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Topography

Topography

Topography, a term in geography, has come to refer to the "lay of the land", or the physiogeographic characteristics of land in terms of elevation, slope, and orientation. "Terrain" is a similar concept, used more to describe the land itself than the study of it. "Relief" is often used to refer to the third dimension of a map whether in actuality (as in a "raised relief" map, or drawn, as with contours, hachures or shading) or the territory it describes. shading Topography is similar to topology, popularly thought of as the mathematical study of surfaces. This may help explain its adoption in the world of geographers. Its actual original meaning, from Greek "topos" (place) and "graphein" (to draw), relates to the description of places rather than broad regions, in topographic surveys. Most 18th and early 19th century national surveys did not record relief across the entire area of coverage, calculating only spot elevations at survey points. The United States Geological Survey (USGS) topographical survey maps included contour representation of relief, and so maps that show relief, especially with exact representation of elevation, came to be called topographic maps (or "topo" maps) in the United States, and the usage has spread internationally. The understanding of topography is critical for a number of reasons. In terms of environmental quality, agriculture, and hydrology, understanding the topography of an area enables the understanding of watershed boundaries, drainage characteristics, water movement, and impacts on water quality. Understanding topography also impinges on soil conservation, especially in agriculture. Contour plowing is an established practice of enabling sustainable agriculture on sloping land, and is the practice of plowing along topographic lines. Topography is critical militarily because it determines the ability of armed forces to take and hold areas, and to move troops and material into and through areas. Topography is important in determining weather patterns. Two areas in fairly close proximity geographically may differ radically in characteristics such as precipitation because of elevation differences or because of a "rain shadow" effect. Tectonic processes and erosional processes are the determiners of topography. Tectonic processes such as orogenies cause land to be elevated, and erosional (and weathering) processes cause land to be worn away to lower elevations.

Geography

)]] Geography is the study of the locational and spatial variation of both natural and human phenomena on Earth. The word derives from the Greek words Ge (γη) or Gaea (γεια), both meaning "Earth", and graphein (γραφειν) meaning "to describe" and "to write". Modern geography is a diverse discipline that draws influence from almost every other arena of knowledge. Geographers engage with other disciplines according to their particular research interests and, while subjects such as biology and economics have a powerful influence, there are geographers who use concepts taken from subjects such as sociology, psychology and sports science, among many others. Within the discipline there have been many long-running tensions among those seeking to define geography - whether as a 'science' or as a 'humanity', as a 'systematic' subject or 'regional' specialism and so forth - which at various times have come close to destroying geography as an academic discipline. Whilst profound differences do exist among geographers, the dual concepts of space and place provide a commonality of interest, which gives the subject a unique identity.

Structure of geography

William Hughes - who taught the geography of the Holy Lands to divinity students at King's College London - defined geography in an address in 1863: :"Mere place names are not geography. To know by heart a whole gazeteer full of them would not, in itself, constitute anyone a geographer. Geography has higher aims than this: it seeks to classify phenomena (alike of the natural and of the political world insofar as it treats of the latter) to compare, to generalise, to ascend from effects to causes and in doing so to trace out the great laws of nature and to mark their influence upon man. In a word, geography is a science, a thing not of mere names, but of argument and reason, of cause and effect." This was a specific rejection of geography as a merely descriptive discipline and also defined it as inclusive of both the physical world and the human. Within the discipline, however, there are many areas of specialism. Modern geographers tend to specialise in one of the broad branches (or sub-branches). However, most introductory geography syllabuses seek to ensure that geographers have at least working knowledge of the main focus of each branch of the subject.

Physical geography

Physical geography (or physiogeography) focuses on geography as an Earth science. It aims to understand the physical layout of the Earth, its weather and global flora and fauna patterns. Many areas of physical geography make use of geology, particularly in the study of weathering and sediment movement. Physical Geography can be divided into the following broad categories:
- Geomorphology
- Hydrology
- Glaciology
- Biogeography
- Climatology
- Pedology (soil study)
- Coastal/Marine studies
- Geodesy
- Palaeogeography
- Environmental Geography and management
- Landscape ecology Exact lines between these different areas are often difficult to draw. Sometimes Oceanography is included as a branch within physical geography, but is now considered a separate subject in its own right. Related topics: Atmosphere - Archipelago - Continent - Desert - Island - Landform - Ocean - Sea - River - Lake - Ecology - Soil - Timeline of geography, paleontology - Geostatistics - Environmental science - Oceanography - Environmental studies

Human geography

Human geography is a branch of geography that focuses on the study of patterns and processes that shape human interaction with various environments. It encompasses human, political, cultural, social, and economic aspects. While the major focus of human geography is not the physical landscape of the Earth (see Physical geography) it is hardly possible to discuss human geography without referring to the physical landscape on which human activities are being played out, and environmental geography is emerging as a link between the two. Human geography can be divided into broad categories, such as:
- Economic geography
- Development geography
- Population geography or Demography
-
- Urban geography
- Social geography
- Behavioral geography
- Cultural geography
- Political geography, including Geopolitics
-
- Historical geography
- Regional science
- Strategic geography
- Military geography
- Feminist geography
- Distinction between these fields of study have become increasingly blurred over time and the above list should not be considered definitive. Related topics: Countries of the world - Country - Nation - State - Personal union - Province - County - City - Municipality - Central place theory - Urban morphology

Socio-environmental geography

During the time of environmental determinism, geography was defined not as the study of spatial relationships, but as the study of how humans and the natural environment interact. Though environmental determinism has died out, there remains a strong tradition of geographers addressing the relationships between people and nature. There are two main subfields of socio-environmental geography:
- cultural and political ecology (CAPE) and
- risk-hazards research.

Cultural and political ecology

Cultural ecology grew out of the work of Carl Sauer in geography and a similar school of thought in anthropology. It examined how human societies adapt themselves to the natural environment. Sustainability science has been one important outgrowth of this tradition. Political ecology arose when some geographers used aspects of critical geography to look at relations of power and how they affect people's use of the environment. For example, an influential study by Michael Watts argued that famines in the Sahel are caused by the changes in the region's political and economic system as a result of colonialism and the spread of capitalism.

Risk-hazards research

Research on hazards began with the work of geographer Gilbert F. White, who sought to understand why people live in disaster-prone floodplains. Since then, the hazards field has expanded to become a multidisciplinary field examining both natural hazards (such as earthquakes) and technological hazards (such as nuclear reactor meltdowns). Geographers studying hazards are interested in both the dynamics of the hazard event and how people and societies deal with it.

Historical geography

This branch seeks to determine how cultural features of the multifarious societies across the planet evolved and came into being. Study of the landscape is one of many key foci in this field - much can be deduced about earlier societies from their impact on their local environment and surroundings. ; What's in a name? Historical geography and the Berkeley School "Historical Geography" can indeed refer to the reciprocal effects of geography and history on each other. But in the United States, it has a more specialized meaning: This is the name given by Carl Ortwin Sauer of the University of California, Berkeley to his program of reorganizing cultural geography (some say all geography) along regional lines, beginning in the first decades of the 20th Century. To Sauer, a landscape and the cultures in it could only be understood if all of its influences through history were taken into account: Physical, cultural, economic, political, environmental. Sauer stressed regional specialization as the only means of gaining expertise on regions of the world. Sauer's philosophy was the principal shaper of American geographic thought in the mid-20th century. Regional specialists remain in academic geography departments to this day. But many geographers feel that it harmed the discipline in the long run: Too much effort was spent on data collection and classification, and too little on analysis and explanation. Studies became more and more area specific as later geographers struggled to find places to make names for themselves. This probably led in turn to the 1950s crisis in Geography which nearly destroyed it as an academic discipline.

History of geography

:See main article: History of geography History of geography The Greeks are the first known culture to actively explore geography as a science and philosophy. Mapping by the Romans as they explored new lands added new techniques. During the Middle Ages, Arabs such as Idrisi, Ibn Batutta, and Ibn Khaldun maintained the Greek and Roman techniques and developed new ones. Following the journeys of Marco Polo, interest in geography spread throughout Europe. The great voyages of exploration in 16th and 17th centuries revived a desire for both accurate geographic detail, and more solid theoretical foundations. This period is also known as Great Geographical Discoveries. By the 18th century, geography had become recognized as a discrete discipline and became part of a typical university curriculum in Europe (especially Paris and Berlin). Over the past two centuries the quantity of knowledge and the number of tools has exploded. There are strong links between geography and the sciences of geology and botany, as well as economics, sociology and demographics. In the West during the 20th century, the discipline of geography went through four major phases: environmental determinism, regional geography, the quantitative revolution, and critical geography.
Geographic techniques
As spatial interrelationships are key to this synoptic science, maps are a key tool. Classical cartography has been joined by a more modern approach to geographical analysis, computer-based geographic information systems (GIS).
- Cartography studies the representation of the Earth's surface with abstract symbols (map making). Although other subdisciplines of geography rely on maps for presenting their analyses, the actual making of maps is abstract enough to be regarded separately. Cartography has grown from a collection of drafting techniques into an actual science. Cartographers must learn cognitive psychology and ergonomics to understand which symbols convey information about the Earth most effectively, and behavioral psychology to induce the readers of their maps to act on the information. They must learn geodesy and fairly advanced mathematics to understand how the shape of the Earth affects the distortion of map symbols projected onto a flat surface for viewing. It can be said, without much controversy, that cartography is the seed from which the larger field of Geography grew. Most geographers will cite a childhood fascination with maps as an early sign they would end up in the field. mathematics
- Geographic Information Systems deals with the storage of information about the Earth for automatic retrieval by a computer, in an accurate manner appropriate to the information's purpose. In addition to all of the other subdisciplines of geography, GIS specialists must understand computer science and database systems. GIS has so revolutionized the field of cartography that nearly all mapmaking is now done with the assistance of some form of GIS software.
- Geographic quantitative methods deal with numerical methods peculiar to (or at least most commonly found in) geography. In addition to spatial analyses, you are likely to find things like cluster analysis, discriminant analysis, and non-parametric statistical tests in geographic studies.
- Geographic qualitative methods, or ethnographic research techniques, are used by human geographers. In cultural geography there is a tradition of employing qualitative research techniques also used in anthropology and sociology. Participant Observation and in-depth interviews provide human geographers with qualitative data. In their study geographers use four interrelated approaches:
- Systematic - Groups geographical knowledge into categories that can be explored globally
- Regional - Examines systematic relationships between categories for a specific region or location on the planet.
- Descriptive - Simply specifies the locations of features and populations.
- Analytical - Asks why we find features and populations in a specific geographic area.
Related fields

Urban and regional planning
Urban planning and regional planning use the science of geography to assist in determining how to develop (or not develop) the land to meet particular criteria, such as safety, beauty, economic opportunities, the preservation of the built or natural heritage, etcetera. The planning of towns, cities and rural areas may be seen as applied geography although it also draws heavily upon the arts, the sciences and lessons of history. Some of the issues facing planning are considered briefly under the headings of rural exodus, urban exodus and Smart Growth.
Regional science
In the 1950s the regional science movement arose, led by Walter Isard to provide a more quantitative and analytical base to geographical questions, in contrast to the more qualitative tendencies of traditional geography programs. Regional Science comprises the body of knowledge in which the spatial dimension plays a fundamental role, such as regional economics, resource management, location theory, urban and regional planning, transport and communication, human geography, population distribution, landscape ecology, and environmental quality.
Reference

See also

- List of geography topics
- Geographical terms
- List of countries
- Geography reference tables
- Map
- Geographical renaming
- Geographic magazines
- National Geographic Society (United States)
- National Geographic Bee (United States)
- Point of Beginning
- Royal Geographical Society (United Kingdom)
External links

- [http://www.confluence.org/ Confluence.org - A work in progress, involving travelling to every point on the globe where the lines of longitude and latitude intersect and taking a photograph in each direction.]
- [http://www.aag.org/ Association of American Geographers]
- [http://www.gisuser.com/ GISuser.com, information-rich portal about GIS]
- [http://www.populationdata.net/ PopulationData.net]
- [http://www.freemaps.de/ Free Maps Germany]
- [http://www.ericdigests.org/1996-4/high.htm Using Literature To Teach Geography in High Schools. ERIC Digest.]
- [http://ericdigests.org/1992-5/geography.htm Teaching Geography at School and Home. ERIC Digest.]
- [http://ericdigests.org/1996-1/geography.htm The National Geography Content Standards. ERIC Digest.]
- [http://www.geo-guide.de Geo-Guide] extensive list of academic resources on geography and earth science
- [http://www.geopium.org Geopium: Geopolitics of Illicit Drugs in Asia]
- [http://www.nationalgeographic.com/ National Geographic Online]
- [http://www.rgs.org Royal Geographical Society]
- [http://www.rcgs.org Royal Canadian Geographical Society]
- [http://www.canadiangeographic.ca Canadian Geographic]
- [http://hypergeo.free.fr Hypergeo : Geographical Encyclopedia]
- [http://www.rare-maps.com/links.cfm Antique and Rare Maps - Art Source International] - Links to rare and antique maps and to cartography resources.
- [http://www.mapinfo.com/ MapInfo GIS Software]
- Category:School subjects als:Geografie ko:지리학 ms:Geografi ja:地理学 simple:Geography th:ภูมิศาสตร์

Landform

A landform comprises a geomorphological unit. Landforms are categorised by characteristics such as elevation, slope, orientation, stratification, rock exposure, and soil type. Landforms by name include berms, mounds, hills, cliffs, valleys, and so forth. Oceans and continents exemplify highest-order landforms. A number of factors, ranging from plate tectonics to erosion and deposition can generate and affect landforms. Biological factors can also influence landforms -- see for example the role of plants in the development of dune systems and salt marshes, and the work of corals and algae in the formation of coral reefs. coral reefs

List of landforms

- alas
- continent
- limestone pavement
- plain and plateau
- rock formations

Erosion landforms

Landforms produced by erosion and weathering usually occur in coastal or fluvial environments, and many appear above under those headings. Some other erosion landforms that do not fall into the above categories include:
- canyon
- cave
- limestone pavement
- tea table
- Deposition landform -- landforms produced by deposition of load or sediment (usually coastal or fluvial).
- Eolian landform - landforms produced by wind weathering.

External links

- [http://www.deh.gov.au/settlements/industry/minerals/booklets/landform/ Landform Design] Category:Geomorphology ja:地形

Elevation

:For other senses of this word, see elevation (disambiguation).

Basic Definition

In geography, the elevation of a geographic location is its height above mean sea level (or some other fixed point). Elevation is mainly used when referring to points on the Earth itself, while altitude is used for points in the air, such as an aircraft.

Determining Elevation

If you are reading a map from home, it is possible you may need to determine the elevation of some place. The main sort of map to use for this purpose is a topographical map. Learning to read a topographic map is relatively easy although assistance may be required for beginners. Image:HaleakalaMap.jpg Example of a topographical map. Haleakala, Hawaii. If you are somewhere and want to find its elevation, you may also survey it. Questions often arise about where to measure elevation from. The elevation of a mountain usually refers to its summit. The elevation of a hill also refers to the summit. A valley's elevation is usually taken from the lowest point but is often taken all over the valley.

Links

- Altitude
- Topographical map
- [http://www.usgs.gov/ US Geographical Survey]
- [http://www.gsi.go.jp/ENGLISH/ Geographical Survey Institute]
- [http://www.thefreedictionary.com/elevation The Free Dictionary, Elevation]
- Category:Physical geography Category:Length

Slope

In mathematics, the slope or the gradient of a straight line (within a Cartesian coordinate system) is a measure for the "steepness" of the line relative to the horizontal axis. With an understanding of algebra and geometry, one can calculate the slope of a straight line; with calculus, one can calculate the slope of the tangent to a curve at a point. The concept of slope, and much of this article, applies directly to grades or gradients in geography and civil engineering.

Definition of slope

The slope of a line in the plane containing the x and y axes is generally represented by the letter m, and is defined as the change in the y coordinate divided by the corresponding change in the x coordinate, between two distinct points on the line. This is described by the following equation: :

m = \frac

(The delta symbol, "Δ", is commonly used in mathematics to mean "difference" or "change".) Given two points (x₁, y₁) and (x₂, y₂), the change in x from one to the other is x₂ - x₁, while the change in y is y₂ - y₁. Substituting both quantities into the above equation obtains the following: :

m = \frac

Since the y-axis is vertical and the x-axis is horizontal by convention, the above equation is often memorized as "rise over run", where Δy is the "rise" and Δx is the "run". Therefore, by convention, m is equal to the change in y, the vertical coordinate, divided by the change in x, the horizontal coordinate; that is, m is the ratio of the changes. This concept is fundamental to algebra, analytic geometry, trigonometry, and calculus. Note that the points chosen and the order in which they are used is irrelevant; the same line will always have the same slope. Other curves have "accelerating" slopes and one can use calculus to determine such slopes.

Example 1

Suppose a line runs through two points: P(13,8) and Q(1,2). By dividing the difference in y-coordinates by the difference in x-coordinates, one can obtain the slope of the line: :

m = \frac = \frac = \frac = \frac = \frac

The slope is 1/2 = 0.5.

Example 2

If a line runs through the points (4, 15) and (3, 21) then: :

m = \frac = \frac = -6

Geometry

The larger the absolute value of a slope, the steeper the line. A horizontal line has slope 0, a 45° rising line has a slope of +1, and a 45° falling line has a slope of -1. The slope of a vertical line is not defined (it does not make sense to define it as +∞, because it might just as well be defined as -∞). The angle θ a line makes with the positive x axis is closely related to the slope m via the tangent function: :

m = \tan\,\theta

and :

\theta = \arctan\,m

(see trigonometry). Two lines are parallel if and only if their slopes are equal or if they both are vertical and therefore undefined; they are perpendicular (i.e. they form a right angle) if and only if the product of their slopes is -1 or one has a slope of 0 and the other is vertical and undefined.

Slope of a road, etc.

perpendicular There are two common ways to describe how steep a road is. One is by the angle in degrees, and the other is by the slope in a percentage. See also mountain railway. The formula for converting a slope in percentage to degrees is: :

\theta = \arctan\frac.

Algebra

If y is a linear function of x, then the coefficient of x is the slope of the line created by plotting the function. Therefore, if the equation of the line is given in the form :

y = mx + b \,

then m is the slope. This form of a line's equation is called the slope-intercept form, because b can be interpreted as the y-intercept of the line, the y-coordinate where the line intersects the y-axis. If the slope m of a line and a point (x₀, y₀) on the line are both known, then the equation of the line can be found using the point-slope formula: :

y - y_0 = m(x - x_0) \,

For example, consider a line running through the points (2, 8) and (3, 20). This line has a slope, m, of (20 - 8) / (3 - 2) = 12. One can then write the line's equation, in point-slope form: y - 8 = 12(x - 2) = 12x - 24; or: y = 12x - 16. The slope of a linear equation in the general form: :

Ax + By + C = 0 \,

is given by the formula: −A/B.

Calculus

The concept of a slope is central to differential calculus. For non-linear functions, the rate of change varies along the curve. The derivative of the function at a point is the slope of the line tangent to the curve at the point, and is thus equal to the rate of change of the function at that point.

Why calculus is necessary

tangent If we let Δx and Δy be the distances (along the x and y axes, respectively) between two points on a curve, then the slope given by the above definition, :

m = \frac

, is the slope of a secant line to a curve. For a line, the secant between any two points is identical to the line itself; however, this is not the case for any other type of curve. For example, the slope of the secant intersecting y = x² at (0,0) and (3,9) is m = (9 - 0) / (3 - 0) = 3 (which happens to be the slope of the tangent at, and only at, x = 1.5). However, by moving the points used in the above formula closer together so that Δy and Δx decrease, the secant line more closely approximates a tangent line to the curve. It follows that the secant line is identical to the tangent line when Δy and Δx equal zero; however, this results in a slope of 0/0, which is an indeterminate form (see also division by zero). The concept of a limit is necessary to calculate this slope; the slope is the limit of Δy / Δx as Δy and Δx approach zero. However, Δx and Δy are interrelated such that it is sufficient to take the limit where only Δx approaches zero. This limit is the derivative of y with respect to x. It may be written (in calculus notation) as dy/dx.

Dimension

In common usage, the dimensions (from Latin "measured out") of an object are the parameters or measurements required to define its shape and size, that is, usually, its height, width, and length. In mathematics, the dimensions of a space are the parameters required to describe a particular object in this space. The dimension of a space is the number of these parameters. For example, locating a city on the Earth requires two parameters: longitude and latitude; the corresponding space has therefore two dimensions and its dimension is two. This space is said to be 2-dimensional (for short 2D). Locating an airplane might require a 3D space by addition of the altitude parameter, or even a 6D space, if one adds the three angles required for defining the orientation of the airplane. The airplane is then considered to have six degrees of freedom. Other dimensions can be supplemented like speed, temperature or even colour or price of the plane. Generalisations of the concept are possible and a number of alternative definitions may be introduced. Units are sometimes associated with each dimension, for instance, meters or feet with altitude or dollars with price. In science fiction, a "dimension" can also refer to an alternate universe or plane of existence. This useage is derived from the fact that to get to the alternate universe/plane of existence requires movement in an dimension beyond the normal 3 space + time.

Physical dimensions

The physical dimensions are the parameters required to answer to the question where and when happened or will happen some event; for instance: When did Napoleon die? — On the 5 May 1821 at Saint Helena (15°56′ S 5°42′ W). They play a fundamental role in our perception of the world around us. According to Immanuel Kant, we actually do not perceive them but they form the frame in which we perceive events; they form the a priori background in which events are perceived.

Spatial dimensions

Classical physics theories describe three physical dimensions: from a particular point in space, the basic directions in which we can move are up/down, left/right, and forward/backward. Movement in any other direction can be expressed in terms of just these three. Moving down is the same as moving up a negative amount. Moving diagonally upward and forward is just as the name of the direction implies; i.e., moving in a linear combination of up and forward. In its simplest form: a line describes one dimension, a plane describes two dimensions, and a cube describes three dimensions. (See Cartesian coordinate system)

Time

Time is often referred to as the "fourth dimension". It is, in essence, one way to measure physical change. It is perceived differently from the three spatial dimensions in that there is only one of it, and that movement seems to occur at a fixed rate and in one direction. The equations used by physics to model reality often do not treat time in the same way that humans perceive it. In particular, the equations of classical mechanics are symmetric with respect to time, and equations of quantum mechanics are typically symmetric if both time and other quantities (such as charge and parity) are reversed. In these models, the perception of time flowing in one direction is an artifact of the laws of thermodynamics (we perceive time as flowing in the direction of increasing entropy). The most well-known treatment of time as a dimension is Einstein's theory of general relativity, which treats perceived space and time as parts of a four-dimensional manifold. Some scientists consider that the fact that Einstein's theory allows time to move at different rates indicates a 2nd time dimension exists.

Additional dimensions

Theories such as string theory predict that the space we live in has in fact many more dimensions (frequently 10, 11 or 26), but that the universe measured along these additional dimensions is subatomic in size. As a result, we perceive only the three spatial dimensions that have macroscopic size.

Units

In the physical sciences and in engineering, the dimension of a physical quantity is the expression of the class of physical unit that such a quantity is measured against. The dimension of speed, for example, is length divided by time. In the SI system, the dimension is given by the seven exponents of the fundamental quantities. See Dimensional analysis.

Mathematical dimensions

In mathematics, no definition of dimension adequately captures the concept in all situations where we would like to make use of it. Consequently, mathematicians have devised numerous definitions of dimension for different types of spaces. All, however, are ultimately based on the concept of the dimension of Euclidean n-space Eⁿ. The point E⁰ is 0-dimensional. The line E¹ is 1-dimensional. The plane E² is 2-dimensional. And in general Eⁿ is n-dimensional. A tesseract is an example of a four-dimensional object. In the rest of this section we examine some of the more important mathematical definitions of dimension.

Hamel dimension

For vector spaces, there is a natural concept of dimension, namely the cardinality of a basis. See Hamel dimension for details.

Manifolds

A connected topological manifold is locally homeomorphic to Euclidean n-space, and the number n is called the manifold's dimension. One can show that this yields a uniquely defined dimension for every connected topological manifold. The theory of manifolds, in the field of geometric topology, is characterised by the way dimensions 1 and 2 are relatively elementary, the high-dimensional cases n > 4 are simplified by having extra space in which to 'work'; and the cases n = 3 and 4 are in some senses the most difficult. This state of affairs was highly marked in the various cases of the Poincaré conjecture, where four different proof methods are applied.

Lebesgue covering dimension

For any topological space, the Lebesgue covering dimension is defined to be n if n is the smallest integer for which the following holds: any open cover has a refinement (a second cover where each element is a subset of an element in the first cover) such that no point is included in more than n + 1 elements. For manifolds, this coincides with the dimension mentioned above. If no such n exists, then the dimension is infinite.

Inductive dimension

The inductive dimension of a topological space may refer to the small inductive dimension or the large inductive dimension, and is based on the analogy that n+1-dimensional balls have n dimensional boundaries, permitting an inductive definition based on the dimension of the boundaries of open sets.

Hausdorff dimension

For sets which are of a complicated structure, especially fractals, the Hausdorff dimension is useful. The Hausdorff dimension is defined for all metric spaces and, unlike the Hamel dimension, can also attain non-integer real values. The upper and lower box dimensions are a variant of the same idea.

Hilbert spaces

Every Hilbert space admits an orthonormal basis, and any two such bases have the same cardinality. This cardinality is called the dimension of the Hilbert space. This dimension is finite if and only if the space's Hamel dimension is finite, and in this case the two dimensions coincide.

Krull dimension of commutative rings

The Krull dimension of a commutative ring, named after Wolfgang Krull (1899 - 1971), is defined to be the maximal number of strict inclusions in an increasing chain of prime ideals in the ring.

Science fiction

Science fiction texts often mention the concept of dimension, when really referring to parallel universes, alternate universes, or other planes of existence, or concepts that are beyond the reader. The word gives a sense of authority to a film, and inspires imagination and awe in the minds of the reader, that one could travel to "another dimension". This concept is derived from the idea that in order to travel to to parallel/alternate universes/planes of existence one must travel in a spatial direction/dimension besides the standard ones. In effect, the other universes/planes are just a small distance away from our own, but the distance is in a fourth (or higher) spatial dimension, not the standard ones.

Anaglyph

This section should be merged into Anaglyph image. To understand how Anaglyph 3D works, you must understand how Sterioscope 3D works. It's basically blending the two images taken eye width apart. The eye thinks it's seeing one normal image by blending together one image we see with both eyes. You blend two slightly different pictures together and look at it. It looks 3D! Anaglyph is just taking a single framed 3D image and making one eye only see one image... the red sees the blue (because the red of the glasses blends in with the red) and the blue sees the red (the blue of the glasses blends in with the blue). This creates a normal stereo graph image without the need of crossing your eyes the whole time you're looking at the image.

3-D film

This section should be merged into 3-D film. In the 1950's, 3D movies were very popular, but since they didn't have color film then they had an entirely different method. They would take two black and white cameras, line them up eye-with apart and take the film from both at the same time. They took two projectors in the projection booth, both aimed at the screen. The left one with the left film with a blue filter over the lense, and the right with the right film and a red filter over the lense, rather than just having one 3D image blended together.

Modern 3-D films

Spy Kids 3D has developed a new craze of 3D, mostly in amusement parks with motion simulators. Spongebob 3D in Paramount's Great America, California has a Polarization 3D movie along with a motion simulator. Shark Boy and Lava Girl is another modern 3D movie, by the same producer of Spy Kids.

More dimensions

- Dimension of an algebraic variety
- Topological dimension
- Isoperimetric dimension
- Poset dimension
- Pointwise dimension
- Lyapunov dimension
- Kaplan-Yorke dimension
- Exterior dimension
- Hurst exponent
- q-dimension; especially:
- Information dimension (corresponding to q=1)
- Correlation dimension (corresponding to q=2)

Map

] A map is a simplified depiction of a space, a navigational aid which highlights relations between objects within that space. Most usually a map is a two-dimensional, geometrically accurate representation of a three-dimensional space. The science and art of map-making is cartography.

Introduction

Map-making dates back to the Stone Age and appears to predate written language by several millennia. One of the oldest surviving maps is painted on a wall of the Catal Huyuk settlement in south-central Anatolia (now Turkey); it dates from about 6200 BC. Harvey 2000, p. 142]. While we tend to think of maps today as products of a rationalistic, scientific world-view, maps also have a mythic quality. Pre-modern maps, and mapping traditions outside the Western tradition, often merge geography with non-scientific cosmography, showing the relationship of the viewer to the universe. Medieval "T-O" maps, for example, show Jerusalem at the centre of the world, and in some cases related the "body" of the Earth to the body of Christ. By contrast, navigational (or "Portolan") charts of the Mediterranean from the same period are remarkably accurate. Even today, maps can be powerful rhetorical tools beyond their purely practical value, and this has been the source of much fruitful map criticism over the last twenty years, notably in the works of J.B. Harley, Mark Monmonier, and Denis Wood. Because maps are abstract representations of the world, they are not neutral documents and must be carefully interpreted. It is, of course, this abstraction that makes them useful. Lewis Carroll made this point humorously in Sylvie and Bruno with his mention of a fictional map that had "the scale of a mile to the mile". A character notes some practical difficulties with this map and states that "we now use the country itself, as its own map, and I assure you it does nearly as well". This conceit is elaborated in a one-paragraph story by Jorge Luis Borges and Adolfo Bioy Casares, generally known in English as "On Exactitude in Science". Road maps are perhaps the most widely used maps today, and form a subset of navigational maps, which also include aeronautical and nautical charts, railroad network maps, and hiking and bicycling maps. Community maps, including [http://GreenMap.org GreenMaps], are growing in importance. In terms of quantity, the largest number of drawn map sheets is probably made up by local surveys, carried out by municipalities, utilities, tax assessors, emergency services providers, and other local agencies. Many national surveying projects have been carried out by the military, such as the British Ordnance Survey (now a civilian government agency internationally renowned for its comprehensively detailed work).

Orientation of maps

Ordnance Survey, England. A classic "T-O" map with Jerusalem at centre, east toward the top, Europe the bottom left and Africa on the right.]] Conventionally, on most geometrically accurate maps text is upright when the map is oriented with the north up, hence north is identified with the top of a sheet. Maps that don't put north at the top: #Polar maps #Dymaxion maps # Some rectangular maps produced in Australia show the south pole at the top. To someone used to seeing the map the other way around, this map may appear to be "upside down". These are primarily intended as novelty and tourist maps. # Other modern maps put south on top, generally either out of a sense of playful confusion or to make a political statement about the North-South divide. # Old maps of Edo show the Japanese imperial palace as the "top," but also at the centre, of the map. Labels on the map are oriented in such a way that you cannot read them properly unless you put the imperial palace above your head. # Medieval European T and O maps such as the Hereford Mappa Mundi were centred on Jerusalem, with East at the top. If a person is located at an identifiable point within the area of such a map, then the map can be oriented in such a way that every point on the map lies in the same direction as the corresponding point in reality. The practice of navigating in this way is orienteering. For a vertically positioned map representing a horizontal area true orientation is not possible, of course, but it is sometimes approximated by putting the forward direction up. Occasionally a map is on a ceiling, correctly showing directions; in that case, looking up we have in clockwise direction forward, left, backward, and right. If the map is prepared on a table, to be attached to the ceiling, then on the table it is a mirror image of a normal map.

Scale and accuracy

Many but not all maps are drawn to a scale, allowing the reader to infer the actual sizes of, and distances between, depicted objects. A larger scale shows more detail, thus requiring a larger map to show the same area. For example, maps designed for the hiker are often scaled at the ratio 1:24,000, meaning that 1 of any unit of measurement on the map corresponds to 24,000 of that same unit in reality; while maps designed for the motorist are often scaled at 1:250,000. Maps which use some quality other than physical area to determine relative size are called cartograms. A famous example of a map without scale is the London Underground map, which best fulfils its purpose by being less physically accurate and more visually communicative to the hurried glance of the commuter. This is not a cartogram (since there is no consistent measure of distance) but a topological map that also depicts approximate bearings. The simple maps shown on some directional road signs are further examples of this kind. In fact, most commercial navigational maps, such as road maps and town plans, sacrifice an amount of accuracy in scale to deliver a greater visual usefulness to its user, for example by exaggerating the width of roads. With the end-user similarly in mind, cartographers will censor the content of the space depicted by a map in order provide a useful tool to that user. For example, a road map may or may not show railroads, and if it does, it may show them less clearly than highways.

World maps and projections

highways Maps of the world or large areas are often either 'political' or 'physical'. The most important purpose of the political map is to show territorial borders; the purpose of the physical is to show features of geography such as mountains, soil type or land use. Geological maps show not only the physical surface, but characteristics of the underlying rock, fault lines, and subsurface structures. Maps that depict the surface of the Earth also use a projection, a way of translating the three-dimensional real surface of the geoid to a two-dimensional picture. Perhaps the best-known world-map projection is the Mercator Projection, originally designed as a form of nautical chart. Airplane pilots use aeronautical charts based on a Lambert conformal conic projection, in which a cone is laid over the section of the earth to be mapped. The cone intersects the sphere (the earth) at one or two parallels which are chosen as standard lines. This allows the pilots to plot a great-circle route approximation on a flat, two-dimensional chart.

Electronic maps

Lambert conformal conic projection.]] Lambert conformal conic projection service called http://www.metrokc.gov/gis/mapportal/mapsets.htm I.map Map Sets]] From the last quarter of the 20th century, the indispensable tool of the cartographer has been the computer. Much of cartography, especially at the data-gathering survey level, has been subsumed by Geographic Information Systems (GIS). Even when GIS is not involved, most cartographers now use a variety of computer graphics programs to generate new maps. Interactive, computerised maps are commercially available, allowing users to zoom in or zoom out (respectively meaning to increase or decrease the scale), sometimes by replacing one map with another of different scale, centred where possible on the same point. In-car satellite navigation systems are computerised maps with route-planning and advice facilities which monitor by satellite the position of the user. From the computer scientist's standpoint, zooming in entails one or a combination of: #replacing the map by a more detailed one #enlarging the same map without enlarging the pixels, hence show more detail #enlarging the same map with the pixels enlarged (replaced by rectangles of pixels); no additional detail is shown, but, depending on the quality of one's vision, possibly more detail can be seen; if a computer display does not show adjacent pixels really separate, but overlapping instead (this does not apply for an LCD, but may apply for a cathode ray tube), then replacing a pixel by a rectangle of pixels does show more detail. A variation of this method is that interpolation is performed. For example:
- Typically (2) applies to a Portable Document Format (PDF) file. The increase in detail is, of course, limited to the information contained in the file: enlargement of a curve may eventually result in a series of standard geometric figures such as straight lines or arcs of circles.
- (2) may apply to text and (3) to the outline of a map feature such as a forest or building.
- (1) may apply to the text (displaying labels for more features), while (2) applies to the rest of the image. Text is not necessarily enlarged when zooming in. Similarly, a road represented by a double line may or may not become wider when one zooms in.
- The map may also have layers which are partly raster graphics and partly vector graphics. For a single raster graphics image (2) applies until the pixels in the image file correspond to the pixels of the display, thereafter (3) applies. The word "map" has also been used to describe places within video games, such as SOCOM II: U.S. Navy SEALs and Counter-Strike, that players choose to compete on, as a synonym for level. See also Webpage (Graphics), PDF (Layers), Mapquest, or Yahoo! Maps.

References

- David Buisseret, ed., Monarchs, Ministers and Maps: The Emergence of Cartography as a Tool of Government in Early Modern Europe. Chicago: University of Chicago Press, 1992, [ISBN 0226079872]
- Miles Harvey, The Island of Lost Maps: A True Story of Cartographic Crime. New York : Random House, 2000. [ISBN 0767908260, cited above; also ISBN 0375501517]
- Mark Monmonier, How to Lie with Maps, [ISBN 0226534219]
- O'Connor, J.J. and E.F. Robertson, [http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Cartography.html The History of Cartography]. Scotland : St. Andrews University, 2002. http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Cartography.html

External links

- Microsoft: MSN [http://maps.msn.com/]
- world atlas (the whole world on the same scale: 4 km = 49 pixels, 82 m / pixel)
- North America (Canada, US, Mexico), US to street level, Mexico only some parts to street level (for the other parts street level scale is available, but it shows blank maps with e.g. only one dot to represent a village)
- Western Europe to street level, see [http://maps.msn.com/(lhseei35lfegqtv3zp4gqg45)/map.aspx?&lats1=50&lons1=17&alts1=10000®n1=1 map of area covered]
- Australia
- Brazil
- [http://www.ab-map.com/ US Map]: ABMap - Interactive World Map and detailed united states street map
- [http://geographersblog.blogspot.com/ Antique Maps Yellow] News and Commentary about Antique Maps (Blog)
- [http://fax.libs.uga.edu/hmaps/ Historical Maps from the Hargrett Library Collection] (University of Georgia) --browse over 1000 maps from as early as 1544-- DjVu format; requires free plugin or JAVA
- [http://texashistory.unt.edu/browse/subject/Texas_Landscape_and_Nature/Geography_and_Maps/ Historical Maps from the Portal to Texas History]
- atlases with maps by country:
- [http://www.populationdata.net/cartes/cartes.html PopulationData.net]
- [http://www.cia.gov/cia/publications/factbook/] The World Factbook
- the same maps but with a faster, cleaner directory: [http://www.lib.utexas.edu/maps/cia04.html]
- [http://go.hrw.com/atlas/norm_htm/ Holt, Rinehart and Winston]
- [http://www.infoplease.com/atlas/ Infoplease]
- [http://www.lib.utexas.edu/maps UT scanned collection] - University of Texas at Austin (also at [http://us.aminet.net/pix/map/] and [http://wuarchive.wustl.edu/aminet/pix/map ]; [http://images.google.com/images?q=aminet%20map thumbnails at Google])
- [http://www.lib.utexas.edu/maps/map_sites/country_sites.html list of links including those to external sites]
- [http://www.aquarius.geomar.de/ Online Map Creation]: Webinterface to GMT mapping package
- new version: [http://www.planiglobe.com/omc_set.html Planiglobe]
- [http://www.mapzones.com MapZones] Map of all countries
- [http://www.multimap.com/ Multimap world atlas]: on UK, US, Canada, Australia and Western Europe more detailed than the rest of the world; the list of scales to select from is location-dependent - also provides geographic coordinates of a location selected from a map.
- MapQuest [http://www.mapquest.com/maps]: on US, Canada and Western Europe more detailed than the rest of the world - the list of scales to select from is not location-dependent: for part of the world several zoom level links lead to an image with just the message "No Data Available".
- [http://www.welt-atlas.de/worldatlas/ Atlas of the World] A world atlas with hundreds of very detailed and elaborate maps from every part of the world
- [http://maps.yahoo.com Yahoo Maps]: on US, Canada, Germany, France, Spain, Italy
- [http://de.maps.yahoo.com/ Yahoo Germany]: on France, UK, Germany, Italy, Spain, Portugal, Austria, Switzerland, Benelux
- [http://www.landkarte-online.net/ Maps of the World] (ger.)
- [http://krak.dk/ krak.dk]: Interactive map of Denmark
- [http://www.routenplaner24.de/ Routenplaner24] Street Maps and Route Planner in Germany
- [http://www.mapsouthpacific.com/ Map South Pacific]: on Polynesia, Melanesia, Micronesia
- [http://www.freeworldmaps.net/ World and Continent Maps]
- [http://www.streetmaps.de/ Street Maps in Switzerland, Austria and Germany]
- [http://plasma.nationalgeographic.com/mapmachine/ MapMachine] (National Geographic/ESRI)
- [http://www.nationalatlas.gov/ National Atlas of the United States]
- [http://nationalatlas.gov/reference.html US state maps]
- [http://www.citoplan.nl/citoplan/img/legenda_groot.gif Example of legend (Cito-Plan city maps)]
- http://www.geocities.com/marcoschmidt.geo/geo-data.html
- [http://nationalmap.usgs.gov/ USGS National Map]
- [http://www.web-routenplaner.de/ Free Maps and Route Planner on Europe]
- [http://www.outdoormountain.com/shopping/maps.php Maps of OutdoorMountain]
- [http://www.sunysb.edu/libmap/libcats.htm Online Map Catalogs in North America and Europe] Lists some good places to search for online maps
- [http://oddens.geog.uu.nl The fascinating world of maps and mapping] Lists all kinds of maps
- [http://www.maphistory.info/collections.html Map collections] Intorductory page to help navigate the online map resources
- [http://www.uidaho.edu/special-collections/Other.Repositories.html A listing of over 5000 websites] describing holdings of manuscripts, archives, rare books, historical photographs, and other primary sources for the research scholar
- [http://www.guiageo.com Brazilian and World Maps] Vector PDF maps of Brazilian states
- [http://www.centamap.com/cent/index.htm Hong Kong Maps] CentaMap
- [http://www.freeroute.de/ FreeRoute Germany]
- [http://www.cgrer.uiowa.edu/servers/servers_references.html#atlases Links to on-line atlases]
- [http://www.links4maps.com Link Directory for Maps] Link directory of map libraries and societies, antique maps and world, regional, travel, historical and other map resources.
- [http://www.matton.com/maps Royalty Free Maps] Royalty Free Vector and Bitmap Maps, available in Illustrator format.
- [http://www.mcwetboy.net/maproom/ Map Room] - a weblog about maps
- [http://www.rare-maps.com/links.cfm Antique and Rare Maps - Art Source International] - Links to rare and antique maps and to cartography resources.
- Google
- [http://earth.google.com Google Earth]: Interactive map of the world
- [http://maps.google.com Google Maps] Google's map system, similar to Mapquest.
- [http://www.flourish.org/upsidedownmap/ The Upsidedown Map Page] Pictures and info about maps which are oriented without North at the top
- [http://en.wikipedia.org/wiki/Wikipedia:Maps Wikipedia:Maps], use of maps on Wikipedia Note that map services with various zoom levels sometimes show a new map item, such as a new bridge, only on a smaller-scale map and not on an available larger scale map, because the latter is not as often updated, see e.g. the Tuas Second Link on [http://www.multimap.com/p/browse.cgi?scale=100000&lon=103.85&lat=1.3&scale=2000000], and larger scales which do not show it. Category:Cartography ja:地図

Contour line

A contour line (also level set, isopleth, isogram or isarithm) for a function of two variables is a curve connecting points where the function has a same particular value. A contour map is a map showing contour lines. The gradient of the function is always perpendicular to the contour lines. When the lines are close together the gradient is large: the variation is steep. If adjacent contour lines are of the same width, the direction of the gradient cannot be determined from the contour lines alone. However if contour lines rotate through three or more widths the direction of the gradient can also be determined from the contour lines. Different types of contour lines are given different names according to the nature of the quantity involved:
- dew point: isodrosotherm.
- economic production: isoquant.
- elevation: contour lines on a topographic map or isohypse
- depth: isobath.
- humidity: isohume.
- liquid precipitation amount: isohyet.
- magnetic declination: isogon
- presence of linguistic feature: isogloss
- pressure: isobar.
- solar radiation: isohel
- temperature: isotherm.
- wind direction: isogon.
- wind speed: isotach.
- drive time: isochrone.
- Frequency of hail storms: isochalaz.
- Time of thawing: isotac.
- Cost of travel time: isodopane. "Iso" can be replaced with "isallo" to give it the meaning of a line connecting points where some variable has changed at the same rate during some time period.

External links

- [http://avc.comm.nsdlib.org/cgi-bin/wiki_print.pl?Drawing_Contour_Plots Drawing Contour Plots]. A lesson plan that deals with drawing various isopleths.
- See also [http://phrontistery.info/contour.html Forthright's Phrontistery] for many more different types of isopleths.
- [http://www.groundwatersoftware.com/newsletter/july00/ Surfer] contour line software Category:Cartography Category:Multivariate calculus ja:等高線

Shade

Shade is the blocking of sunlight (in particular direct sunshine) by any object, and also the shadow created by that object. It may refer to blocking of sunlight by a roof, a tree, an umbrella, a window shade or blind, curtains, or other objects. Shade is an important issue in temperate and tropical zones for providing cooling and shelter from the sun. Providing certain configurations of shading is an important passive solar technique. This may be done with overhangs, with shade trees, or with vines. Shading using non-living materials blocks the sun, but also results in sunlight being absorbed and re-radiated as heat, or in sunlight being reflected as glare. Green plants, on the other hand, not only absorb a significant portion of the sunlight to invest as energy in photosynthesis to produce sugar, but also actively transpire, producing an additional cooling effect. ---- vines Shading is a process used in drawing for depicting levels of darkness on paper by applying more pressure with a drawing implement for darker areas, and less pressure for lighter areas. There are various techniques of shading including cross hatching where perpendicular lines of varying closeness are drawn in a grid pattern to shade an area. The closer the lines are together, the darker the area appears and vice versa. The term has been recently generalized to mean that shaders are applied. Light patterns, such as objects having light areas and shaded areas, help when creating the illusion of depth on paper and on computer screens. See also 3D computer graphics / Reflection and shading models. ---- A Shade in the esoteric or spiritual sense can be one of many things. A spiritual or emotional imprint left on a person, place or thing. A "presence" that which is seen out of the corner of the eye or known (sometimes physically felt) only under conditions and times. They can come in many forms, anything from a deceased (or living for that matter) person's or animal's imprint left on something or someone, to something invoked into the "possession" of an object and even sometimes a person. ---- In marketing, price shading is a variant of variable pricing in which sales people are given the authority to vary the price by a certain amount or percentage. Category:Drawing

Surface

:For other senses of this word, see surface (disambiguation). surface (disambiguation) In mathematics (topology), a surface is a two-dimensional manifold. Examples arise in three-dimensional space as the boundaries of three-dimensional solid objects. The surface of a fluid object, such as a rain drop or soap bubble, is an idealisation. To speak of the surface of a snowflake, which has a great deal of fine structure, is to go beyond the simple mathematical definition. For the nature of real surfaces see surface tension, surface chemistry, surface energy, roughness.

Definition

In what follows, all surfaces are considered to be second-countable 2-dimensional manifolds. More precisely: a topological surface (with boundary) is a Hausdorff space in which every point has an open neighbourhood homeomorphic to either an open subset of E² (Euclidean 2-space) or an open subset of the closed half of E². The set of points which have an open neighbourhood homeomorphic to Eⁿ is called the interior of the manifold; it is always non-empty. The complement of the interior, is called the boundary; it is a (1)-manifold, or union of closed curves. A surface with empty boundary is said to be closed if it is compact, and open if it is not compact.

Classification of closed surfaces

There is a complete classification of closed (i.e compact without boundary) connected, surfaces up to homeomorphism. Any such surface falls into one of two infinite collections:
- Spheres with g handles attached (called g-fold tori). These are orientable surfaces with Euler characteristic 2-2g, also called surfaces of genus g.
- Spheres with k projective planes attached. These are non-orientable surfaces with Euler characteristic 2-k. Therefore Euler characteristic and orientability describe a compact surfaces up to homeomorphism (and if surfaces are smooth then up to diffeomorphism).

Compact surfaces

Compact surfaces with boundary are just these with one or more removed open disks whose closures are disjoint.

Embeddings in R³

A compact surface can be embedded in R³ if it is orientable or if it has nonempty boundary. It is a consequence of the Whitney embedding theorem that any surface can be embedded in R⁴.

Differential geometry

A simple review of the embedding of a surface in n dimensions, and a computation of the area of such a surface, is provided in the article volume form. Metric properties of Riemann surfaces are briefly reviewed in the article Poincaré metric.

Some models

To make some models of various surfaces, attach the sides of these squares (A with A, B with B) so that the directions of the arrows match: Image:SphereAsSquare.png|sphere Image:ProjectivePlaneAsSquare.png|real projective plane Image:KleinBottleAsSquare.png|Klein bottle Image:TorusAsSquare.png|torus

Fundamental polygon

Each closed surface can be constructed from an even sided oriented polygon, called a fundamental polygon by pairwise identification of its edges. This construction can be represented as a string of length 2n of n distinct symbols where each symbol appears twice with exponent either +1 or -1. The exponent -1 signifies that the corresponding edge has the orientation opposing the one of the fundamental polygon. The above models can be described as follows:
- sphere:

A A^

- projective plane:

A A

- Klein bottle:

A B A^ B

- torus:

A B A^ B^

(See the main article fundamental polygon for details.)

Connected sum of surfaces

Given two surfaces M and M', their connected sum M # M' is obtained by removing a disk in each of them and gluing them along the newly formed boundary components. We use the following notation.
- sphere: S
- torus: T
- Klein bottle: K
- Projective plane: P Facts:
- S # S = S
- S # M = M
- P # P = K
- P # K = P # T We use a shorthand natation: nM = M # M # ... # M (n-times) with 0M = S. Closed surfaces are classified as follows:
- gT (g-fold torus): orientable surface of genus g, for

g \ge 0

.
- gP (g-fold projective plane): non-orientable surface of genus g, for

g \ge 1

Algebraic surface

This notion of a surface is distinct from the notion of an algebraic surface. A non-singular complex projective algebraic curve is a smooth surface. Algebraic surfaces over the complex number field have dimension 4 when considered as a real manifold.

External links

- [http://xahlee.org/surface/gallery.html Math Surfaces Gallery, with 60 ~surfaces and Java Applet for live rotation viewing] Category:Surfaces Category:Geometric topology ja:表面

Surveying

:For the American space program, see Surveyor program Surveying is the art and science of accurately determining the position of points and the distances between them. These points are usually, but not exclusively, associated with positions on the surface of the Earth, and are often used to establish land boundaries for ownership or governmental purposes. In order to accomplish their objective, surveyors use elements of engineering, physics, mathematics and law. Surveying has been an essential element in the development of the human environment since the beginning of recorded history and it is a requirement in the planning and execution of nearly every form of construction. Its most familiar modern uses are in the fields of transport, building and construction, communications, mapping, and the definition of legal boundaries for land ownership.

Method

The simplest method for measuring height is with an altimeter, which is basically a barometer—using air pressure as an indication of height. But for surveying more precision is needed. The basic tool is a theodolite, set on a tripod, with which one can measure angles (horizontal and vertical), combined with triangulation. Starting from a benchmark, a position with known location and elevation, the distance and angles to the unknown point are measured. A more modern instrument is a total station, which is basically a theodolite with an electronic distance measurement device (EDM). Still more modern is the use of satellite positioning systems, such as a Global Positioning System (GPS). Though GPS systems have increased the speed of surveying, they are still only accurate to about 20 mm. It is because of this that EDMDs have not been completely phased out. Robotics allows surveyors to gather precise measurements without extra workers to look through and turn the telescope or record data. A faster way to measure (no obstacles) is with a helicopter with laser echolocation, combined with GPS to determine the height of the helicopter. To increase precision, beacons are placed on the ground (about 20 km apart). This method reaches a precision of about 5 cm. With the triangulation method, first, one needs to know the horizontal distance to the object. If this is not known or cannot be measured directly, it is determined as explained in the triangulation article. Then the height of an object can be determined by measuring the angle between the horizontal plane and the line through that point at a known distance and the top of the object. In order to determine the height of a mountain, one should do this from sea level (the plane of reference), but here the distances can be too great and the mountain may not be visible. So it is done in steps, first determining the position of one point, then moving to that point and doing a relative measurement, and so on until the mountain top is reached.

Origins

Surveying techniques have existed throughout much of recorded history. In Ancient Egypt, when the Nile River overflowed its banks and washed out farm boundaries, boundaries were re-established through the application of simple geometry. The nearly perfect squareness and north-south orientation of the Great Pyramid of Giza, built c. 2700 BC, affirm the ancient Egyptians' command of surveying.
- The Egyptian land register (3000 BC).
- In Rome, the tax register of conquered lands(300 AD).
- In England, The Domesday Book by William the Conqueror(1086)
- covered all England
- contained names of the land owners, area, land quality, and specific information of the area's content and habitants.
- did not include maps showing exact locations
- Continental Europe's Cadastre was created in 1808
- founded by Napoleon I (Bonaparte), "A good cadastre will be my greatest achievement in my civil law", Napoleon I
- contained numbers of the parcels of land (or just land), land usage, names etc., and value of the land
- 100 million parcels of land, triangle survey, measurable survey, map scale: 1:2500 and 1:1250
- spread fast around Europe, but faced problems especially in Mediterranean countries, Balkan, and Eastern Europe due to cadastre upkeep costs and troubles. A cadastre loses its value if register and maps are not constantly updated. Large scale surveys are a necessary pre-requisite to map-making. In the late 1780s, a team from the Ordnance Survey of Great Britain, originally under General William Roy began the Principal Triangulation of Britain using the specially built Ramsden theodolite.

Types of surveys

Ramsden theodolite]]
- ALTA/ACSM survey: a surveying standard jointly proposed by the American Land Title Association and the American Congress on Surveying and Mapping that incorporates elements of the boundary survey, mortgage survey, and topographic survey. ALTA/ACSM surveys, frequently shortened to ALTA surveys, are often required for real estate transactions.
- Boundary survey: the actual positions of existing marks on land (typically iron rods or concrete monuments in the ground, but also tacks in trees, pipes, and manholes) are measured, and a map, or plat, is drawn from the data.
- Deformation survey: a survey to determine if a structure or object is changing shape or moving. The three-dimensional positions of specific points on an object are determined, a period of time is allowed to pass, these positions are then re-measured and calculated, and a comparison between the two sets of positions is made.
- Draw lot: one lot from a plat is drawn, with any easements and setbacks that may be on it.
- Foundation survey: the position of the house is measured before it is finished being built.
- Mortgage survey: a simple survey that generally determines land boundaries and building locations. Mortgage surveys are required by title companies and lending institutions when they provide financing to show that there are no structures encroaching on the property and that the position of structures is generally within zoning and building code requirements. Mortgage surveys are not sufficiently accurate for use in building new structures.
- Physical survey: the finished house and driveway are measured, and all markers on the boundary are indicated. This is recorded when the lot is sold.
- Plot plan: a proposal for a house or other building and driveway or parking lot are added to a draw lot.
- Subdivision plat: a plot or map based on a survey of a parcel of land. Lines are drawn inside it, indicating the location of roads and lots. Plats are usually discussed back and forth between the developer and the surveyor until they are agreed upon, at which point pins are driven into the ground to mark the lot corners and curve ends, and the plat is recorded in the cadastre (USA, elsewhere) or land registry (UK).
- Topographic survey: a survey that measures the elevation of points on a particular piece of land, and presents them as contours on a plot.
- Hydrographic survey: a survey conducted with the purpose of mapping the seabed for navigation, engineering, or resource management purposes. Products of such surveys are nautical charts. See hydrography.
- Construction surveying (otherwise 'lay-out' or 'setting-out'): the process of establishing and marking the position and detailed layout of new structures such as roads or buildings for subsequent construction. In this sense, surveying may be regarded as a sub-discipline of civil engineering.

Surveying as a career

The basic principles of surveying have changed little over the ages, but the tools used by surveyors have evolved tremendously. Engineering, especially civil engineering, depends heavily on the surveyor. Whenever there are roads, dams, retaining walls, bridges or residential areas to be built, surveyors are involved. They determine the boundaries of private property and the boundaries of various lines of political divisions. They also provide advice and data for geographical information systems (GIS), computer databases that contain data on land features and boundaries. Surveyors must have a thorough knowledge of algebra, basic calculus, geometry, and trigonometry. They must also know the laws that deal with surveys, property, and contracts. In addition, they must be able to use delicate instruments with accuracy and precision. In most states of the U.S., surveying is recognized as a distinct profession apart from engineering. Licensing requirements vary by state. In the past, experience gained through an apprenticeship, together with passing a series of state-administered examinations, was required to attain licensure. Nowadays, many states require a Bachelor of Science in Surveying, or a Bachelor of Science in Civil Engineering with additional coursework in surveying, in addition to experience and examination requirements. Registered surveyors usually denote themselves with the letters P.S. (professional surveyor), L.S. (land surveyor), or P.L.S. (professional land surveyor), following their names, depending upon the dictates of their particular state of registration.

Topography

See also

Geography

Structure of geography

Physical geography

Human geography

Socio-environmental geography

Cultural and political ecology

Risk-hazards research

Historical geography

History of geography

Geographic techniques

Related fields

Urban and regional planning

Regional science

Reference

See also

External links

Landform

See also

List of landforms

Slope landforms

Coastal and oceanic landforms

Fluvial landforms

Mountain and glacial landforms

Volcanic landforms

Erosion landforms

External links

Elevation

Basic Definition

Determining Elevation

Links

Slope

Definition of slope

Example 1

Example 2

Geometry

Slope of a road, etc.

Algebra

Calculus

Why calculus is necessary

See also

Dimension

Physical dimensions

Spatial dimensions

Time

Additional dimensions

Units

Mathematical dimensions

Hamel dimension

Manifolds

Lebesgue covering dimension

Inductive dimension

Hausdorff dimension

Hilbert spaces

Krull dimension of commutative rings

Science fiction

Anaglyph

3-D film

Modern 3-D films

More dimensions

See also

Degrees of freedom

Other

Further reading

Map

Introduction

Orientation of maps

Scale and accuracy

World maps and projections

Electronic maps

References

See also

External links

Contour line

External links

Shade

Surface

Definition

Classification of closed surfaces

Embeddings in R³