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69-454: Specific charts are used for each phase of a flight and may vary from a map of a particular airport facility to an overview of the instrument routes covering an entire continent (e.g., global navigation charts), and many types in between. Visual flight charts are categorized according to their scale , which is proportional to the size of the area covered by one map. The amount of detail is necessarily reduced when larger areas are represented on

138-610: A {\displaystyle a} is the radius of the sphere, λ {\displaystyle \lambda } is the longitude from the central meridian of the projection (here taken as the Greenwich meridian at λ = 0 {\displaystyle \lambda =0} ) and φ {\displaystyle \varphi } is the latitude. Note that λ {\displaystyle \lambda } and φ {\displaystyle \varphi } are in radians (obtained by multiplying

207-546: A δ φ {\displaystyle a\,\delta \varphi } where a {\displaystyle a} is the radius of the sphere and φ {\displaystyle \varphi } is in radian measure. The lines PM and KQ are arcs of parallel circles of length ( a cos ⁡ φ ) δ λ {\displaystyle (a\cos \varphi )\delta \lambda } with λ {\displaystyle \lambda } in radian measure. In deriving

276-419: A meridian distance of about 10 km and over an east-west line of about 8 km. Thus a plan of New York City accurate to one metre or a building site plan accurate to one millimetre would both satisfy the above conditions for the neglect of curvature. They can be treated by plane surveying and mapped by scale drawings in which any two points at the same distance on the drawing are at the same distance on

345-400: A point property of the projection at P it suffices to take an infinitesimal element PMQK of the surface: in the limit of Q approaching P such an element tends to an infinitesimally small planar rectangle. Normal cylindrical projections of the sphere have x = a λ {\displaystyle x=a\lambda } and y {\displaystyle y} equal to

414-408: A "scale") to represent it. The second distinct concept of scale applies to the variation in scale across a map. It is the ratio of the mapped point's scale to the nominal scale. In this case 'scale' means the scale factor (also called point scale or particular scale ). If the region of the map is small enough to ignore Earth's curvature, such as in a town plan, then a single value can be used as

483-471: A bundle of Western Han -era (202 BCE to 9 CE) counting rods made of bones was unearthed from Qianyang County in Shaanxi . The use of counting rods must predate it; Sunzi ( c. 544 to c. 496 BCE), a military strategist at the end of Spring and Autumn period of 771 BCE to 5th century BCE, mentions their use to make calculations to win wars before going into the battle; Laozi (died 531 BCE), writing in

552-421: A checker counting board diagram, with the order of magnitude symbols "千百十一分厘毛" (thousand, hundred, ten, unit, tenth, hundredth, thousandth). Examples: Rod numerals are a positional numeral system made from shapes of counting rods. Positive numbers are written as they are and the negative numbers are written with a slant bar at the last digit. The vertical bar in the horizontal forms 6–9 are drawn shorter to have

621-525: A clear distinction of the intrinsic projection scaling and the reduction scaling. From this point we ignore the RF and work with the projection map. Consider a small circle on the surface of the Earth centred at a point P at latitude φ {\displaystyle \varphi } and longitude λ {\displaystyle \lambda } . Since the point scale varies with position and direction

690-428: A constant separation on the ground. While a map may display a graphical bar scale, the scale must be used with the understanding that it will be accurate on only some lines of the map. (This is discussed further in the examples in the following sections.) Let P be a point at latitude φ {\displaystyle \varphi } and longitude λ {\displaystyle \lambda } on

759-487: A facsimile from a page of Yongle Encyclopedia . He arranged 七萬一千八百二十四 as He treated the Chinese order numbers as place value markers, and 七一八二四 became place value decimal number. He then wrote the rod numerals according to their place value: In Japan, mathematicians put counting rods on a counting board, a sheet of cloth with grids, and used only vertical forms relying on the grids. An 18th-century Japanese mathematics book has

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828-471: A function of latitude only. Therefore, the infinitesimal element PMQK on the sphere projects to an infinitesimal element P'M'Q'K' which is an exact rectangle with a base δ x = a δ λ {\displaystyle \delta x=a\,\delta \lambda } and height  δ y {\displaystyle \delta y} . By comparing the elements on sphere and projection we can immediately deduce expressions for

897-441: A map. The foundations for quantitative map scaling goes back to ancient China with textual evidence that the idea of map scaling was understood by the second century BC. Ancient Chinese surveyors and cartographers had ample technical resources used to produce maps such as counting rods , carpenter's square 's, plumb lines , compasses for drawing circles, and sighting tubes for measuring inclination. Reference frames postulating

966-449: A map. When an aircraft is flying under instrument flight rules (IFR), the pilot will often have no visual reference to the ground, and must therefore rely on external (e.g. GPS or VOR ) aids in order to navigate. Although in some situations air traffic control may issue radar vectors to direct an aircraft's path, this is usually done to facilitate traffic flow, and will not be the sole means of navigating to an important point, such as

1035-503: A nascent coordinate system for identifying locations were hinted by ancient Chinese astronomers that divided the sky into various sectors or lunar lodges. The Chinese cartographer and geographer Pei Xiu of the Three Kingdoms period created a set of large-area maps that were drawn to scale. He produced a set of principles that stressed the importance of consistent scaling, directional measurements, and adjustments in land measurements in

1104-612: A number of wood scripts from a tomb in Hubei dating from the period of the Han dynasty (206 BCE to 220 CE). On one of the wooden scripts was written: "当利二月定算𝍥". This is one of the earliest examples of using counting-rod numerals in writing. A square lacquer box, dating from c. 168 BCE, containing a square chess board with the TLV patterns, chessmen, counting rods, and other items, was excavated in 1972, from Mawangdui M3, Changsha, Hunan Province. In 1976,

1173-527: A ratio: if the scale is an inch to two miles and the map user can see two villages that are about two inches apart on the map, then it is easy to work out that the villages are about four miles apart on the ground. A lexical scale may cause problems if it expressed in a language that the user does not understand or in obsolete or ill-defined units. For example, a scale of one inch to a furlong (1:7920) will be understood by many older people in countries where Imperial units used to be taught in schools. But

1242-418: A scale of one pouce to one league may be about 1:144,000, depending on the cartographer 's choice of the many possible definitions for a league, and only a minority of modern users will be familiar with the units used. A small-scale map cover large regions, such as world maps , continents or large nations. In other words, they show large areas of land on a small space. They are called small scale because

1311-453: A separation along the line to the bar scale does not give a distance related to the true distance in any simple way. (But see addendum ). Even if a distance along this line of constant planar angle could be worked out, its relevance is questionable since such a line on the projection corresponds to a complicated curve on the sphere. For these reasons bar scales on small-scale maps must be used with extreme caution. The Mercator projection maps

1380-441: A separation from a parallel to the bar scale we must divide the bar scale distance by this factor to obtain the distance between the points when measured along the parallel (which is not the true distance along a great circle ). On a line at a bearing of say 45 degrees ( β = 45 ∘ {\displaystyle \beta =45^{\circ }} ) the scale is continuously varying with latitude and transferring

1449-431: A smaller area. Maps that show an extensive area are "small scale" maps. This can be a cause of confusion. Mapping large areas causes noticeable distortions because it significantly flattens the curved surface of the earth. How distortion gets distributed depends on the map projection . Scale varies across the map , and the stated map scale is only an approximation. This is discussed in detail below. The region over which

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1518-647: A true positional numeral system with digits for 1–9 and a blank for 0, from the Warring states period (circa 475 BCE) to the 16th century. Chinese arithmeticians used counting rods well over two thousand years ago. In 1954, forty-odd counting rods of the Warring States period (5th century BCE to 221 BCE) were found in Zuǒjiāgōngshān (左家公山) Chu Grave No.15 in Changsha , Hunan . In 1973, archeologists unearthed

1587-422: Is commonly illustrated by the impossibility of smoothing an orange peel onto a flat surface without tearing and deforming it. The only true representation of a sphere at constant scale is another sphere such as a globe . Given the limited practical size of globes, we must use maps for detailed mapping. Maps require projections. A projection implies distortion: A constant separation on the map does not correspond to

1656-488: Is conformal since it is constructed to preserve angles and its scale factor is isotropic, a function of latitude only: Mercator does preserve shape in small regions. Definition: on a conformal projection with an isotropic scale, points which have the same scale value may be joined to form the isoscale lines . These are not plotted on maps for end users but they feature in many of the standard texts. (See Snyder pages 203—206.) There are two conventions used in setting down

1725-518: Is no standard: The terms are sometimes used in the absolute sense of the table, but other times in a relative sense. For example, a map reader whose work refers solely to large-scale maps (as tabulated above) might refer to a map at 1:500,000 as small-scale. In the English language, the word large-scale is often used to mean "extensive". However, as explained above, cartographers use the term "large scale" to refer to less extensive maps – those that show

1794-439: Is said to be conformal if the angle between a pair of lines intersecting at a point P is the same as the angle between the projected lines at the projected point P', for all pairs of lines intersecting at point P. A conformal map has an isotropic scale factor. Conversely isotropic scale factors across the map imply a conformal projection. Isotropy of scale implies that small elements are stretched equally in all directions, that

1863-454: Is the shape of a small element is preserved. This is the property of orthomorphism (from Greek 'right shape'). The qualification 'small' means that at some given accuracy of measurement no change can be detected in the scale factor over the element. Since conformal projections have an isotropic scale factor they have also been called orthomorphic projections . For example, the Mercator projection

1932-446: Is useful to note that The following examples illustrate three normal cylindrical projections and in each case the variation of scale with position and direction is illustrated by the use of Tissot's indicatrix . The equirectangular projection , also known as the Plate Carrée (French for "flat square") or (somewhat misleadingly) the equidistant projection, is defined by where

2001-532: The Federal Aviation Administration . Scale (map) The scale of a map is the ratio of a distance on the map to the corresponding distance on the ground. This simple concept is complicated by the curvature of the Earth 's surface, which forces scale to vary across a map. Because of this variation, the concept of scale becomes meaningful in two distinct ways. The first way is the ratio of

2070-507: The abacus flourished , counting rods were abandoned except in Japan, where rod numerals developed into a symbolic notation for algebra . Counting rods represent digits by the number of rods, and the perpendicular rod represents five. To avoid confusion, vertical and horizontal forms are alternately used. Generally, vertical rod numbers are used for the position for the units, hundreds, ten thousands, etc., while horizontal rod numbers are used for

2139-476: The parallel scale is denoted by k ( λ , φ ) {\displaystyle k(\lambda ,\,\varphi )} . Definition: if the point scale depends only on position, not on direction, we say that it is isotropic and conventionally denote its value in any direction by the parallel scale factor k ( λ , φ ) {\displaystyle k(\lambda ,\varphi )} . Definition: A map projection

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2208-534: The representative fraction is relatively small. Large-scale maps show smaller areas in more detail, such as county maps or town plans might. Such maps are called large scale because the representative fraction is relatively large. For instance a town plan, which is a large-scale map, might be on a scale of 1:10,000, whereas the world map, which is a small scale map, might be on a scale of 1:100,000,000. The following table describes typical ranges for these scales but should not be considered authoritative because there

2277-428: The Earth's surface) and bearing (on the map) is not universally observed, many writers using the terms almost interchangeably. Definition: the point scale at P is the ratio of the two distances P'Q' and PQ in the limit that Q approaches P. We write this as where the notation indicates that the point scale is a function of the position of P and also the direction of the element PQ. Definition: if P and Q lie on

2346-458: The RF (or principal scale) gives the actual circumference of the Earth. The bar scale on the map is also drawn at the true scale so that transferring a separation between two points on the equator to the bar scale will give the correct distance between those points. The same is true on the meridians. On a parallel other than the equator the scale is sec ⁡ φ {\displaystyle \sec \varphi } so when we transfer

2415-591: The Tissot diagram each infinitesimal circular element preserves its shape but is enlarged more and more as the latitude increases. Lambert's equal area projection maps the sphere to a finite rectangle by the equations where a, λ {\displaystyle \lambda } and φ {\displaystyle \varphi } are as in the previous example. Since y ′ ( φ ) = cos ⁡ φ {\displaystyle y'(\varphi )=\cos \varphi }

2484-593: The Warring States period, said "a good calculator doesn't use counting rods". The Book of Han (finished 111 CE) recorded: "they calculate with bamboo, diameter one fen, length six cun, arranged into a hexagonal bundle of two hundred seventy one pieces". At first, calculating rods were round in cross-section, but by the time of the Sui dynasty (581 to 618 CE) mathematicians used triangular rods to represent positive numbers and rectangular rods for negative numbers . After

2553-483: The actual printed (or viewed) maps. If the definition of point scale in the previous section is in terms of the projection map then we can expect the scale factors to be close to unity. For normal tangent cylindrical projections the scale along the equator is k=1 and in general the scale changes as we move off the equator. Analysis of scale on the projection map is an investigation of the change of k away from its true value of unity. Actual printed maps are produced from

2622-424: The counting board, they are positional. [REDACTED]   [REDACTED] is an exact image of the counting rod number 405 on a table top or floor. The value of a number depends on its physical position on the counting board. A 9 at the rightmost position on the board stands for 9. Moving the batch of rods representing 9 to the left one position (i.e., to the tens place) gives 9[] or 90. Shifting left again to

2691-628: The counting rods, after its replacement by abacus . One variation of horizontal rod numerals, the Suzhou numerals is still in use for book-keeping and in herbal medicine prescription in Chinatowns in some parts of the world. Unicode 5.0 includes counting rod numerals in their own block in the Supplementary Multilingual Plane (SMP) from U+1D360 to U+1D37F. The code points for the horizontal digits 1–9 are U+1D360 to U+1D368 and those for

2760-634: The degree measure by a factor of π {\displaystyle \pi } /180). The longitude λ {\displaystyle \lambda } is in the range [ − π , π ] {\displaystyle [-\pi ,\pi ]} and the latitude φ {\displaystyle \varphi } is in the range [ − π / 2 , π / 2 ] {\displaystyle [-\pi /2,\pi /2]} . Since y ′ ( φ ) = 1 {\displaystyle y'(\varphi )=1}

2829-407: The earth can be regarded as flat depends on the accuracy of the survey measurements. If measured only to the nearest metre, then curvature of the earth is undetectable over a meridian distance of about 100 kilometres (62 mi) and over an east-west line of about 80 km (at a latitude of 45 degrees). If surveyed to the nearest 1 millimetre (0.039 in), then curvature is undetectable over

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2898-433: The equations of any given projection. For example, the equirectangular cylindrical projection may be written as Here we shall adopt the first of these conventions (following the usage in the surveys by Snyder). Clearly the above projection equations define positions on a huge cylinder wrapped around the Earth and then unrolled. We say that these coordinates define the projection map which must be distinguished logically from

2967-556: The first century CE, stated "(when using subtraction) subtract same signed numbers, add different signed numbers, subtract a positive number from zero to make a negative number, and subtract a negative number from zero to make a positive number". Later, a go stone was sometimes used to represent zero. This alternation of vertical and horizontal rod numeral form is very important to understanding written transcription of rod numerals on manuscripts correctly. For instance, in Licheng suanjin, 81

3036-403: The ground. True ground distances are calculated by measuring the distance on the map and then multiplying by the inverse of the scale fraction or, equivalently, simply using dividers to transfer the separation between the points on the map to a bar scale on the map. As proved by Gauss ’s Theorema Egregium , a sphere (or ellipsoid) cannot be projected onto a plane without distortion. This

3105-402: The major axis to the minor axis is sec ⁡ φ {\displaystyle \sec \varphi } . Clearly the area of the ellipse increases by the same factor. It is instructive to consider the use of bar scales that might appear on a printed version of this projection. The scale is true (k=1) on the equator so that multiplying its length on a printed map by the inverse of

3174-535: The map. The distortion ellipse is known as Tissot's indicatrix . The example shown here is the Winkel tripel projection , the standard projection for world maps made by the National Geographic Society . The minimum distortion is on the central meridian at latitudes of 30 degrees (North and South). (Other examples ). The key to a quantitative understanding of scale is to consider an infinitesimal element on

3243-404: The modern horizontal bar). The method for using counting rods for mathematical calculation was called rod calculation or rod calculus (筹算). Rod calculus can be used for a wide range of calculations, including finding the value of π , finding square roots , cube roots , or higher order roots , and solving a system of linear equations . Before the introduction of a written zero, a space

3312-429: The position from which an aircraft commences its approach to landing. Charts used for IFR flights contain an abundance of information regarding locations of waypoints , known as " fixes ", which are defined by measurements from electronic beacons of various types, as well as the routes connecting these waypoints. Only limited topographic information is found on IFR charts, although the minimum safe altitudes available on

3381-482: The previous section gives For the calculation of the point scale in an arbitrary direction see addendum . The figure illustrates the Tissot indicatrix for this projection. On the equator h=k=1 and the circular elements are undistorted on projection. At higher latitudes the circles are distorted into an ellipse given by stretching in the parallel direction only: there is no distortion in the meridian direction. The ratio of

3450-452: The projection map by a constant scaling denoted by a ratio such as 1:100M (for whole world maps) or 1:10000 (for such as town plans). To avoid confusion in the use of the word 'scale' this constant scale fraction is called the representative fraction (RF) of the printed map and it is to be identified with the ratio printed on the map. The actual printed map coordinates for the equirectangular cylindrical projection are This convention allows

3519-400: The projection of the circle on the projection will be distorted. Tissot proved that, as long as the distortion is not too great, the circle will become an ellipse on the projection. In general the dimension, shape and orientation of the ellipse will change over the projection. Superimposing these distortion ellipses on the map projection conveys the way in which the point scale is changing over

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3588-787: The routes are shown. En-route low- and high-altitude charts are published with a scale that depends upon the density of navigation information required in the vicinity. Information from IFR charts is often programmed into a flight management system or autopilot , which eases the task of following (or deviating from) a flight plan. Terminal procedure publications such as standard terminal arrival plates, standard instrument departure plates and other documentation provide detailed information for arrival, departure and taxiing at each approved airport having instrument capabilities of some sort. Aeronautical charts may be purchased at fixed-base operators (FBOs), internet supply sources, or catalogs of aeronautical gear. They may also be viewed online from

3657-598: The same character height. A circle (〇) is used for 0. Many historians think it was imported from Indian numerals by Gautama Siddha in 718, but some think it was created from the Chinese text space filler "□", and others think that the Indians acquired it from China, because it resembles a Confucian philosophical symbol for "nothing". In the 13th century, Southern Song mathematicians changed digits for 4, 5, and 9 to reduce strokes. The new horizontal forms eventually transformed into Suzhou numerals . Japanese continued to use

3726-413: The same meridian ( α = 0 ) {\displaystyle (\alpha =0)} , the meridian scale is denoted by h ( λ , φ ) {\displaystyle h(\lambda ,\,\varphi )} . Definition: if P and Q lie on the same parallel ( α = π / 2 ) {\displaystyle (\alpha =\pi /2)} ,

3795-445: The scale factors are The calculation of the point scale in an arbitrary direction is given below . Counting rods Counting rods (筭) are small bars, typically 3–14 cm (1" to 6") long, that were used by mathematicians for calculation in ancient East Asia . They are placed either horizontally or vertically to represent any integer or rational number . The written forms based on them are called rod numerals . They are

3864-401: The scale factors are: In the mathematical addendum it is shown that the point scale in an arbitrary direction is also equal to sec ⁡ φ {\displaystyle \sec \varphi } so the scale is isotropic (same in all directions), its magnitude increasing with latitude as sec ⁡ φ {\displaystyle \sec \varphi } . In

3933-428: The scale factors on parallels and meridians. (The treatment of scale in a general direction may be found below .) Note that the parallel scale factor k = sec ⁡ φ {\displaystyle k=\sec \varphi } is independent of the definition of y ( φ ) {\displaystyle y(\varphi )} so it is the same for all normal cylindrical projections. It

4002-424: The scale without causing measurement errors. In maps covering larger areas, or the whole Earth, the map's scale may be less useful or even useless in measuring distances. The map projection becomes critical in understanding how scale varies throughout the map. When scale varies noticeably, it can be accounted for as the scale factor. Tissot's indicatrix is often used to illustrate the variation of point scale across

4071-418: The size of the generating globe to the size of the Earth. The generating globe is a conceptual model to which the Earth is shrunk and from which the map is projected . The ratio of the Earth's size to the generating globe's size is called the nominal scale (also called principal scale or representative fraction ). Many maps state the nominal scale and may even display a bar scale (sometimes merely called

4140-607: The sphere (or ellipsoid ). Let Q be a neighbouring point and let α {\displaystyle \alpha } be the angle between the element PQ and the meridian at P: this angle is the azimuth angle of the element PQ. Let P' and Q' be corresponding points on the projection. The angle between the direction P'Q' and the projection of the meridian is the bearing β {\displaystyle \beta } . In general α ≠ β {\displaystyle \alpha \neq \beta } . Comment: this precise distinction between azimuth (on

4209-461: The sphere to a rectangle (of infinite extent in the y {\displaystyle y} -direction) by the equations where a, λ {\displaystyle \lambda \,} and φ {\displaystyle \varphi \,} are as in the previous example. Since y ′ ( φ ) = a sec ⁡ φ {\displaystyle y'(\varphi )=a\sec \varphi }

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4278-513: The sphere. The figure shows a point P at latitude φ {\displaystyle \varphi } and longitude λ {\displaystyle \lambda } on the sphere. The point Q is at latitude φ + δ φ {\displaystyle \varphi +\delta \varphi } and longitude λ + δ λ {\displaystyle \lambda +\delta \lambda } . The lines PK and MQ are arcs of meridians of length

4347-532: The tens, thousands, hundred thousands etc. It is written in Sunzi Suanjing that "one is vertical, ten is horizontal". Red rods represent positive numbers and black rods represent negative numbers . Ancient Chinese clearly understood negative numbers and zero (leaving a blank space for it), though they had no symbol for the latter. The Nine Chapters on the Mathematical Art , which was mainly composed in

4416-416: The terrain that was being mapped. Map scales may be expressed in words (a lexical scale), as a ratio, or as a fraction. Examples are: In addition to the above many maps carry one or more (graphical) bar scales . For example, some modern British maps have three bar scales, one each for kilometres, miles and nautical miles. A lexical scale in a language known to the user may be easier to visualise than

4485-415: The third position (to the hundreds place) gives 9[][] or 900. Each time one shifts a number one position to the left, it is multiplied by 10. Each time one shifts a number one position to the right, it is divided by 10. This applies to single-digit numbers or multiple-digit numbers. Song dynasty mathematician Jia Xian used hand-written Chinese decimal orders 步十百千萬 as rod numeral place value, as evident from

4554-463: The traditional forms. Examples: In Japan, Seki Takakazu developed the rod numerals into symbolic notation for algebra and drastically improved Japanese mathematics . After his period, the positional numeral system using Chinese numeral characters was developed, and the rod numerals were used only for the plus and minus signs . A fraction was expressed with rod numerals as two rod numerals one on top of another (without any other symbol, like

4623-522: The vertical digits 1–9 are U+1D369 to U+1D371. The former are called unit digits and the latter are called tens digits , which is opposite of the convention described above. The Unicode Standard states that the orientation of the Unicode characters follows Song dynasty convention, which differs from Han dynasty practice which represented digits as vertical lines, and tens as horizontal lines. Zero should be represented by U+3007 (〇, ideographic number zero) and

4692-562: Was transcribed as [REDACTED] [REDACTED] , and 108 was transcribed as [REDACTED] [REDACTED] ; it is clear that the latter clearly had a blank zero on the "counting board" (i.e., floor or mat), even though on the written transcription, there was no blank. In the same manuscript, 405 was transcribed as [REDACTED]   [REDACTED] , with a blank space in between for obvious reasons, and could in no way be interpreted as "45" [REDACTED] [REDACTED] . In other words, transcribed rod numerals may not be positional, but on

4761-455: Was used to indicate no units, and the rotation of the character in the subsequent unit column, by 90°, adopted, to help reduce the ambiguity in record values calculated on the rods. For example 107 (𝍠 𝍧) and 17 (𝍩𝍧) would be distinguished by rotation, though multiple zero units could lead to ambiguity, eg. 1007 (𝍩 𝍧) , and 10007 (𝍠 𝍧). Once written zero came into play, the rod numerals had become independent, and their use indeed outlives

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