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Cartesian coordinate system
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Illustration of a Cartesian coordinate plane. Four points are marked and labeled with their coordinates: (2, 3) in green, (−3, 1) in red, (−1.5, −2.5) in blue, and the origin (0, 0) in purple.
A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis (plural axes) of the system, and the point where they meet is its origin, at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.
One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines). In general, n Cartesian coordinates (an element of real n-space) specify the point in an n-dimensional Euclidean space for any dimension n. These coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes.
Cartesian coordinate system with a circle of radius 2 centered at the origin marked in red. The equation of a circle is (x − a)2 + (y − b)2 = r2 where a and b are the coordinates of the center (a, b) and r is the radius.
The invention of Cartesian coordinates in the 17th century by René Descartes (Latinized name: Cartesius) revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes (such as curves) can be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, may be described as the set of all points whose coordinates x and y satisfy the equation x2 + y2 = 4.
Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory and more. A familiar example is the concept of the graph of a function. Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering and many more. They are the most common coordinate system used in computer graphics, computer-aided geometric design and other geometry-related data processing.
Contents
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1 History
2 Description
2.1 One dimension
2.2 Two dimensions
2.3 Three dimensions
2.4 Higher dimensions
2.5 Generalizations
3 Notations and conventions
3.1 Quadrants and octants
4 Cartesian formulae for the plane
4.1 Distance between two points
4.2 Euclidean transformations
4.2.1 Translation
4.2.2 Rotation
4.2.3 Reflection
4.2.4 Glide reflection
4.2.5 General matrix form of the transformations
4.2.6 Affine transformation
4.2.7 Scaling
4.2.8 Shearing
5 Orientation and handedness
5.1 In two dimensions
5.2 In three dimensions
6 Representing a vector in the standard basis
7 Applications
8 See also
9 References
10 Sources
11 Further reading
12 External links
History[edit]
The adjective Cartesian refers to the French mathematician and philosopher René Descartes who published this idea in 1637. It was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery.[1] The French cleric Nicole Oresme, used constructions similar to Cartesian coordinates well before the time of Descartes and Fermat.[citation needed].
Both Descartes and Fermat used a single axis in their treatments and have a variable length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes' La Géométrie was translated into Latin in 1649 by Frans van Schooten and his students. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes' work.[2]
The development of the Cartesian coordinate system would play a fundamental role in the development of the calculus by Isaac Newton and Gottfried Wilhelm Leibniz.[3] The two-coordinate description of the plane was later generalized into the concept of vector spaces.[4]
Many other coordinate systems have been developed since Descartes, such as the polar coordinates for the plane, and the spherical and cylindrical coordinates for three-dimensional space.
Description[edit]
One dimension [edit]
Main article: number line
Choosing a Cartesian coordinate system for a one-dimensional space – that is, for a straight line—involves choosing a point O of the line (the origin), a unit of length, and an orientation for the line. An orientation chooses which of the two half-lines determined by O is the positive, and which is negative; we then say that the line "is oriented" (or "points") from the negative half towards the positive half. Then each point P of the line can be specified by its distance from O, taken with a + or − sign depending on which half-line contains P.
A line with a chosen Cartesian system is called a number line. Every real number has a unique location on the line. Conversely, every point on the line can be interpreted as a number in an ordered continuum such as the real numbers.
Two dimensions [edit]
Further information: two-dimensional space
A Cartesian coordinate system in two dimensions (also called a rectangular coordinate system or an orthogonal coordinate system[5]) is defined by an ordered pair of perpendicular lines (axes), a single unit of length for both axes, and an orientation for each axis. The point where the axes meet is taken as the origin for both, thus turning each axis into a number line. For any point P, a line is drawn through P perpendicular to each axis, and the position where it meets the axis is interpreted as a number. The two numbers, in that chosen order, are the Cartesian coordinates of P. The reverse construction allows one to determine the point P given its coordinates.
The first and second coordinates are called the abscissa and the ordinate of P, respectively; and the point where the axes meet is called the origin of the coordinate system. The coordinates are usually written as two numbers in parentheses, in that order, separated by a comma, as in (3, -10.5). Thus the origin has coordinates (0,0), and the points on the positive half-axes, one unit away from the origin, have coordinates (1,0) and (0,1).
In mathematics, physics, and engineering, the first axis is usually defined or depicted as horizontal and oriented to the right, and the second axis is vertical and oriented upwards. (However, in some computer graphics contexts, the ordinate axis may be oriented downwards.) The origin is often labeled O, and the two coordinates are often denoted by the letters X and Y, or x and y. The axes may then be referred to as the X-axis and Y-axis. The choices of letters come from the original convention, which is to use the latter part of the alphabet to indicate unknown values. The first part of the alphabet was used to designate known values.
A Euclidean plane with a chosen Cartesian coordinate system Cartesian plane. In a Cartesian plane one can define canonical representatives of certain geometric figures, such as the unit circle (with radius equal to the length unit, and center at the origin), the unit square (whose diagonal has endpoints at (0,0) and (1,1)), the unit hyperbola, and so on.
The two axes divide the plane into four right angles, called quadrant. The quadrants may be named or numbered in various ways, but the quadrant where all coordinates are positive is usually called the first quadrant.
If the coordinates of a point are (x,y), then its distances from the X-axis and from the Y-axis are |y| and |x|. respectively; where |...| denotes the absolute value of a number.
Three dimensions [edit]
Further information: three-dimensional space
A three dimensional Cartesian coordinate system, with origin O and axis lines X, Y and Z, oriented as shown by the arrows. The tick marks on the axes are one length unit apart. The black dot shows the point with coordinates x = 2, y = 3, and z = 4, or (2, 3, 4).
A Cartesian coordinate system for a three-dimensional space consists of an ordered triplet of lines (the axes) that go through a common point (the origin), and are pair-wise perpendicular; an orientation for each axis; and a single unit of length for all three axes. As in the two-dimensional case, each axis becomes a number line. For any point P of space, one considers a plane through P perpendicular to each coordinate axis, and interprets the point where that plane cuts the axis as a number. The Cartesian coordinates of P are those three numbers, in the chosen order The reverse construction determines the point P given its three coordinates.
Alternatively, each coordinate of a point P can be taken as the distance from P to the plane defined by the other two axes, with the sign determined by the orientation of the corresponding axis.
Each pair of axes defines a coordinate plane. These planes divide space into eight trihedra, called octants.
The coordinates are usually written as three numbers (or algebraic formulas) surrounded by parentheses and separated by commas, as in (3,-2.5,1) or (t,u+v,π/2). Thus, the origin has coordinates (0,0,0), and the unit points on the three axes are (1,0,0), (0,1,0), and (0,0,1).
There are no standard names for the coordinates in the three axes. The coordinates are often denoted by the letters X, Y, and Z (or x, y, and z), in which case the lines are called the X-, Y-, and Z-axis, respectively. Then the coordinate planes can be referred to as the XY-, YZ-, and XZ-planes.
In mathematics, physics, and engineering contexts, the first two axes are often defined or depicted as horizontal, with the third axis pointing up. In that case the third coordinate may be called height or altitude. The orientations are usually chosen so that the 90 degree angle from the first axis to the second axis looks counter-clockwise when seen from the point (0,0,1); a convention that is commonly called the right hand rule.
The coordinate surfaces of the Cartesian coordinates (x, y, z). The z-axis is vertical and the x-axis is highlighted in green. Thus, the red plane shows the points with x = 1, the blue plane shows the points with z = 1, and the yellow plane shows the points with y = −1. The three surfaces intersect at the point P (shown as a black sphere) with the Cartesian coordinates (1, −1, 1).
Higher dimensions[edit]
A Euclidean plane with a chosen Cartesian system is called a Cartesian plane. Since Cartesian coordinates are unique and non-ambiguous, the points of a Cartesian plane can be identified with pairs of real numbers; that is with the Cartesian product R 2 = R × R {\displaystyle \mathbb {R} ^{2}=\mathbb {R} \times \mathbb {R} } , where R {\displaystyle \mathbb {R} } is the set of all reals. In the same way, the points in any Euclidean space of dimension n be identified with the tuples (lists) of n real numbers, that is, with the Cartesian product R n {\displaystyle \mathbb {R} ^{n}} .
Generalizations[edit]
The concept of Cartesian coordinates generalizes to allow axes that are not perpendicular to each other, and/or different units along each axis. In that case, each coordinate is obtained by projecting the point onto one axis along a direction that is parallel to the other axis (or, in general, to the hyperplane defined by all the other axes). In such an oblique coordinate system the computations of distances and angles must be modified from that in standard Cartesian systems, and many standard formulas (such as the Pythagorean formula for the distance) do not hold (see affine plane).
Notations and conventions[edit]
The Cartesian coordinates of a point are usually written in parentheses and separated by commas, as in (10, 5) or (3, 5, 7). The origin is often labelled with the capital letter O. In analytic geometry, unknown or generic coordinates are often denoted by the letters (x, y) in the plane, and (x, y, z) in three-dimensional space. This custom comes from a convention of algebra, which uses letters near the end of the alphabet for unknown values (such as were the coordinates of points in many geometric problems), and letters near the beginning for given quantities.
These conventional names are often used in other domains, such as physics and engineering, although other letters may be used. For example, in a graph showing how a pressure varies with time, the graph coordinates may be denoted p and t. Each axis is usually named after the coordinate which is measured along it; so one says the x-axis, the y-axis, the t-axis, etc.
Another common convention for coordinate naming is to use subscripts, as (x1, x2, ..., xn) for the n coordinates in an n-dimensional space, especially when n is greater than 3 or unspecified. Some authors prefer the numbering (x0, x1, ..., xn−1). These notations are especially advantageous in computer programming: by storing the coordinates of a point as an array, instead of a record, the subscript can serve to index the coordinates.
In mathematical illustrations of two-dimensional Cartesian systems, the first coordinate (traditionally called the abscissa) is measured along a horizontal axis, oriented from left to right. The second coordinate (the ordinate) is then measured along a vertical axis, usually oriented from bottom to top. Young children learning the Cartesian system, commonly learn the order to read the values before cementing the x, y, z axis concepts, by starting with 2D mnemonics (e.g. 'Walk along the hall then up the stairs' akin to straight across the x axis then up vertically along the y axis).[6]
Computer graphics and image processing, however, often use a coordinate system with the y-axis oriented downwards on the computer display. This convention developed in the 1960s (or earlier) from the way that images were originally stored in display buffers.
For three-dimensional systems, a convention is to portray the xy-plane horizontally, with the z-axis added to represent height (positive up). Furthermore, there is a convention to orient the x-axis toward the viewer, biased either to the right or left. If a diagram (3D projection or 2D perspective drawing) shows the x- and y-axis horizontally and vertically, respectively, then the z-axis should be shown pointing "out of the page" towards the viewer or camera. In such a 2D diagram of a 3D coordinate system, the z-axis would appear as a line or ray pointing down and to the left or down and to the right, depending on the presumed viewer or camera perspective. In any diagram or display, the orientation of the three axes, as a whole, is arbitrary. However, the orientation of the axes relative to each other should always comply with the right-hand rule, unless specifically stated otherwise. All laws of physics and math assume this right-handedness, which ensures consistency.
For 3D diagrams, the names "abscissa" and "ordinate" are rarely used for x and y, respectively. When they are, the z-coordinate is sometimes called the applicate. The words abscissa, ordinate and applicate are sometimes used to refer to coordinate axes rather than the coordinate values.[5]
Quadrants and octants[edit]
Main articles: octant (solid geometry) and quadrant (plane geometry)
The four quadrants of a Cartesian coordinate system
The axes of a two-dimensional Cartesian system divide the plane into four infinite regions, called quadrants,[5] each bounded by two half-axes. These are often numbered from 1st to 4th and denoted by Roman numerals: I (where the signs of the two coordinates are I (+,+), II (−,+), III (−,−), and IV (+,−). When the axes are drawn according to the mathematical custom, the numbering goes counter-clockwise starting from the upper right ("north-east") quadrant.
Similarly, a three-dimensional Cartesian system defines a division of space into eight regions or octants,[5] according to the signs of the coordinates of the points. The convention used for naming a specific octant is to list its signs, e.g. (+ + +) or (− + −). The generalization of the quadrant and octant to an arbitrary number of dimensions is the orthant, and a similar naming system applies.
Cartesian formulae for the plane[edit]
Distance between two points[edit]
The Euclidean distance between two points of the plane with Cartesian coordinates ( x 1 , y 1 ) {\displaystyle (x_{1},y_{1})} and ( x 2 , y 2 ) {\displaystyle (x_{2},y_{2})} is
d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 . {\displaystyle d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}.}
This is the Cartesian version of Pythagoras's theorem. In three-dimensional space, the distance between points ( x 1 , y 1 , z 1 ) {\displaystyle (x_{1},y_{1},z_{1})} and ( x 2 , y 2 , z 2 ) {\displaystyle (x_{2},y_{2},z_{2})} is
d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 + ( z 2 − z 1 ) 2 , {\displaystyle d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}},}
which can be obtained by two consecutive applications of Pythagoras' theorem.[7]
Euclidean transformations[edit]
The Euclidean transformations or Euclidean motions are the (bijective) mappings of points of the Euclidean plane to themselves which preserve distances between points. There are four types of these mappings (also called isometries): translations, rotations, reflections and glide reflections.[8]
Translation[edit]
Translating a set of points of the plane, preserving the distances and directions between them, is equivalent to adding a fixed pair of numbers (a, b) to the Cartesian coordinates of every point in the set. That is, if the original coordinates of a point are (x, y), after the translation they will be
( x ′ , y ′ ) = ( x + a , y + b ) . {\displaystyle (x',y')=(x+a,y+b).}
Rotation[edit]
To rotate a figure counterclockwise around the origin by some angle θ {\displaystyle \theta } is equivalent to replacing every point with coordinates (x,y) by the point with coordinates (x',y'), where
x ′ = x cos  θ − y sin  θ {\displaystyle x'=x\cos \theta -y\sin \theta }
y ′ = x sin  θ + y cos  θ . {\displaystyle y'=x\sin \theta +y\cos \theta .}
Thus:
( x ′ , y ′ ) = ( ( x cos  θ − y sin  θ ) , ( x sin  θ + y cos  θ ) ) . {\displaystyle (x',y')=((x\cos \theta -y\sin \theta \,),(x\sin \theta +y\cos \theta \,)).}
Reflection[edit]
If (x, y) are the Cartesian coordinates of a point, then (−x, y) are the coordinates of its reflection across the second coordinate axis (the y-axis), as if that line were a mirror. Likewise, (x, −y) are the coordinates of its reflection across the first coordinate axis (the x-axis). In more generality, reflection across a line through the origin making an angle θ {\displaystyle \theta } with the x-axis, is equivalent to replacing every point with coordinates (x, y) by the point with coordinates (x′,y′), where
x ′ = x cos  2 θ + y sin  2 θ {\displaystyle x'=x\cos 2\theta +y\sin 2\theta }
y ′ = x sin  2 θ − y cos  2 θ . {\displaystyle y'=x\sin 2\theta -y\cos 2\theta .}
Thus: ( x ′ , y ′ ) = ( ( x cos  2 θ + y sin  2 θ ) , ( x sin  2 θ − y cos  2 θ ) ) . {\displaystyle (x',y')=((x\cos 2\theta +y\sin 2\theta \,),(x\sin 2\theta -y\cos 2\theta \,)).}
Glide reflection[edit]
A glide reflection is the composition of a reflection across a line followed by a translation in the direction of that line. It can be seen that the order of these operations does not matter (the translation can come first, followed by the reflection).
General matrix form of the transformations[edit]
These Euclidean transformations of the plane can all be described in a uniform way by using matrices. The result ( x ′ , y ′ ) {\displaystyle (x',y')} of applying a Euclidean transformation to a point ( x , y ) {\displaystyle (x,y)} is given by the formula
( x ′ , y ′ ) = ( x , y ) A + b {\displaystyle (x',y')=(x,y)A+b}
where A is a 2×2 orthogonal matrix and b = (b1, b2) is an arbitrary ordered pair of numbers;[9] that is,
x ′ = x A 11 + y A 21 + b 1 {\displaystyle x'=xA_{11}+yA_{21}+b_{1}}
y ′ = x A 12 + y A 22 + b 2 , {\displaystyle y'=xA_{12}+yA_{22}+b_{2},}
where
A = ( A 11 A 12 A 21 A 22 ) . {\displaystyle A={\begin{pmatrix}A_{11}&A_{12}\\A_{21}&A_{22}\end{pmatrix}}.} [Note the use of row vectors for point coordinates and that the matrix is written on the right.]
To be orthogonal, the matrix A must have orthogonal rows with same Euclidean length of one, that is,
A 11 A 21 + A 12 A 22 = 0 {\displaystyle A_{11}A_{21}+A_{12}A_{22}=0}
and
A 11 2 + A 12 2 = A 21 2 + A 22 2 = 1. {\displaystyle A_{11}^{2}+A_{12}^{2}=A_{21}^{2}+A_{22}^{2}=1.}
This is equivalent to saying that A times its transpose must be the identity matrix. If these conditions do not hold, the formula describes a more general affine transformation of the plane provided that the determinant of A is not zero.
The formula defines a translation if and only if A is the identity matrix. The transformation is a rotation around some point if and only if A is a rotation matrix, meaning that
A 11 A 22 − A 21 A 12 = 1. {\displaystyle A_{11}A_{22}-A_{21}A_{12}=1.}
A reflection or glide reflection is obtained when,
A 11 A 22 − A 21 A 12 = − 1. {\displaystyle A_{11}A_{22}-A_{21}A_{12}=-1.}
Assuming that translation is not used transformations can be combined by simply multiplying the associated transformation matrices.
Affine transformation[edit]
Another way to represent coordinate transformations in Cartesian coordinates is through affine transformations. In affine transformations an extra dimension is added and all points are given a value of 1 for this extra dimension. The advantage of doing this is that point translations can be specified in the final column of matrix A. In this way, all of the euclidean transformations become transactable as matrix point multiplications. The affine transformation is given by:
( A 11 A 21 b 1 A 12 A 22 b 2 0 0 1 ) ( x y 1 ) = ( x ′ y ′ 1 ) . {\displaystyle {\begin{pmatrix}A_{11}&A_{21}&b_{1}\\A_{12}&A_{22}&b_{2}\\0&0&1\end{pmatrix}}{\begin{pmatrix}x\\y\\1\end{pmatrix}}={\begin{pmatrix}x'\\y'\\1\end{pmatrix}}.} [Note the matrix A from above was transposed. The matrix is on the left and column vectors for point coordinates are used.]
Using affine transformations multiple different euclidean transformations including translation can be combined by simply multiplying the corresponding matrices.
Scaling[edit]
An example of an affine transformation which is not a Euclidean motion is given by scaling. To make a figure larger or smaller is equivalent to multiplying the Cartesian coordinates of every point by the same positive number m. If (x, y) are the coordinates of a point on the original figure, the corresponding point on the scaled figure has coordinates
( x ′ , y ′ ) = ( m x , m y ) . {\displaystyle (x',y')=(mx,my).}
If m is greater than 1, the figure becomes larger; if m is between 0 and 1, it becomes smaller.
Shearing[edit]
A shearing transformation will push the top of a square sideways to form a parallelogram. Horizontal shearing is defined by:
( x ′ , y ′ ) = ( x + y s , y ) {\displaystyle (x',y')=(x+ys,y)}
Shearing can also be applied vertically:
( x ′ , y ′ ) = ( x , x s + y ) {\displaystyle (x',y')=(x,xs+y)}
Orientation and handedness[edit]
Main article: orientation (mathematics)
See also: right-hand rule and axes conventions
In two dimensions[edit]
The right-hand rule
Fixing or choosing the x-axis determines the y-axis up to direction. Namely, the y-axis is necessarily the perpendicular to the x-axis through the point marked 0 on the x-axis. But there is a choice of which of the two half lines on the perpendicular to designate as positive and which as negative. Each of these two choices determines a different orientation (also called handedness) of the Cartesian plane.
The usual way of orienting the axes, with the positive x-axis pointing right and the positive y-axis pointing up (and the x-axis being the "first" and the y-axis the "second" axis) is considered the positive or standard orientation, also called the right-handed orientation.
A commonly used mnemonic for defining the positive orientation is the right-hand rule. Placing a somewhat closed right hand on the plane with the thumb pointing up, the fingers point from the x-axis to the y-axis, in a positively oriented coordinate system.
The other way of orienting the axes is following the left hand rule, placing the left hand on the plane with the thumb pointing up.
When pointing the thumb away from the origin along an axis towards positive, the curvature of the fingers indicates a positive rotation along that axis.
Regardless of the rule used to orient the axes, rotating the coordinate system will preserve the orientation. Switching any two axes will reverse the orientation, but switching both will leave the orientation unchanged.
In three dimensions[edit]
Fig. 7 – The left-handed orientation is shown on the left, and the right-handed on the right.
Fig. 8 – The right-handed Cartesian coordinate system indicating the coordinate planes.
Once the x- and y-axes are specified, they determine the line along which the z-axis should lie, but there are two possible directions on this line. The two possible coordinate systems which result are called 'right-handed' and 'left-handed'. The standard orientation, where the xy-plane is horizontal and the z-axis points up (and the x- and the y-axis form a positively oriented two-dimensional coordinate system in the xy-plane if observed from above the xy-plane) is called right-handed or positive.
3D Cartesian coordinate handedness
The name derives from the right-hand rule. If the index finger of the right hand is pointed forward, the middle finger bent inward at a right angle to it, and the thumb placed at a right angle to both, the three fingers indicate the relative directions of the x-, y-, and z-axes in a right-handed system. The thumb indicates the x-axis, the index finger the y-axis and the middle finger the z-axis. Conversely, if the same is done with the left hand, a left-handed system results.
Figure 7 depicts a left and a right-handed coordinate system. Because a three-dimensional object is represented on the two-dimensional screen, distortion and ambiguity result. The axis pointing downward (and to the right) is also meant to point towards the observer, whereas the "middle"-axis is meant to point away from the observer. The red circle is parallel to the horizontal xy-plane and indicates rotation from the x-axis to the y-axis (in both cases). Hence the red arrow passes in front of the z-axis.
Figure 8 is another attempt at depicting a right-handed coordinate system. Again, there is an ambiguity caused by projecting the three-dimensional coordinate system into the plane. Many observers see Figure 8 as "flipping in and out" between a convex cube and a concave "corner". This corresponds to the two possible orientations of the coordinate system. Seeing the figure as convex gives a left-handed coordinate system. Thus the "correct" way to view Figure 8 is to imagine the x-axis as pointing towards the observer and thus seeing a concave corner.
Representing a vector in the standard basis[edit]
A point in space in a Cartesian coordinate system may also be represented by a position vector, which can be thought of as an arrow pointing from the origin of the coordinate system to the point.[10] If the coordinates represent spatial positions (displacements), it is common to represent the vector from the origin to the point of interest as r {\displaystyle \mathbf {r} } . In two dimensions, the vector from the origin to the point with Cartesian coordinates (x, y) can be written as:
r = x i + y j {\displaystyle \mathbf {r} =x\mathbf {i} +y\mathbf {j} }
where i = ( 1 0 ) {\displaystyle \mathbf {i} ={\begin{pmatrix}1\\0\end{pmatrix}}} , and j = ( 0 1 ) {\displaystyle \mathbf {j} ={\begin{pmatrix}0\\1\end{pmatrix}}} are unit vectors in the direction of the x-axis and y-axis respectively, generally referred to as the standard basis (in some application areas these may also be referred to as versors). Similarly, in three dimensions, the vector from the origin to the point with Cartesian coordinates ( x , y , z ) {\displaystyle (x,y,z)} can be written as:[11]
r = x i + y j + z k {\displaystyle \mathbf {r} =x\mathbf {i} +y\mathbf {j} +z\mathbf {k} }
where k = ( 0 0 1 ) {\displaystyle \mathbf {k} ={\begin{pmatrix}0\\0\\1\end{pmatrix}}} is the unit vector in the direction of the z-axis.
There is no natural interpretation of multiplying vectors to obtain another vector that works in all dimensions, however there is a way to use complex numbers to provide such a multiplication. In a two dimensional cartesian plane, identify the point with coordinates (x, y) with the complex number z = x + iy. Here, i is the imaginary unit and is identified with the point with coordinates (0, 1), so it is not the unit vector in the direction of the x-axis. Since the complex numbers can be multiplied giving another complex number, this identification provides a means to "multiply" vectors. In a three dimensional cartesian space a similar identification can be made with a subset of the quaternions.
Applications[edit]
Cartesian coordinates are an abstraction that have a multitude of possible applications in the real world. However, three constructive steps are involved in superimposing coordinates on a problem application. 1) Units of distance must be decided defining the spatial size represented by the numbers used as coordinates. 2) An origin must be assigned to a specific spatial location or landmark, and 3) the orientation of the axes must be defined using available directional cues for all but one axis.
Consider as an example superimposing 3D Cartesian coordinates over all points on the Earth (i.e. geospatial 3D). What units make sense? Kilometers are a good choice, since the original definition of the kilometer was geospatial...10,000 km equalling the surface distance from the Equator to the North Pole. Where to place the origin? Based on symmetry, the gravitational center of the Earth suggests a natural landmark (which can be sensed via satellite orbits). Finally, how to orient X-, Y- and Z-axis directions? The axis of Earth's spin provides a natural direction strongly associated with "up vs. down", so positive Z can adopt the direction from geocenter to North Pole. A location on the Equator is needed to define the X-axis, and the prime meridian stands out as a reference direction, so the X-axis takes the direction from geocenter out to [ 0 degrees longitude, 0 degrees latitude ]. Note that with 3 dimensions, and two perpendicular axes directions pinned down for X and Z, the Y-axis is determined by the first two choices. In order to obey the right-hand rule, the Y-axis must point out from the geocenter to [ 90 degrees longitude, 0 degrees latitude ]. So what are the geocentric coordinates of the Empire State Building in New York City? Using [ longitude = −73.985656, latitude = 40.748433 ], Earth radius = 40,000/2π, and transforming from spherical --> Cartesian coordinates, you can estimate the geocentric coordinates of the Empire State Building, [ x, y, z ] = [ 1330.53 km, –4635.75 km, 4155.46 km ]. GPS navigation relies on such geocentric coordinates.
In engineering projects, agreement on the definition of coordinates is a crucial foundation. One cannot assume that coordinates come predefined for a novel application, so knowledge of how to erect a coordinate system where there is none is essential to applying René Descartes' ingenious thinking.
While spatial apps employ identical units along all axes, in business and scientific apps, each axis may have different units of measurement associated with it (such as kilograms, seconds, pounds, etc.). Although four- and higher-dimensional spaces are difficult to visualize, the algebra of Cartesian coordinates can be extended relatively easily to four or more variables, so that certain calculations involving many variables can be done. (This sort of algebraic extension is what is used to define the geometry of higher-dimensional spaces.) Conversely, it is often helpful to use the geometry of Cartesian coordinates in two or three dimensions to visualize algebraic relationships between two or three of many non-spatial variables.
The graph of a function or relation is the set of all points satisfying that function or relation. For a function of one variable, f, the set of all points (x, y), where y = f(x) is the graph of the function f. For a function g of two variables, the set of all points (x, y, z), where z = g(x, y) is the graph of the function g. A sketch of the graph of such a function or relation would consist of all the salient parts of the function or relation which would include its relative extrema, its concavity and points of inflection, any points of discontinuity and its end behavior. All of these terms are more fully defined in calculus. Such graphs are useful in calculus to understand the nature and behavior of a function or relation.
See also[edit]
Horizontal and vertical
Jones diagram, which plots four variables rather than two
Orthogonal coordinates
Polar coordinate system
Spherical coordinate system
References[edit]
Jump up ^ Bix, Robert A.; D'Souza, Harry J. "Analytic geometry". Encyclopædia Britannica. Retrieved 2017-08-06.
Jump up ^ Burton 2011, p. 374
Jump up ^ A Tour of the Calculus, David Berlinski
Jump up ^ Axler, Sheldon. Linear Algebra Done Right - Springer. p. 1. doi:10.1007/978-3-319-11080-6.
^ Jump up to: a b c d "Cartesian orthogonal coordinate system". Encyclopedia of Mathematics. Retrieved 2017-08-06.
Jump up ^ "Charts and Graphs: Choosing the Right Format". www.mindtools.com. Retrieved 2017-08-29.
Jump up ^ Hughes-Hallett, Deborah; McCallum, William G.; Gleason, Andrew M. (2013). Calculus : Single and Multivariable (6 ed.). John wiley. ISBN 978-0470-88861-2.
Jump up ^ Smart 1998, Chap. 2
Jump up ^ Brannan, Esplen & Gray 1998, pg. 49
Jump up ^ Brannan, Esplen & Gray 1998, Appendix 2, pp. 377–382
Jump up ^ David J. Griffiths (1999). Introduction to Electrodynamics. Prentice Hall. ISBN 0-13-805326-X.
Sources[edit]
Brannan, David A.; Esplen, Matthew F.; Gray, Jeremy J. (1998), Geometry, Cambridge: Cambridge University Press, ISBN 0-521-59787-0
Burton, David M. (2011), The History of Mathematics/An Introduction (7th ed.), New York: McGraw-Hill, ISBN 978-0-07-338315-6
Smart, James R. (1998), Modern Geometries (5th ed.), Pacific Grove: Brooks/Cole, ISBN 0-534-35188-3
Further reading[edit]
Descartes, René (2001). Discourse on Method, Optics, Geometry, and Meteorology. Translated by Paul J. Oscamp (Revised ed.). Indianapolis, IN: Hackett Publishing. ISBN 0-87220-567-3. OCLC 488633510.
Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers (1st ed.). New York: McGraw-Hill. pp. 55–79. LCCN 59-14456. OCLC 19959906.
Margenau H, Murphy GM (1956). The Mathematics of Physics and Chemistry. New York: D. van Nostrand. LCCN 55-10911.
Moon P, Spencer DE (1988). "Rectangular Coordinates (x, y, z)". Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions (corrected 2nd, 3rd print ed.). New York: Springer-Verlag. pp. 9–11 (Table 1.01). ISBN 978-0-387-18430-2.
Morse PM, Feshbach H (1953). Methods of Theoretical Physics, Part I. New York: McGraw-Hill. ISBN 0-07-043316-X. LCCN 52-11515.
Sauer R, Szabó I (1967). Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag. LCCN 67-25285.
External links[edit]
Cartesian Coordinate System
"Cartesian coordinates". PlanetMath.
MathWorld description of Cartesian coordinates
Coordinate Converter – converts between polar, Cartesian and spherical coordinates
Coordinates of a point Interactive tool to explore coordinates of a point
open source JavaScript class for 2D/3D Cartesian coordinate system manipulation
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text:
The Indian War of Independence (book)
From Wikipedia, the free encyclopedia
Jump to: navigation, search
For the main article on the 1857 revolt, see Indian Rebellion of 1857.
The Indian War of Independence
Author
Vinayak Damodar Savarkar
Country
India
Language
Marathi, English
Genre
History (Nationalist)
Publisher
Sethani Kampani, Bombay (reprint, India)
Publication date
1909, 1949 (reprint, India)
Published in English
1909
The Indian War of Independence is an Indian nationalist history of the 1857 revolt by Vinayak Damodar Savarkar that was first published in 1909.[1][2] The book, initially written in Marathi, was penned by Savarkar in response to celebrations in Britain of the 50th anniversary of the 1857 Indian uprising with records from India Office archives and the whole project received support from Indian nationalists in Britain including the likes of Madame Cama, V.V.S. Iyer and M.P.T. Acharya, as well as Indian students who had dared not show their support or sympathy for India House openly.[3] Published during Savarkar's stay in London at the India House, the book was influenced by histories of the French Revolution and the American Revolution, as much as it sought to bring the Indian movement to public attention in Britain as well as to inspire nationalist revolution in India.[4]
The book, which describes the 1857 revolt as a unified and national uprising of India as a nation against British authority,[5] was seen at the time as highly inflammatory, and the Marathi edition was banned in British India even before its publication.[6] Publication of the English translation faltered after British printers and publishing houses were warned by the Home Office of its highly seditious content, while the British foreign office brought pressure on the French Government to prevent its publication from Paris.[6] It was ultimately printed in the Netherlands in 1909, with the British government not tracing it until too late.[3][6][7] The copies were printed with false dust wrappers purporting to be copies of The Pickwick Papers and other literary classics, and large quantities were shipped to India where it quickly became a bible of political extremists.[6] It was excluded from the catalogue of the British Library to prevent Indian students from accessing it. In India, the book remained banned till the end of The Raj forty years later.[6]
The Indian War of Independence is considered to be an influential work in Indian history and nationalist writing,[8] and also one of Savarkar's most influential works in developing and framing ideas of masculine Hinduism.[9] While some erstwhile and modern histories draw similar conclusions as the Savarkar,[10] others, including R.C. Majumdar, disagreed with Savarkar's conclusions in his book on the national and unified character of the mutiny.[2][10][11]
References[edit]
Jump up ^ Savarkar, Vinayak Damodar (10 May 1909). The Indian War of Independence of 1857. London. Retrieved 9 November 2017.
^ Jump up to: a b Vohra 2000, p. 70
^ Jump up to: a b Yadav 1992, p. 14
Jump up ^ "V.D. Savarkar and The Indian War of Independence, 1857". University of California, Irvine. Retrieved 2008-06-20.
Jump up ^ Misra 2004, p. 184
^ Jump up to: a b c d e Hopkirk 2001, p. 45
Jump up ^ "Mutiny at the Margins". University of Edinburgh. Retrieved 2008-06-20.
Jump up ^ Dirks 2001, p. 127
Jump up ^ Bannerjee 2005, p. 50
^ Jump up to: a b Hasan 1998, p. 149
Jump up ^ Nanda 1965, p. 701
Further reading[edit]
Bannerjee, Sikata (2005), Make Me a Man!: Masculinity, Hinduism, and Nationalism in India, SUNY press, ISBN 0-7914-6367-2 .
Dirks, Nicholas B (2001), Castes of Mind: Colonialism and the Making of Modern India, Princeton University Press, ISBN 0-691-08895-0 .
Hasan, Farhad (1998), Social Scientist, Vol. 26, No. 1/4 (Jan. - Apr., 1998), pp. 148-151 .
Hopkirk, Peter (2001), On Secret Service East of Constantinople, Oxford Paperbacks, ISBN 0-19-280230-5 .
Misra, Amalendu (2004), Identity and Religion: Foundations of Anti-Islamism in India, SAGE, ISBN 0-7619-3226-7 .
Nanda, Krishan (1965), The Western Political Quarterly, Vol. 18, No. 3 (Sep., 1965), pp. 700-701, University of Utah on behalf of the Western Political Science Association .
Vohra, Ranbir (2000), The Making of India: A Historical Survey, M.E. Sharpe, ISBN 0-7656-0712-3 .
Yadav, B.D (1992), M.P.T. Acharya, Reminiscences of an Indian Revolutionary, Anmol Publications Pvt ltd, ISBN 81-7041-470-9 .
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Facial symmetry
From Wikipedia, the free encyclopedia
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This article is about the physiognomical concept. For other uses, see Symmetry (disambiguation).
While symmetrical faces (leftmost image) are perceived to be attractive, completely symmetric faces (a and b) are disconcerting and are not perceived as normal.
Facial symmetry is one specific measure of bodily asymmetry. Along with traits such as averageness and youthfulness it influences judgments of aesthetic traits of physical attractiveness and beauty.[1] For instance, in mate selection, people have been shown to have a preference of symmetry.[2][3] This is because it is seen an indicator of health and genetic fitness,[3] but also as holding adaptation qualities; reflecting the ability to withstand the changes in their environments.[4]
Facial symmetry has been suggested as a possible physical manifestation of the 'big-five' personality traits.[5] For example, it is found that extraversion and openness are strongly associated with the symmetry of the face.[5] Hormones such as testosterone and estrogen are believed to be associated with developmental processes and growth of facial features during puberty and as a result are hypothesized to be the cause for individual differences in the implications associated with facial symmetry.[6]
Facial bilateral symmetry is measured via fluctuating asymmetry of the face comparing random differences in facial features of the two sides of the face that develop and accumulate throughout one's lifetime as a result of stressors.[7][8]
Contents
1 Facial symmetry
2 Relationship to personality
3 Facial averageness vs. symmetry
4 See also
5 References
6 Further reading
7 External links
Facial symmetry[edit]
Facial symmetry has been shown to have an effect of ratings of attractiveness in human faces.[1][3] More symmetrical faces are perceived as more attractive in both males and females, although facial symmetry plays a larger role in judgments of attractiveness concerning female faces.[9] A wide variety of methods have been used to examine the claim that facial symmetry plays a role in judgments of beauty.
Blending of multiple faces to create a composite[1] and face-half mirroring[10] have been among the techniques used. While studies employing the first method produced results that indicate that more symmetrical faces are perceived as more attractive, studies applying the later method have indicated that humans prefer slight asymmetry. Also studies show that nearly symmetrical faces are considered highly attractive compared to unsymmetrical ones, yet more specifically, people tend to find a face unattractive if a person has an unsymmetrical nose, and effect of unsymmetrical lips do not really affect people on judging attractiveness.
The most conspicuous directional asymmetries are sometimes only temporary.[11] For example, during speech, most people (76%) tend to express greater amplitude of movement on the right side of their mouth. This is most likely caused by the uneven strengths of contralateral neural connections between the left hemisphere of the brain (linguistic localization) and the right side of the face.[12]
Conclusions derived from face mirroring, however, have been called into question, because it has been shown that mirroring face-halves creates artificial features. For example, if the nose of an individual is slightly bent to the right side, then mirroring the right side of the face will lead to an over-sized nose, while mirroring the left side will lead to an unnaturally small nose.[12]
Evolutionary theorists in biology and psychology argue that more symmetric faces are preferred because symmetry is a possible honest sign of superior genetic quality and developmental stability.[1][13] However it is possible that high facial symmetry in an individual is not due to their superior genetics but due to a lack of exposure to stressors during development. This potentially confounding source of facial symmetry is not explored in the literature. The notion that facial symmetry is viewed by humans as an indicator of 'good genes' is supported by experiments demonstrating that more symmetrical faces are rated as healthier than less symmetrical faces.[3][9] For example, evidence has found that ratings of a person’s facial symmetry is positively associated with ratings of healthiness of their facial skin.[14] Additionally, asymmetry can be fluctuated by increasing health challenges during growth. As a result, phenotypic quality and how well an individual's genome can fight disease and uphold normal development can be indicated by symmetry.[12] Furthermore, studies suggest that facial fluctuation asymmetry, non symmetrical faces, may also be associated with an individuals physiological, psychological and emotional distress.[15]
Additionally current studies indicate that face preferences in adults can be predicted by the number of infections in childhood. People who had more infections as children prefer more sex-typical and symmetric faces because those faces indicate being healthy. Especially recent infections trigger this preference.[16]
Original
Facial symmetry is also a valid marker of cognitive aging.[8] Progressive changes occurring throughout life in the soft tissues of the face will cause more prominent facial asymmetry in older faces.[12] Therefore, symmetrical transformation of older faces generally increases their attractiveness while symmetrical transformation in young adults and children will decrease their attractiveness.[12] Males with more symmetric faces in old age have higher intelligence and are more efficient at information processing than males with less symmetric faces. This is possibly due to the better genes leading to more resistance against stressors in life which is then reflected in less accumulated fluctuating asymmetry.[8]
Non-scientific theories of attraction and symmetry abound. Symmetry of each face is evaluated by human brain via the facial recognition process. This fast and effective detection is difficult with computer programs.[17][18] For instance, the faces that were considered to possess a strong aesthetic appeal were the ones where the conjunction of similar or corresponding facial features were, to some extent, contrasting.[19] Likewise shared features echoism, proportions and similarity to people the person bonded to early in life (prima copulism) have been suggested as relevant.[20]
Relationship to personality[edit]
Research indicates that facial symmetry is linked to the 'big-five' model of personality. The most consistent finding is that facial symmetry is positively correlated with extraversion, indicating that individuals with more symmetric faces are also more extroverted.[5][21][22] More symmetrical faces are also judged to be lower on neuroticism but higher on conscientiousness and agreeableness.[23] More symmetrical faces are also more likely to have more desirable social attributes assigned to them, such as sociable, intelligent or lively.[22]
However, the relationship of facial symmetry and the 'big-five' personality model remains somewhat unclear with regard to neuroticism, openness, agreeableness and conscientiousness. While the previously described studies lead to the expectations that more symmetrical faces will be higher on extraversion, openness, agreeableness and conscientiousness, and lower on neuroticism, these expectations were not confirmed empirically. Openness and agreeableness appear to be significantly negatively related to facial symmetry, while neuroticism and conscientiousness do not seem to be linked to facial symmetry. The relationship of facial symmetry and the 'big-five' personality model requires further examination in order to allow for complete conclusions to be drawn.[5]
Recent study shows the association of trustworthiness and with symmetry on the face. People are biologically evolved to detect the deceptive intention of other people by observing the asymmetry in the face. When people are lying, their facial muscles become imbalanced.[24]
Facial averageness vs. symmetry[edit]
According to the theory of sexual selection, facial symmetry plays a large role in what we perceive as attractiveness.[25] However there is question whether changes in either symmetry or averageness alone increase facial attractiveness. Experiments show that symmetry and averageness make independent contributions to attractiveness, but when the contribution of symmetry is excluded, averageness remains a significant predictor of attractiveness.[26] In addition, research has shown that when one couples facial symmetry with averageness, attractiveness ratings rise, suggesting these two influences are invariably linked.[27]
See also[edit]
Beauty
Symmetry in nature
Patterns in nature
Physical attractiveness
References[edit]
^ Jump up to: a b c d Grammer, K., & Thornhill, R. (1994). Human (Homo sapiens) facial attractiveness and sexual selection: the role of symmetry and averageness. Journal of comparative psychology (Washington, D.C. : 1983), 108(3), 233–42.
Jump up ^ Rhodes, Gillian; Zebrowitz, Leslie A. (2002). Facial Attractiveness: Evolutionary, Cognitive, and Social Perspectives. Ablex. ISBN 1-56750-636-4.
^ Jump up to: a b c d Jones, B. C., Little, A. C., Tiddeman, B. P., Burt, D. M., & Perrett, D. I. (2001). Facial symmetry and judgements of apparent health Support for a “‘ good genes ’” explanation of the attractiveness – symmetry relationship, 22, 417–429.
Jump up ^ Fink, B., Neave, N., Manning, J. T., & Grammer, K. (2006). Facial symmetry and judgements of attractiveness, health and personality. Personality and Individual Differences, 41(3), 491-499.
^ Jump up to: a b c d Fink, B., Neave, N., Manning, J. T., & Grammer, K. (2005). Facial symmetry and the “big-five” personality factors. Personality and Individual Differences, 39(3), 523–529.
Jump up ^ Gangestad, S., & Thornhill, R. (2003). Facial masculinity and fluctuating asymmetry. Evolution and Human Behavior, 24(4), 231–241.
Jump up ^ Penton-Voak, I. S., Jones, B. C., Little, a C., Baker, S., Tiddeman, B., Burt, D. M., & Perrett, D. I. (2001). Symmetry, sexual dimorphism in facial proportions and male facial attractiveness. Proceedings: Biological Sciences, 268(1476), 1617–23.
^ Jump up to: a b c Penke, L., Bates, T. C., Gow, A. J., Pattie, A., Starr, J. M., Jones, B. C., Perrett, D. I., et al. (2009). Symmetric faces are a sign of successful cognitive aging. Evolution and Human Behavior, 30(6), 429–437.
^ Jump up to: a b Rhodes, G., Proffitt, F., Grady, J. M., & Sumich, A. (1998). "Facial symmetry and the perception of beauty." Psychonomic Bulletin & Review, 5(4), 659–669.
Jump up ^ Kowner, R. (1996). "Facial asymmetry and attractiveness judgment in developmental perspective." Journal of Experimental Psychology. Human perception and performance, 22(3), 662–75.
Jump up ^ "Symmetric Shape Deformation Considering Facial Features and Attractiveness Improvement" by Kim Jung-Sik, Shin Il-Kyu, Soo-Mi Choi, Dept. of Computer Science and Engineering, Vol. 16 No. 2 pp. 29–37, Sejong University
^ Jump up to: a b c d e Perrett, D. I., Burt, D. M., Penton-Voak, I. S., Lee, K. J., Rowland, D. a, & Edwards, R. (1999). "Symmetry and Human Facial Attractiveness." Evolution and Human Behavior, 20(5), 295–307.
Jump up ^ Scheib, J. E., Gangestad, S. W., & Thornhill, R. (1999). "Facial attractiveness, symmetry and cues of good genes." Proceedings: Biological Sciences / The Royal Society, 266(1431), 1913–17.
Jump up ^ Jones, B.C.; Little, A.C.; Feinberg, D.R.; Penton-Voak, I.S.; Tiddeman, B.P.; Perrett, D.I. "The relationship between shape symmetry and perceived skin condition in male facial attractiveness". Evolution and Human Behavior. 25 (1): 24–30. doi:10.1016/s1090-5138(03)00080-1.
Jump up ^ Shackelford, T. K., & Larsen, R. J. (1997). "Facial asymmetry as an indicator of psychological, emotional, and physiological distress." Journal of Personality and Social Psychology, 72(2), 456–466.
Jump up ^ de Barra, Mícheál; DeBruine, Lisa M.; Jones, Benedict C.; Mahmud, Zahid Hayat; Curtis, Valerie A. (November 2013). "Illness in childhood predicts face preferences in adulthood". Evolution and Human Behavior. 34 (6): 384–389. doi:10.1016/j.evolhumbehav.2013.07.001.
Jump up ^ Chunyu L, Changshui Z, Fang W, Pingfan Y. "Principle [sic] component analysis based symmetry detection." Acta Electronica Sinica 1999;27(5):25–8.
Jump up ^ Enquist M, Arak A. "Symmetry, beauty and evolution." Nature 2002;372: 169–72.
Jump up ^ Witherspoon, Gary (1977). Language and Art in the Navajo Universe. Ann Arbor: University of Michigan Press. p. 198. ISBN 0-472-08966-8.
Jump up ^ Malin, Suzi (2004). Love at First Sight. DK Adult. ISBN 0-7566-0401-X.
Jump up ^ Pound, N., Penton-Voak, I. S., & Brown, W. M. (2007). "Facial symmetry is positively associated with self-reported extraversion." Personality and Individual Differences, 43(6), 1572–1582.
^ Jump up to: a b Fink, B., Neave, N., Manning, J. T., & Grammer, K. (2006). "Facial symmetry and judgements of attractiveness, health and personality." Personality and Individual Differences, 41(3), 491–499.
Jump up ^ Noor, F., & Evans, D. C. (2003). "The effect of facial symmetry on perceptions of personality and attractiveness." Journal of Research in Personality, 37(4), 339–347.
Jump up ^ D W, Bava S, Reis V A. 2003. "Relationship between facial asymmetry and judging trustworthiness in faces". University of California.8(3), 225–232
Jump up ^ https://www.sciencedaily.com/articles/f/facial_symmetry.htm
Jump up ^ Byatt, G., Rhodes, G., Sumich, A. Psychological Science (January 1999), 10 (1), pg. 52-58
Jump up ^ http://psycnet.apa.org/journals/com/108/3/233/ Grammer. K., & Thornhill. R. (1994). Human (homo sapiens) facial attractiveness and sexual selection: the role of symmetry and averageness. Journal of comparative psychology, 108(3), 233.
Fink, B.; Manning, J. T.; Neave, N.; Grammer, K. (2004). "Second to fourth digit ratio and facial asymmetry". Evolution and Human Behavior. 25 (2): 125–132. doi:10.1016/S1090-5138(03)00084-9.
Oral diagnosis 402 by Dr.Gihane Gharib Madkour, Lecturer of Oral Medicine, Oral Diagnosis & Periodontology, Cairo University
Symmetric Shape Deformation Considering Facial Features and Attractiveness Improvement by Kim Jung-Sik, Shin Il-Kyu, Soo-Mi Choi, Dept. of Computer Science and Engineering, Vol.16 No.2 P.29~37 Sejong University
Chunyu L, Changshui Z, Fang W, Pingfan Y. Principle component analysis based symmetry detection. Acta Electronica Sinica 1999;27(5):25–8.
Enquist M, Arak A. Symmetry, beauty and evolution. Nature 2002;372: 169–72.
Further reading[edit]
Fink, B.; Grammer, K.; Thornhill, R. (2001). "Human (Homo sapiens) facial attractiveness in relation to skin texture and color". Journal of Comparative Psychology. 115 (1): 92–99. doi:10.1037/0735-7036.115.1.92. PMID 11334223.
Johnston, V.; Hagel, R.; Franklin, M.; Fink, B.; Grammer, K. (2001). "Male facial attractiveness: evidence for hormone-mediated adaptive design". Evolution and Human Behavior. 22 (4): 251. doi:10.1016/S1090-5138(01)00066-6.
External links[edit]
"Teeth, Beauty, Biology, and Health" – a class handout on dentition and facial symmetry at the College of Dentistry – University of Illinois at Chicago.
"Symmetry measurement tool" – Symmetry measurement tool
"Chimeric Face Experiment" – Perform a facial symmetry and attractiveness experiment by uploading a photo
FaceResearch – Online studies on facial symmetry by researchers affiliated with University of Aberdeen (Scotland) School of Psychology, and University of St. Andrews (Scotland).
"A facial symmetry plugin for the GIMP"—Try experimenting with facial symmetry, using open source software.
"[1]" – perfect symmetry
Retrieved from "https://en.wikipedia.org/w/index.php?title=Facial_symmetry&oldid=831362246"
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