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From Wikipedia

Pie chart

A pie chart (or a circle graph) is a circularchart divided into sectors, illustrating proportion. In a pie chart, the arc length of each sector (and consequently its central angle and area), is proportional to the quantity it represents. When angles are measured with 1 turn as unit then a number of percent is identified with the same number of centiturns. Together, the sectors create a full disk. It is named for its resemblance to a pie which has been sliced. The earliest known pie chart is generally credited to William Playfair's Statistical Breviary of 1801.

The pie chart is perhaps the most ubiquitous statistical chart in the business world and the mass media. However, it has been criticized, and some recommend avoiding it, pointing out in particular that it is difficult to compare different sections of a given pie chart, or to compare data across different pie charts. Pie charts can be an effective way of displaying information in some cases, in particular if the intent is to compare the size of a slice with the whole pie, rather than comparing the slices among them. Pie charts work particularly well when the slices represent 25 to 50% of the data, but in general, other plots such as the bar chart or the dot plot, or non-graphical methods such as tables, may be more adapted for representing certain information.It also shows the frequency within certain groups of information.

Example

The following example chart is based on preliminary results of the election for the European Parliament in 2004. The table lists the number of seats allocated to each party group, along with the derived percentage of the total that they each make up. The values in the last column, the derived central angle of each sector, is found by multiplying the percentage by 360°.

*Because of rounding, these totals do not add up to 100 and 360.

The size of each central angle is proportional to the size of the corresponding quantity, here the number of seats. Since the sum of the central angles has to be 360°, the central angle for a quantity that is a fraction Q of the total is 360Q degrees. In the example, the central angle for the largest group (European People's Party (EPP)) is 135.7° because 0.377 times 360, rounded to one decimal place(s), equals 135.7.

Use, effectiveness and visual perception

Pie charts are common in business and journalism, perhaps because they are perceived as being less "geeky" than other types of graph. However statisticians generally regard pie charts as a poor method of displaying information, and they are uncommon in scientific literature. One reason is that it is more difficult for comparisons to be made between the size of items in a chart when area is used instead of length and when different items are shown as different shapes. Stevens' power law states that visual area is perceived with a power of 0.7, compared to a power of 1.0 for length. This suggests that length is a better scale to use, since perceived differences would be linearly related to actual differences.

Further, in research performed at AT&T Bell Laboratories, it was shown that comparison by angle was less accurate than comparison by length. This can be illustrated with the diagram to the right, showing three pie charts, and, below each of them, the corresponding bar chart representing the same data. Most subjects have difficulty ordering the slices in the pie chart by size; when the bar chart is used the comparison is much easier.. Similarly, comparisons between data sets are easier using the bar chart. However, if the goal is to compare a given category (a slice of the pie) with the total (the whole pie) in a single chart and the multiple is close to 25 or 50 percent, then a pie chart can often be more effective than a bar graph.

Variants and similar charts

Polar area diagram

The polar area diagram is similar to a usual pie chart, except sectors are equal angles and differ rather in how far each sector extends from the center of the circle. The polar area diagram is used to plot cyclic phenomena (e.g., count of deaths by month). For example, if the count of deaths in each month for a year are to be plotted then there will be 12 sectors (one per month) all with the same angle of 30 degrees each. The radius of each sector would be proportional to the square root of the death count for the month, so the area of a sector represents the number of deaths in a month. If the death count in each month is subdivided by cause of death, it is possible to make multiple comparisons on one diagram, as is clearly seen in the form of polar area diagram famously developed by Florence Nightingale.

The first known use of polar area diagrams was by André-Michel Guerry, which he called courbes circulaires, in an 1829 paper showing seasonal and daily variation in wind direction over the year and births and deaths by hour of the day. Léon Lalanne later used a polar diagram to show the frequency of wind directions around compass points in 1843. The wind rose is still used by meteorologists. Nightingale published her rose diagram in 1858. The name "coxcomb" is sometimes used erroneously. This was the name Nightingale used to refer to a book containing the diagrams rather than the diagrams themselves. It has been suggested that most of Nightingale's early reputation was built on her ability to give clear and concise presentations of data.

Spie chart

A useful variant of the polar area chart is the spie chart designed by Feitelson . This superimposes a normal pie chart with a modified polar area chart to permit the comparison of a set of data at two different states. For the first state, for example time 1, a normal pie chart is drawn. For the second state, the angles of the slices are the same as in the original pie chart, and the radii vary according to the change in the value of each variable. In addition to comparing a partition at two times (e.g. this year's budget distribution with last year's budget distribution), this is useful for visualizing hazards for population groups (e.g. the distribution of age and gener groups among road casualties compared with these groups's sizes in the general population). The R Graph Gallery provides an example.

Multi-level pie chart

Multi-level pie chart, also known as a radial tree c

Complex plane

In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis. It can be thought of as a modified Cartesian plane, with the real part of a complex number represented by a displacement along the x-axis, and the imaginary part by a displacement along the y-axis.

The complex plane is sometimes called the Argand plane because it is used in Argand diagrams. These are named after Jean-Robert Argand (1768-1822), although they were first described by Norwegian-Danish land surveyor and mathematician Caspar Wessel (1745-1818). Argand diagrams are frequently used to plot the positions of the poles and zeroes of a function in the complex plane.

The concept of the complex plane allows a geometric interpretation of complex numbers. Under addition, they add like vectors. The multiplication of two complex numbers can be expressed most easily in polar coordinates– the magnitude or modulus of the product is the product of the two absolute values, or moduli, and the angle or argument of the product is the sum of the two angles, or arguments. In particular, multiplication by a complex number of modulus 1 acts as a rotation.

Notational conventions

In complex analysis the complex numbers are customarily represented by the symbol z, which can be separated into its real (x) and imaginary (y) parts, like this:

z = x + iy\, for example: z = 4 + i5,

where x and y are real numbers, and i is the imaginary unit. In this customary notation the complex number z corresponds to the point (x, y) in the Cartesian plane.

In the Cartesian plane the point (x, y) can also be represented in polar coordinates as

(x, y) = (r\cos\theta, r\sin\theta)\qquad\left(r = \sqrt{x^2+y^2}; \quad \theta=\arctan\frac{y}{x}\right).\,

In the Cartesian plane it may be assumed that the arctangent takes values from −Ï€/2 to Ï€/2 (in radians), and some care must be taken to define the real arctangent function for points (x, y) when x≤ 0. In the complex plane these polar coordinates take the form

z = x + iy = |z|\left(\cos\theta + i\sin\theta\right) = |z|e^{i\theta}\,

where

|z| = \sqrt{x^2+y^2}; \quad \theta = \arg(z) = -i\ln\frac{z}.\,

Here |z| is the absolute value or modulus of the complex number z; θ, the argument of z, is usually taken on the interval 0 ≤ θ< 2π; and the last equality (to |z|eiθ) is taken from Euler's formula. Notice that the argument of z is multi-valued, because the complex exponential function is periodic, with period 2πi. Thus, if θ is one value of arg(z), the other values are given by arg(z) = θ + 2nπ, where n is any integer ≠ 0.

The theory of contour integration comprises a major part of complex analysis. In this context the direction of travel around a closed curve is important – reversing the direction in which the curve is traversed multiplies the value of the integral by −1. By convention the positive direction is counterclockwise. For example, the unit circle is traversed in the positive direction when we start at the point z = 1, then travel up and to the left through the point z = i, then down and to the left through −1, then down and to the right through −i, and finally up and to the right to z = 1, where we started.

Almost all of complex analysis is concerned with complex functions– that is, with functions that map some subset of the complex plane into some other (possibly overlapping, or even identical) subset of the complex plane. Here it is customary to speak of the domain of f(z) as lying in the z-plane, while referring to the range or image of f(z) as a set of points in the w-plane. In symbols we write

z = x + iy;\qquad f(z) = w = u + iv\,

and often think of the function f as a transformation of the z-plane (with coordinates (x, y)) into the w-plane (with coordinates (u, v)).

Stereographic projections

It can be useful to think of the complex plane as if it occupied the surface of a sphere. Given a sphere of unit radius, place it's center at the origin of the complex plane, oriented so that the equator on the sphere coincides with the unit circle in the plane, and the north pole is "above" the plane.

We can establish a one-to-one correspondence between the points on the surface of the sphere minus the north pole and the points in the complex plane as follows. Given a point in the plane, draw a straight line connecting it with the north pole on the sphere. That line will intersect the surface of the sphere in exactly one other point. The point z = 0 will be projected onto the south pole of the sphere. Since the interior of the unit circle lies inside the sphere, that entire region (|z| < 1) will be mapped onto the southern hemisphere. The unit circle itself (|z| = 1) will be mapped


From Yahoo Answers

Question:

Answers:http://math.rice.edu/~pcmi/sphere/drg_txt.html

Question:Hey, I was wondering how you would find the trigonometric values (sin, cos, etc.) of a 505 degree angle? I subtracted 360 from that and got 145, but that still isn't one of the values included in the unit circle diagrams... any help would be appreciated, thanks. no, but it was 145, not 135, which is my problem...

Answers:sin is square root of 2 over 2 cos is negative square root of 2 over 2 tan is -1 for the sin and cos, the square root is just over the top number maybe this will help http://images.google.com/imgres?imgurl=http://dcr.csusb.edu/LearningCenter/LCimages/UnitCircle.gif&imgrefurl=http://dcr.csusb.edu/LearningCenter/tutssl.html&h=575&w=732&sz=52&hl=en&start=1&um=1&tbnid=b6OyxY_TLZ5-KM:&tbnh=111&tbnw=141&prev=/images%3Fq%3Dunit%2Bcircle%26um%3D1%26hl%3Den%26sa%3DN

Question:ok, i dont have a diagram so, im going to explain this the best i can, if you dont undestand i am sorry, ok so i have a unit cirle with, the (phata) angle is 180 degrees, i have to find the cosin , and sin of it. i think they are both 1, is that right and can anyone explain it if im not?

Answers:cos 180 = -1 negative because cos is negative in quadrant II sin 180 = 0

Question:I am completing my LDAF but I'm really stuck on this question - Draw a diagram illustrating the "vicious circle" by which people who have a learning disability are denied oppurtunities to reach their potential. Does anyone know where I can find this diagram or information about it?

Answers:Dont know if this is any good, but it is all I could find http://www.ldaf.org.uk/index.html?LDAF_Session=ec77d934f3d7fc7ee65d86a761bd4a2d http://uk.search.yahoo.com/search?p=LDAF&ei=UTF-8&fr=FP-tab-web-t340&fl=0&x=wrt&meta=vc%3DcountryUK http://www.bldpb.com/blackpool/ldaf/understand_abuse.pdf hope this helps :)

From Youtube

Unit Circle (CAST diagram) trigonometry A level Mathematics.MP4 :

Exact trig angles in degrees unit circle and CAST diagram odd and even functions.MP4 :