator map. 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 (for example).
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 (grid formed by lines of latitude and longitude), 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 a nautical atlas composed of several large-scale sheets in the cylindrical equidistant projection as a way to minimize distortion of directions. If these sheets were brought to the same scale and assembled an approximation of the Mercator projection would be obtained (1537).
English mathematician Edward Wright (c. 1558–1615), who 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.
Uses[edit]
The Mercator projection portrays Greenland as larger than Australia; in actuality, Australia is more than three and a half times as large as Greenland.
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 appears larger than Africa, when in reality Africa's area is 14 times greater and Greenland's is comparable to Algeria's alone. Africa also appears to be roughly the same size as Europe, when in reality Africa is nearly 3 times larger.
Alaska takes as much area on the map as Brazil, when Brazil's area is nearly five 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 (and would be infinitely large on a complete map), although it is actually the fifth in area.
The Mercator projection is still used commonly for navigation. On the other hand, because of great land area distortions, it is not well suited for general world maps. Therefore, Mercator himself used the equal-area sinusoidal projection to show relative areas. However, despite such distortions, Mercator projection was, especially in the late 19th and early 20th centuries, perhaps the most common projection used in world maps, despite being much criticized for this use.[1][2][3][4] Because of its very common usage, it has been supposed to have influenced on people's view of the world,[5] and because it shows countries near the Equator as too small when compared to those of Europe and North America, it has been supposed to cause people to consider those countries as less important.[6] As a result of these criticisms, most 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 he promoted is a specific parameterization of the cylindrical equal-area projection. In response, a 1989 resolution by seven North American geographical groups deprecated the use of cylindrical projections for general purpose world maps, which would include both the Mercator and the Gall–Peters.[7]
Web Mercator[edit]
Main article: Web Mercator
Many major online street mapping services (Bing Maps, OpenStreetMap, Google Maps, MapQuest, Yahoo! Maps, and others) use a variant of the Mercator projection for their map images[8] called Web Mercator or Google Web Mercator. 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 variant projection's near-conformality.
The major online street mapping services tiling systems display most of the world at the lowest zoom level as a single square image, excluding the polar regions by truncation at latitudes of φmax = ±85.05113°. (See below.) Latitude values outside this range are mapped using a different relationship that doesn't diverge at φ = ±90°.[citation needed]
Mathematics of the Mercator projection[edit]
The spherical model[edit]
Although the surface of Earth is best modelled by an oblate ellipsoid of revolution, for small scale maps the ellipsoid is approximated by a sphere of radius a. Many different methods exist for calculating a. The simplest include (a) the equatorial radius of the ellipsoid, (b) the arithmetic or geometric mean of the semi-axes of the ellipsoid, (c) the radius of the sphere having the same volume as the ellipsoid.[9] The range of all possible choices is about 35 km, but for small scale (large region) applications the variation may be ignored, and mean values of 6,371 km and 40,030 km may be taken for the radius and circumference respectively. These are the values used for numerical examples in later sections. Only high-accuracy cartography on large scale maps requires an ellipsoidal model.
Cylindrical projections[edit]
The spherical approximation of Earth with radius a can be modelled by a smaller sphere of radius R, called the globe in this section. The globe determines the scale of the map. The various cylindrical projections specify how the geographic detail is transferred from the globe to a cylinder tangential to it at the equator. The cylinder is then unrolled to give the planar map.[10][11] The fraction R/a is called the representative fraction (RF) or the principal scale of the projection. For example, a Mercator map printed in a book might have an equatorial width of 13.4 cm corresponding to a globe radius of 2.13 cm and an RF of approximately 1/300M (M is used as an abbreviation for 1,000,000 in writing an RF) whereas Mercator's original 1569 map has a width of 198 cm corresponding to a globe radius of 31.5 cm and an RF of about 1/20M.
A cylindrical map projection is specified by formulae linking the geographic coordinates of latitude φ and longitude λ to Cartesian coordinates on the map with origin on the equator and x-axis along the equator. By construction, all points on the same meridian lie on the same generator[12] of the cylinder at a constant value of x, but the distance y along the generator (measured from the equator) is an arbitrary[13] function of latitude, y(φ). In general this function does not describe the geometrical projection (as of light rays onto a screen) from the centre of the globe to the cylinder, which is only one of an unlimited number of ways to conceptually project a cylindrical map.
Since the cylinder is tangential to the globe at the equator, the scale factor between globe and cylinder is unity on the equator but nowhere else. In particular since the radius of a parallel, or circle of latitude, is R cos φ, the corresponding parallel on the map must have been stretched by a factor of 1/cos φ = sec φ. This scale factor on the parallel is conventionally denoted by k and the corresponding scale factor on the meridian is denoted by h.[14]
Small element geometry[edit]
The relations between y(φ) and properties of the projection, such as the transformation of angles and the variation in scale, follow from the geometry of corresponding small elements on the globe and map. The figure below shows a point P at latitude φ and longitude λ on the globe and a nearby point Q at latitude φ + δφ and longitude λ + δλ. The vertical lines PK and MQ are arcs of meridians of length Rδφ.[15] The horizontal lines PM and KQ are arcs of parallels of length R(cos φ)δλ.[16] The corresponding points on the projection define a rectangle of width δx and height δy.
For small elements, the angle PKQ is approximately a right angle and therefore
tan  α ≈ R cos  φ δ λ R δ φ , tan  β = δ x δ y , {\displaystyle \tan \alpha \approx {\frac {R\cos \varphi \,\delta \lambda }{R\,\delta \varphi }},\qquad \qquad \tan \beta ={\frac {\delta x}{\delta y}},}
The previously mentioned scaling factors from globe to cylinder are given by
parallel scale factor k ( φ ) = P ′ M ′ P M = δ x R cos  φ δ λ , {\displaystyle \quad k(\varphi )\;=\;{\frac {P'M'}{PM}}\;=\;{\frac {\delta x}{R\cos \varphi \,\delta \lambda }},}
meridian scale factor h ( φ ) = P ′ K ′ P K = δ y R δ φ . {\displaystyle \quad h(\varphi )\;=\;{\frac {P'K'}{PK}}\;=\;{\frac {\delta y}{R\delta \varphi \,}}.}
Since the meridians are mapped to lines of constant x we must have x = R(λ − λ0) and δx = Rδλ, (λ in radians). Therefore, in the limit of infinitesimally small elements
tan  β = R sec  φ y ′ ( φ ) tan  α , k = sec  φ , h = y ′ ( φ ) R . {\displaystyle \tan \beta ={\frac {R\sec \varphi }{y'(\varphi )}}\tan \alpha \,,\qquad k=\sec \varphi \,,\qquad h={\frac {y'(\varphi )}{R}}.}
Derivation of the Mercator projection[edit]
The choice of the function y(φ) for the Mercator projection is determined by the demand that the projection be conformal, a condition which can be defined in two equivalent ways:
Equality of angles. The condition that a sailing course of constant azimuth α on the globe is mapped into a constant grid bearing β on the map. Setting α = β in the above equations gives y′(φ) = R sec φ.
Isotropy of scale factors. This is the statement that the point scale factor is independent of direction so that small shapes are preserved by the projection. Setting h = k in the above equations again gives y′(φ) = R sec φ.
Integrating the equation
y ′ ( φ ) = R sec  φ , {\displaystyle y'(\varphi )=R\sec \varphi ,}
with y(0) = 0, by using integral tables[17] or elementary methods,[18] gives y(φ). Therefore,
x = R ( λ − λ 0 ) , y = R ln  [ tan  ( π 4 + φ 2 ) ] . {\displaystyle x=R(\lambda -\lambda _{0}),\qquad y=R\ln \left[\tan \left({\frac {\pi }{4}}+{\frac {\varphi }{2}}\right)\right].}
In the first equation λ0 is the longitude of an arbitrary central meridian usually, but not always, that of Greenwich (i.e., zero). The difference (λ − λ0) is in radians.
The function y(φ) is plotted alongside φ for the case R = 1: it tends to infinity at the poles. The linear y-axis values are not usually shown on printed maps; instead some maps show the non-linear scale of latitude values on the right. More often than not the maps show only a graticule of selected meridians and parallels
Inverse transformations[edit]
λ = λ 0 + x R , φ = 2 tan − 1  [ exp  ( y R ) ] − π 2 . {\displaystyle \lambda =\lambda _{0}+{\frac {x}{R}},\qquad \varphi =2\tan ^{-1}\left[\exp \left({\frac {y}{R}}\right)\right]-{\frac {\pi }{2}}\,.}
The expression on the right of the second equation defines the Gudermannian function; i.e., φ = gd(y/R): the direct equation may therefore be written as y = R·gd−1(φ).[17]
Alternative expressions[edit]
There are many alternative expressions for y(φ), all derived by elementary manipulations.[18]
y = R 2 ln  [ 1 + sin  φ 1 − sin  φ ] = R ln  [ 1 + sin  φ cos  φ ] = R ln  ( sec  φ + tan  φ ) = R tanh − 1  ( sin  φ ) = R sinh − 1  ( tan  φ ) = R sgn  ( φ ) cosh − 1  ( sec  φ ) = R gd − 1  ( φ ) . {\displaystyle {\begin{aligned}y&=&{\frac {R}{2}}\ln \left[{\frac {1+\sin \varphi }{1-\sin \varphi }}\right]&=&{R}\ln \left[{\frac {1+\sin \varphi }{\cos \varphi }}\right]&=R\ln \left(\sec \varphi +\tan \varphi \right)\\[2ex]&=&R\tanh ^{-1}\left(\sin \varphi \right)&=&R\sinh ^{-1}\left(\tan \varphi \right)&=R\operatorname {sgn} (\varphi )\cosh ^{-1}\left(\sec \varphi \right)=R\operatorname {gd} ^{-1}(\varphi ).\end{aligned}}}
Corresponding inverses are:
φ = sin − 1  ( tanh  y R ) = tan − 1  ( sinh  y R ) = sgn  ( y ) sec − 1  ( cosh  y R ) = gd  y R . {\displaystyle \varphi =\sin ^{-1}\left(\tanh {\frac {y}{R}}\right)=\tan ^{-1}\left(\sinh {\frac {y}{R}}\right)=\operatorname {sgn} (y)\sec ^{-1}\left(\cosh {\frac {y}{R}}\right)=\operatorname {gd} {\frac {y}{R}}.}
For angles expressed in degrees:
x = π R ( λ ∘ − λ 0 ∘ ) 180 , y = R ln  [ tan  ( 45 + φ ∘ 2 ) ] . {\displaystyle x={\frac {\pi R(\lambda ^{\circ }-\lambda _{0}^{\circ })}{180}},\qquad \quad y=R\ln \left[\tan \left(45+{\frac {\varphi ^{\circ }}{2}}\right)\right].}
The above formulae are written in terms of the globe radius R. It is often convenient to work directly with the map width W = 2πR. For example, the basic transformation equations become
x = W 2 π ( λ − λ 0 ) , y = W 2 π ln  [ tan  ( π 4 + φ 2 ) ] . {\displaystyle x={\frac {W}{2\pi }}\left(\lambda -\lambda _{0}\right),\qquad \quad y={\frac {W}{2\pi }}\ln \left[\tan \left({\frac {\pi }{4}}+{\frac {\varphi }{2}}\right)\right].}
Truncation and aspect ratio[edit]
The ordinate y of the Mercator projection becomes infinite at the poles and the map must be truncated at some latitude less than ninety degrees. This need not be done symmetrically. Mercator's original map is truncated at 80°N and 66°S with the result that European countries were moved towards the centre of the map. The aspect ratio of his map is 198/120 = 1.65. Even more extreme truncations have been used: a Finnish school atlas was truncated at approximately 76°N and 56°S, an aspect ratio of 1.97.
Much web based mapping uses a zoomable version of the Mercator projection with an aspect ratio of unity. In this case the maximum latitude attained must correspond to y = ±W/2, or equivalently y/R = π. Any of the inverse transformation formulae may be used to calculate the corresponding latitudes:
φ = tan − 1  [ sinh  ( y R ) ] = tan − 1  [ sinh  π ] = tan − 1  [ 11.5487 ] = 85.05113 ∘ . {\displaystyle \varphi =\tan ^{-1}\left[\sinh \left({\frac {y}{R}}\right)\right]=\tan ^{-1}\left[\sinh \pi \right]=\tan ^{-1}\left[11.5487\right]=85.05113^{\circ }.}
Scale factor[edit]
The figure comparing the infinitesimal elements on globe and projection shows that when α=β the triangles PQM and P′Q′M′ are similar so that the scale factor in an arbitrary direction is the same as the parallel and meridian scale factors:
δ s ′ δ s = P ′ Q ′ P Q = P ′ M ′ P M = k = P ′ K ′ P K = h = sec  φ . {\displaystyle {\frac {\delta s'}{\delta s}}={\frac {P'Q'}{PQ}}={\frac {P'M'}{PM}}=k={\frac {P'K'}{PK}}=h=\sec \varphi .}
This result holds for an arbitrary direction: the definition of isotropy of the point scale factor. The graph shows the variation of the scale factor with latitude. Some numerical values are listed below.
at latitude 30° the scale factor is k = sec 30° = 1.15,
at latitude 45° the scale factor is k = sec 45° = 1.41,
at latitude 60° the scale factor is k = sec 60° = 2,
at latitude 80° the scale factor is k = sec 80° = 5.76,
at latitude 85° the scale factor is k = sec 85° = 11.5
Working from the projected map requires the scale factor in terms of the Mercator ordinate y (unless the map is provided with an explicit latitude scale). Since ruler measurements can furnish the map ordinate y and also the width W of the map then y/R = 2πy/W and the scale factor is determined using one of the alternative forms for the forms of the inverse transformation:
k = sec  φ = cosh  ( y R ) = cosh  ( 2 π y W ) . {\displaystyle k=\sec \varphi =\cosh \left({\frac {y}{R}}\right)=\cosh \left({\frac {2\pi y}{W}}\right).}
The variation with latitude is sometimes indicated by multiple bar scales as shown below and, for example, on a Finnish school atlas. The interpretation of such bar scales is non-trivial. See the discussion on distance formulae below.
Area scale[edit]
The area scale factor is the product of the parallel and meridian scales hk = sec2φ. For Greenland, taking 73° as a median latitude, hk = 11.7. For Australia, taking 25° as a median latitude, hk = 1.2. For Great Britain, taking 55° as a median latitude, hk = 3.04.
Distortion[edit]
Tissot's indicatrices on the Mercator projection
The classic way of showing the distortion inherent in a projection is to use Tissot's indicatrix. Nicolas Tissot noted that the scale factors at a point on a map projection, specified by the numbers h and k, define an ellipse at that point. For cylindrical projections, the axes of the ellipse are aligned to the meridians and parallels.[14][19][20] For the Mercator projection, h = k, so the ellipses degenerate into circles with radius proportional to the value of the scale factor for that latitude. These circles are rendered on the projected map with extreme variation in size, indicative of Mercator's scale variations.
Accuracy[edit]
One measure of a map's accuracy is a comparison of the length of corresponding line elements on the map and globe. Therefore, by construction, the Mercator projection is perfectly accurate, k = 1, along the equator and nowhere else. At a latitude of ±25° the value of sec φ is about 1.1 and therefore the projection may be deemed accurate to within 10% in a strip of width 50° centred on the equator. Narrower strips are better: sec 8° = 1.01, so a strip of width 16° (centred on the equator) is accurate to within 1% or 1 part in 100. Similarly sec 2.56° = 1.001, so a strip of width 5.12° (centred on the equator) is accurate to within 0.1% or 1 part in 1,000. Therefore, the Mercator projection is adequate for mapping countries close to the equator.
Secant projection[edit]
In a secant (in the sense of cutting) Mercator projection the globe is projected to a cylinder which cuts the sphere at two parallels with latitudes ±φ1. The scale is now true at these latitudes whereas parallels between these latitudes are contracted by the projection and their scale factor must be less than one. The result is that deviation of the scale from unity is reduced over a wider range of latitudes.
An example of such a projection is
x = 0.99 R λ y = 0.99 R ln  tan ( π 4 + φ 2 ) k = 0.99 sec  φ . {\displaystyle x=0.99R\lambda \qquad y=0.99R\ln \tan \!\left({\frac {\pi }{4}}+{\frac {\varphi }{2}}\right)\qquad k\;=0.99\sec \varphi .}
The scale on the equator is 0.99; the scale is k = 1 at a latitude of approximately ±8° (the value of φ1); the scale is k = 1.01 at a latitude of approximately ±11.4°. Therefore, the projection has an accuracy of 1%, over a wider strip of 22° compared with the 16° of the normal (tangent) projection. This is a standard technique of extending the region over which a map projection has a given accuracy.
Generalization to the ellipsoid[edit]
When the Earth is modelled by an ellipsoid (of revolution) the Mercator projection must be modified if it is to remain conformal. The transformation equations and scale factor for the non-secant version are[21]
x = R ( λ − λ 0 ) , y = R ln  [ tan  ( π 4 + φ 2 ) ( 1 − e sin  φ 1 + e sin  φ ) e 2 ] , k = sec  φ 1 − e 2 sin 2  φ . {\displaystyle {\begin{aligned}x&=R\left(\lambda -\lambda _{0}\right),\\y&=R\ln \left[\tan \left({\frac {\pi }{4}}+{\frac {\varphi }{2}}\right)\left({\frac {1-e\sin \varphi }{1+e\sin \varphi }}\right)^{\frac {e}{2}}\right],\\k&=\sec \varphi {\sqrt {1-e^{2}\sin ^{2}\varphi }}.\end{aligned}}}
The scale factor is unity on the equator, as it must be since the cylinder is tangential to the ellipsoid at the equator. The ellipsoidal correction of the scale factor increases with latitude but it is never greater than e2, a correction of less than 1%. (The value of e2 is about 0.006 for all reference ellipsoids.) This is much smaller than the scale inaccuracy, except very close to the equator. Only accurate Mercator projections of regions near the equator will necessitate the ellipsoidal corrections.
Formulae for distance[edit]
Converting ruler distance on the Mercator map into true (great circle) distance on the sphere is straightforward along the equator but nowhere else. One problem is the variation of scale with latitude, and another is that straight lines on the map (rhumb lines), other than the meridians or the equator, do not correspond to great circles.
The distinction between rhumb (sailing) distance and great circle (true) distance was clearly understood by Mercator. (See Legend 12 on the 1569 map.) He stressed that the rhumb line distance is an acceptable approximation for true great circle distance for courses of short or moderate distance, particularly at lower latitudes. He even quantifies his statement: "When the great circle distances which are to be measured in the vicinity of the equator do not exceed 20 degrees of a great circle, or 15 degrees near Spain and France, or 8 and even 10 degrees in northern parts it is convenient to use rhumb line distances".
For a ruler measurement of a short line, with midpoint at latitude φ, where the scale factor is k = sec φ = 1/cos φ:
True distance = rhumb distance ≅ ruler distance × cos φ / RF. (short lines)
With radius and great circle circumference equal to 6,371 km and 40,030 km respectively an RF of 1/300M, for which R = 2.12 cm and W = 13.34 cm, implies that a ruler measurement of 3 mm. in any direction from a point on the equator corresponds to approximately 900 km. The corresponding distances for latitudes 20°, 40°, 60° and 80° are 846 km, 689 km, 450 km and 156 km respectively.
Longer distances require various approaches.
On the equator[edit]
Scale is unity on the equator (for a non-secant projection). Therefore, interpreting ruler measurements on the equator is simple:
True distance = ruler distance / RF (equator)
For the above model, with RF = 1/300M, 1 cm corresponds to 3,000 km.
On other parallels[edit]
On any other parallel the scale factor is sec φ so that
Parallel distance = ruler distance × cos φ / RF (parallel).
For the above model 1 cm corresponds to 1,500 km at a latitude of 60°.
This is not the shortest distance between the chosen endpoints on the parallel because a parallel is not a great circle. The difference is small for short distances but increases as λ, the longitudinal separation, increases. For two points, A and B, separated by 10° of longitude on the parallel at 60° the distance along the parallel is approximately 0.5 km greater than the great circle distance. (The distance AB along the parallel is (a cos φ) λ. The length of the chord AB is 2(a cos φ) sin λ/2. This chord subtends an angle at the centre equal to 2arcsin(cos φ sin λ/2) and the great circle distance between A and B is 2a arcsin(cos φ sin λ/2).) In the extreme case where the longitudinal separation is 180°, the distance along the parallel is one half of the circumference of that parallel; i.e., 10,007.5 km. On the other hand, the geodesic between these points is a great circle arc through the pole subtending an angle of 60° at the center: the length of this arc is one sixth of the great circle circumference, about 6,672 km. The difference is 3,338 km so the ruler distance measured from the map is quite misleading even after correcting for the latitude variation of the scale factor.
On a meridian[edit]
A meridian of the map is a great circle on the globe but the continuous scale variation means ruler measurement alone cannot yield the true distance between distant points on the meridian. However, if the map is marked with an accurate and finely spaced latitude scale from which the latitude may be read directly—as is the case for the Mercator 1569 world map (sheets 3, 9, 15) and all subsequent nautical charts—the meridian distance between two latitudes φ1 and φ2 is simply
m 12 = a | φ 1 − φ 2 | . {\displaystyle m_{12}=a|\varphi _{1}-\varphi _{2}|.}
If the latitudes of the end points cannot be determined with confidence then they can be found instead by calculation on the ruler distance. Calling the ruler distances of the end points on the map meridian as measured from the equator y1 and y2, the true distance between these points on the sphere is given by using any one of the inverse Mercator formulæ:
m 12 = a | tan − 1  [ sinh  ( y 1 R ) ] − tan − 1  [ sinh  ( y 2 R ) ] | , {\displaystyle m_{12}=a\left|\tan ^{-1}\left[\sinh \left({\frac {y_{1}}{R}}\right)\right]-\tan ^{-1}\left[\sinh \left({\frac {y_{2}}{R}}\right)\right]\right|,}
where R may be calculated from the width W of the map by R = W/2π. For example, on a map with R = 1 the values of y = 0, 1, 2, 3 correspond to latitudes of φ = 0°, 50°, 75°, 84° and therefore the successive intervals of 1 cm on the map correspond to latitude intervals on the globe of 50°, 25°, 9° and distances of 5,560 km, 2,780 km, and 1,000 km on the Earth.
On a rhumb[edit]
A straight line on the Mercator map at angle α to the meridians is a rhumb line. When α = π/2 or 3π/2 the rhumb corresponds to one of the parallels; only one, the equator, is a great circle. When α = 0 or π it corresponds to a meridian great circle (if continued around the Earth). For all other values it is a spiral from pole to pole on the globe intersecting all meridians at the same angle, and is thus not a great circle.[18] This section discusses only the last of these cases.
If α is neither 0 nor π then the above figure of the infinitesimal elements shows that the length of an infinitesimal rhumb line on the sphere between latitudes φ; and φ + δφ is a sec α δφ. Since α is constant on the rhumb this expression can be integrated to give, for finite rhumb lines on the Earth:
r 12 = a sec  α | φ 1 − φ 2 | = a sec  α Δ φ . {\displaystyle r_{12}=a\sec \alpha \,|\varphi _{1}-\varphi _{2}|=a\,\sec \alpha \;\Delta \varphi .}
Once again, if Δφ may be read directly from an accurate latitude scale on the map, then the rhumb distance between map points with latitudes φ1 and φ2 is given by the above. If there is no such scale then the ruler distances between the end points and the equator, y1 and y2, give the result via an inverse formula:
r 12 = a sec  α | tan − 1  sinh  ( y 1 R ) − tan − 1  sinh  ( y 2 R ) | . {\displaystyle r_{12}=a\sec \alpha \left|\tan ^{-1}\sinh \left({\frac {y_{1}}{R}}\right)-\tan ^{-1}\sinh \left({\frac {y_{2}}{R}}\right)\right|.}
These formulæ give rhumb distances on the sphere which may differ greatly from true distances whose determination requires more sophisticated calculations.[22]
See also[edit]
Atlas portal
Cartography
Central cylindrical projection - sometimes erroneously described as the method of construction of the Mercator projection
Gall–Peters projection
Jordan Transverse Mercator
List of map projections
Mercator 1569 world map
Nautical chart
Rhumbline network
Tissot's indicatrix
Transverse Mercator projection
Universal Transverse Mercator coordinate system
Notes[edit]
Jump up ^ Kellaway, G.P. (1946). Map Projections p. 37–38. London: Methuen & Co. LTD. (According to this source, it had been claimed that the Mercator projection was used for "imperialistic motives"
Jump up ^ Abelson, C.E. (1954). Common Map Projections s. 4. Sevenoaks: W.H. Smith & Sons.
Jump up ^ Chamberlin, Wellman (1947). The Round Earth on Flat Paper s. 99. Washington, D.C.: The National Geographic Society.
Jump up ^ Fisher, Irving (1943). "A World Map on a Regular Icosahedron by Gnomonic Projection." Geographical Review 33 (4): 605.
Jump up ^ "Mercator Projection vs. Peters Projection, part 2". Matt T. Rosenberg, about.com.
Jump up ^ "Mercator Projection vs. Peters Projection, part 1". Matt T. Rosenberg, about.com.
Jump up ^ American Cartographer. 1989. 16(3): 222–223.
Jump up ^ http://groups.google.com/group/Google-Maps-API/msg/8222b18e7921f6e6
Jump up ^ Maling, pages 77–79.
Jump up ^ Snyder, Working manual pp 37—95.
Jump up ^ Snyder, Flattening the Earth.
Jump up ^ A generator of a cylinder is a straight line on the surface parallel to the axis of the cylinder.
Jump up ^ The function y(φ) is not completely arbitrary: it must be monotonic increasing and antisymmetric (y(−φ) = −y(φ), so that y(0)=0): it is normally continuous with a continuous first derivative.
^ Jump up to: a b Snyder. Working Manual, page 20.
Jump up ^ R is the radius of the globe and φ is measured in radians.
Jump up ^ λ is measured in radians.
^ Jump up to: a b NIST. See Sections 4.26#ii and 4.23#viii
^ Jump up to: a b c Osborne Chapter 2.
Jump up ^ Snyder, Flattening the Earth, pp 147—149
Jump up ^ More general example of Tissot's indicatrix: the Winkel tripel projection.
Jump up ^ Osborne, Chapters 5, 6
Jump up ^ See great-circle distance, the Vincenty's formulae or Mathworld.
References[edit]
Maling, Derek Hylton (1992), Coordinate Systems and Map Projections (second ed.), Pergamon Press, ISBN 0-08-037233-3 .
Monmonier, Mark (2004), Rhumb Lines and Map Wars: A Social History of the Mercator Projection (Hardcover ed.), Chicago: The University of Chicago Press, ISBN 0-226-53431-6
Olver, F. W.J.; Lozier, D.W.; Boisvert, R.F.; et al., eds. (2010), NIST Handbook of Mathematical Functions, Cambridge University Press
Osborne, Peter (2013), The Mercator Projections, doi:10.5281/zenodo.35392. (Supplements: Maxima files and Latex code and figures)
Rapp, Richard H (1991), Geometric Geodesy, Part I
Snyder, John P (1993), Flattening the Earth: Two Thousand Years of Map Projections, University of Chicago Press, ISBN 0-226-76747-7
Snyder, John P. (1987), Map Projections – A Working Manual. U.S. Geological Survey Professional Paper 1395, United States Government Printing Office, Washington, D.C. This paper can be downloaded from USGS pages. It gives full details of most projections, together with interesting introductory sections, but it does not derive any of the projections from first principles.
External links[edit]
Wikimedia Commons has media related to Mercator projections.
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.
Web Mercator: Non-Conformal, Non-Mercator (Noel Zinn, Hydrometronics LLC)
Mercator's Projection at University of British Columbia
Mercator's Projection at Wolfram MathWorld
Google Maps Coordinates
[show]
v
t
e
Map projection
History
List
Portal
[show]
By surface
Cylindrical
Mercator-conformal
Gauss–Krüger
Transverse Mercator
Equal-area
Balthasart
Behrmann
Gall–Peters
Hobo–Dyer
Lambert
Smyth equal-surface
Trystan Edwards
Cassini
Central
Equirectangular
Gall stereographic
Miller
Space-oblique Mercator
Web Mercator
Pseudocylindrical
Eckert II
Eckert IV
Eckert VI
Goode homolosine
Kavrayskiy VII
Mollweide
Sinusoidal
Tobler hyperelliptical
Wagner VI
Conical
Albers
Equidistant
Lambert conformal
Pseudoconical
Bonne
Bottomley
Polyconic
Werner
Azimuthal
(planar)
General perspective
Gnomonic
Orthographic
Stereographic
Equidistant
Lambert equal-area
Pseudoazimuthal
Aitoff
Hammer
Wiechel
Winkel tripel
[show]
By metric
Conformal
Adams hemisphere-in-a-square
Gauss–Krüger
Guyou hemisphere-in-a-square
Lambert conformal conic
Mercator
Peirce quincuncial
Stereographic
Transverse Mercator
Equal-area
Bonne
Sinusoidal
Werner
Bottomley
Sinusoidal
Werner
Cylindrical
Balthasart
Behrmann
Gall–Peters
Hobo–Dyer
Lambert cylindrical equal-area
Smyth equal-surface
Trystan Edwards
Tobler hyperelliptical
Collignon
Mollweide
Albers
Briesemeister
Eckert II
Eckert IV
Eckert VI
Hammer
Lambert azimuthal equal-area
Quadrilateralized spherical cube
Equidistant in
some aspect
Conic
Equirectangular
Sinusoidal
Two-point
Werner
Gnomonic
Gnomonic
Loxodromic
Loximuthal
Mercator
Retroazimuthal
(Mecca or Qibla)
Craig
Hammer
Littrow
[show]
By construction
Compromise
Chamberlin trimetric
Kavrayskiy VII
Miller cylindrical
Robinson
Van der Grinten
Wagner VI
Winkel tripel
Hybrid
Goode homolosine
HEALPix
Perspective
Planar
Gnomonic
Orthographic
Stereographic
Central cylindrical
Polyhedral
Cahill Butterfly
Dymaxion
Quadrilateralized spherical cube
Waterman butterfly
[show]
See also
Latitude
Longitude
Tissot's indicatrix
Retrieved from "https://en.wikipedia.org/w/index.php?title=Mercator_projection&oldid=825915010"
Categories:
Map projections
Conformal projections
Hidden categories:
Articles containing Latin-language text
All articles with unsourced statements
Articles with unsourced statements from February 2017
Navigation menu
Personal tools
Not logged in
Talk
Contributions
Create account
Log in
Namespaces
Article
Talk
Variants
Views
Read
Edit
View history
More
Search
Navigation
Main page
Contents
Featured content
Current events
Random article
Donate to Wikipedia
Wikipedia store
Interaction
Help
About Wikipedia
Community portal
Recent changes
Contact page
Tools
What links here
Related changes
Upload file
Special pages
Permanent link
Page information
Wikidata item
Cite this page
Print/export
Create a book
Download as PDF
Printable version
In other projects
Wikimedia Commons
Languages
العربية
Asturianu
Български
Català
Čeština
Dansk
Deutsch
Español
Esperanto
فارسی
Français
Gaeilge
한국어
हिन्दी
Hrvatski
Bahasa Indonesia
Íslenska
Italiano
עברית
Magyar
മലയാളം
Bahasa Melayu
Nederlands
日本語
Norsk
Polski
Português
Русский
Simple English
Српски / srpski
Srpskohrvatski / српскохрватски
Suomi
Svenska
ไทย
Türkçe
Українська
粵語
中文
Edit links
This page was last edited on 16 February 2018, at 03:48.
Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.
Privacy policy
About Wikipedia
Disclaimers
Contact Wikipedia
Developers
Cookie statement
Mobile view
Enable previews

values: mw-page-base noprint

values: mw-head-base noprint

values: firstHeading firstHeading en

values: siteSub noprint

values: jump-to-nav mw-jump

value: img

attrs: href class

attrs: class style

values:
/wiki/Cylindrical_map_projection
mw-redirect
Cylindrical map projection

values:
/wiki/Conformal_projection
mw-redirect
Conformal projection

values: Edit section: Properties and historical details

values:
/wiki/Polar_region
mw-redirect
Polar region

values:
/wiki/Rhumb_lines
mw-redirect
Rhumb lines

values:
/wiki/Marine_navigation
mw-redirect
Marine navigation

values:
/wiki/Parallel_ruler
mw-redirect
Parallel ruler

values:
/wiki/Atlas_(geography)
mw-redirect
Atlas (geography)

values:
/wiki/Equal-area_projection
mw-redirect
Equal-area projection

values: This claim needs references to reliable sources. (February 2017)

values:
noprint Inline-Template Template-Fact
white-space:nowrap;

values: Edit section: Mathematics of the Mercator projection

values:
/wiki/Map_projections#Cylindrical
mw-redirect
Map projections

values: display:block; line-height:1em; margin:0 0.1em;

values: display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;

values: display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;

values: vertical-align: -2.505ex; width:41.492ex; height:6.176ex;

values: vertical-align: -2.505ex; width:33.61ex; height:6.176ex;

values: vertical-align: -2.505ex; width:28.092ex; height:6.176ex;

values: vertical-align: -2.671ex; width:54.1ex; height:6.509ex;

values: Edit section: Derivation of the Mercator projection

values: {\displaystyle y'(\varphi )=R\sec \varphi ,}

values: vertical-align: -0.838ex; width:15.959ex; height:3.009ex;

values:
/wiki/Integral_table
mw-redirect
Integral table

values: vertical-align: -1.838ex; width:44.972ex; height:4.843ex;

values: vertical-align: -2.005ex; width:47.009ex; height:5.009ex;

attrs: columnalign rowspacing columnspacing displaystyle

values: right left right left right left right left right left right left

values: 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em

values: vertical-align: -5.171ex; width:89.02ex; height:11.509ex;

values: vertical-align: -2.005ex; width:73.413ex; height:5.009ex;

values: vertical-align: -2.505ex; width:52.633ex; height:6.509ex;

values: vertical-align: -1.838ex; width:51.202ex; height:5.176ex;

values: Edit section: Truncation and aspect ratio

values: vertical-align: -2.005ex; width:69.271ex; height:5.009ex;

values: vertical-align: -2.338ex; width:48.848ex; height:6.009ex;

values: vertical-align: -2.505ex; width:38.132ex; height:6.176ex;

values: World Scale from DMA Series 1150 map.png

values: vertical-align: -1.838ex; width:62.736ex; height:4.843ex;

values: Edit section: Generalization to the ellipsoid

values:
/wiki/Conformal_mapping
mw-redirect
Conformal mapping

values: vertical-align: -7.338ex; width:42.357ex; height:15.843ex;

values: {\displaystyle m_{12}=a|\varphi _{1}-\varphi _{2}|.}

values: vertical-align: -0.838ex; width:18.175ex; height:2.843ex;

values: vertical-align: -2.005ex; width:49.481ex; height:5.176ex;

values: vertical-align: -0.838ex; width:36.872ex; height:2.843ex;

values: vertical-align: -2.005ex; width:50.117ex; height:5.176ex;

values: display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle

values: margin:0.5em 0 0.5em 1em;border:solid #aaa 1px

values: Universal Transverse Mercator coordinate system

values:
nofollow
external text
http://dlmf.nist.gov/4.26#ii

values:
nofollow
external text
http://dlmf.nist.gov/4.23#viii

values:
nofollow
external text
http://dlmf.nist.gov

values:
nofollow
external text
//doi.org/10.5281%2Fzenodo.35392

values:
nofollow
external text
https://zenodo.org/record/35561

values:
nofollow
external text
https://zenodo.org/record/35562

values:
nofollow
external text
http://hdl.handle.net/1811/24333

values:
nofollow
external text
https://pubs.er.usgs.gov/pubs/pp/pp1395

values:
nofollow
external text
http://wilhelmkruecken.de/index.htm

values:
row
navbox-group
width:1%;font-weight: normal;

values: Lambert cylindrical equal-area projection

values:
/wiki/Polyconic_projection
mw-redirect
Polyconic projection

values: nowraplinks collapsible collapsed navbox-subgroup

values: nowraplinks hlist collapsible collapsed navbox-inner

values: Category:Articles containing Latin-language text

values: Category:All articles with unsourced statements

values: Category:Articles with unsourced statements from February 2017

values: Discussion about edits from this IP address [ctrl-alt-n]

values: A list of edits made from this IP address [ctrl-alt-y]

values: You are encouraged to create an account and log in; however, it is not mandatory

values: You're encouraged to log in; however, it's not mandatory. [ctrl-alt-o]

values:
/wiki/Mercator_projection
View the content page [ctrl-alt-c]
c

values: Discussion about the content page [ctrl-alt-t]

values: p-namespaces navigation vectorTabs p-namespaces-label

values:
p-variants
navigation
vectorMenu emptyPortlet
p-variants-label

values: Guidance on how to use and edit Wikipedia

values: About the project, what you can do, where to find things

values: A list of recent changes in the wiki [ctrl-alt-r]

values: portal navigation p-interaction p-interaction-label

values: List of all English Wikipedia pages containing links to this page [ctrl-alt-j]

values: Recent changes in pages linked from this page [ctrl-alt-k]

values:
/wiki/Wikipedia:File_Upload_Wizard
Upload files [ctrl-alt-u]
u

values: A list of all special pages [ctrl-alt-q]

values: Permanent link to this revision of the page

values: Link to connected data repository item [ctrl-alt-g]

values: Printable version of this page [ctrl-alt-p]

values: إسقاط مركاتور – Arabic

values: Меркаторова проекция – Bulgarian

values: سیستم تصویر مرکاتور – Persian

values: 메르카토르 도법 – Korean

values: मर्केटर प्रक्षेप – Hindi

values: Proiezione cilindrica centrografica modificata di Mercatore – Italian

values: היטל מרקטור – Hebrew

values: മെർക്കാറ്റർ പ്രക്ഷേപം – Malayalam

values: メルカトル図法 – Japanese

values: Odwzorowanie walcowe równokątne – Polish

values: Проекция Меркатора – Russian

values: Меркаторова пројекција – Serbian

values: เส้นโครงแผนที่แบบเมอร์เคเตอร์ – Thai

values: Проекція Меркатора – Ukrainian

values: 麥卡托投影法 – Chinese

values:
suggestions
display: none; font-size: 13px;

values: mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject page-Mercator_projection rootpage-Mercator_projection skin-vector action-view