Cylindrical coordinates are a generalization of two-dimensional polar coordinates to three dimensions by superposing a height () axis. Unfortunately, there are a number of different notations used for the other two coordinates. Either or is used to refer to the radial coordinate and either or to the azimuthal coordinates. Arfken (1985), for instance, uses , while Beyer (1987) uses . In this work, the notation is used.The following table summarizes notational conventions used by a number of authors.(radial, azimuthal, vertical)referencethis work, Beyer (1987, p. 212)(Rr, Ttheta, Zz)SetCoordinates[Cylindrical] in the Wolfram Language package VectorAnalysis` Arfken (1985, p. 95)Moon and Spencer (1988, p. 12)Korn and Korn (1968, p. 60)Morse and Feshbach (1953)In terms of the Cartesian coordinates ,(1)(2)(3)where , , , and the inverse tangent must be suitably defined to take the correct quadrant of into..
The polar coordinates (the radial coordinate) and (the angular coordinate, often called the polar angle) are defined in terms of Cartesian coordinates by(1)(2)where is the radial distance from the origin, and is the counterclockwise angle from the x-axis. In terms of and ,(3)(4)(Here, should be interpreted as the two-argument inverse tangent which takes the signs of and into account to determine in which quadrant lies.) It follows immediately that polar coordinates aren't inherently unique; in particular, will be precisely the same polar point as for any integer . What's more, one often allows negative values of under the assumption that is plotted identically to .The expression of a point as an ordered pair is known as polar notation, the equation of a curve expressed in polar coordinates is known as a polar equation, and a plot of a curve in polar coordinates is known as a polar plot.In much the same way that Cartesian curves can be plotted on..
A coordinate system composed of intersecting surfaces. If the intersections are all at right angles, then the curvilinear coordinates are said to form an orthogonal coordinate system. If not, they form a skew coordinate system.A general metric has a line element(1)where Einstein summation is being used. Orthogonal coordinates are defined as those with a diagonal metric so that(2)where is the Kronecker delta and is a so-called scale factor. Orthogonal curvilinear coordinates therefore have a simple line element(3)(4)which is just the Pythagorean theorem, sothe differential vector is(5)or(6)where the scale factors are(7)and(8)(9)Equation (◇) may therefore be re-expressed as(10)
A system for specifying points using coordinates measured in some specified way. The simplest coordinate system consists of coordinate axes oriented perpendicularly to each other, known as Cartesian coordinates. Depending on the type of problem under consideration, coordinate systems possessing special properties may allow particularly simple solution.In three dimensions, so-called right-handed coordinate systems (left figure) are usually chosen by convention, although left-handed coordinate systems (right figure) are also encountered occasionally.
Parametric equations are a set of equations that express a set of quantities as explicit functions of a number of independent variables, known as "parameters." For example, while the equation of a circle in Cartesian coordinates can be given by , one set of parametric equations for the circle are given by(1)(2)illustrated above. Note that parametric representations are generally nonunique, so the same quantities may be expressed by a number of different parameterizations. A single parameter is usually represented with the parameter , while the symbols and are commonly used for parametric equations in two parameters.Parametric equations provide a convenient way to represent curves and surfaces, as implemented, for example, in the Wolfram Language commands ParametricPlot[x, y, t, t1, t2] and ParametricPlot3D[x, y, z, u, u1, u2, v, v1, v2]. Unsurprisingly, curves and surfaces obtained by way of parametric equation representations..
There are several different definitions of conical coordinates defined by Morse and Feshbach (1953), Byerly (1959), Arfken (1970), and Moon and Spencer (1988). The system defined in the Wolfram Language is(1)(2)(3)where . Byerly (1959) uses a system which is essentially the same coordinate system as above, but replacing with , with , and with . Moon and Spencer (1988) use instead of .The above equations give(4)(5)(6)The scale factors are(7)(8)(9)The Laplacian is(10)The Helmholtz differential equationis separable in conical coordinates.
(1)(2)(3)where , , and .(4)(5)(6)The scale factors are(7)(8)(9)The Laplacian is(10)The Helmholtz differential equationis separable.
Given a reference triangle and a point , the triple , with , and representing the distances from to the vertices of the reference triangle, is the tripolar coordinates of .The tripolar coordinates satisfy(Euler 1786).Given , the number of points having tripolar coordinates satisfying depends on , and being the sides of a triangle (two points), a degenerate triangle (one point) or not a triangle (zero points) (Bottema 1987)The following table summarizes the tripolar coordinated for a number of named centers.centertripolar coordinatesincenter triangle centroid circumcenter orthocenter symmedian point
A system of curvilinear coordinates. There are several different conventions for the orientation and designation of these coordinates. Arfken (1970) defines coordinates such that(1)(2)(3)In this work, following Morse and Feshbach (1953), the coordinates are used instead. In this convention, the traces of the coordinate surfaces of the -plane are confocal parabolas with a common axis. The curves open into the negative x-axis; the curves open into the positive x-axis. The and curves intersect along the y-axis.(4)(5)(6)where , , and . The scale factors are(7)(8)(9)Laplace's equation is(10)The Helmholtz differential equationis separable in parabolic cylindrical coordinates.
The confocal ellipsoidal coordinates, called simply "ellipsoidal coordinates" by Morse and Feshbach (1953) and "elliptic coordinates" by Hilbert and Cohn-Vossen (1999, p. 22), are given by the equations(1)(2)(3)where , , and . These coordinates correspond to three confocal quadrics all sharing the same pair of foci. Surfaces of constant are confocal ellipsoids, surfaces of constant are one-sheeted hyperboloids, and surfaces of constant are two-sheeted hyperboloids (Hilbert and Cohn-Vossen 1999, pp. 22-23). For every , there is a unique set of ellipsoidal coordinates. However, specifies eight points symmetrically located in octants.Solving for , , and gives(4)(5)(6)The Laplacian is(7)where(8)Another definition is(9)(10)(11)where(12)(Arfken 1970, pp. 117-118). Byerly (1959, p. 251) uses a slightly different definition in which the Greek variables are replaced by their..
A system of curvilinear coordinates in which two sets of coordinate surfaces are obtained by revolving the parabolas of parabolic cylindrical coordinates about the x-axis, which is then relabeled the z-axis. There are several notational conventions. Whereas is used in this work, Arfken (1970) uses .The equations for the parabolic coordinates are(1)(2)(3)where , , and . To solve for , , and , examine(4)(5)(6)so(7)and(8)(9)We therefore have(10)(11)(12)The scale factors are(13)(14)(15)The line element is(16)and the volume element is(17)The Laplacian is(18)(19)(20)The Helmholtz differential equationis separable in parabolic coordinates.
The polar angle on a sphere measured from the north pole instead of the equator. The angle in spherical coordinates is the colatitude. It is related to the latitude by .
Given a reference triangle , the trilinear coordinates of a point with respect to are an ordered triple of numbers, each of which is proportional to the directed distance from to one of the side lines. Trilinear coordinates are denoted or and also are known as homogeneous coordinates or "trilinears." Trilinear coordinates were introduced by Plücker in 1835. Since it is only the ratio of distances that is significant, the triplet of trilinear coordinates obtained by multiplying a given triplet by any nonzero constant describes the same point, so(1)For simplicity, the three polygon vertices , , and of a triangle are commonly written as , , and , respectively.Trilinear coordinates can be normalized so that they give the actual directed distances from to each of the sides. To perform the normalization, let the point in the above diagram have trilinear coordinates and lie at distances , , and from the sides , , and , respectively. Then..
An orthogonal coordinate system is a system of curvilinear coordinates in which each family of surfaces intersects the others at right angles. Orthogonal coordinates therefore satisfy the additional constraint that(1)where is the Kronecker delta. Therefore, the line element becomes(2)(3)and the volume element becomes(4)(5)(6)(7)(8)where the latter is the Jacobian.The gradient of a function is given in orthogonal curvilinear coordinates by(9)(10)the divergence is(11)and the curl is(12)(13)For surfaces of first degree, the only three-dimensional coordinate system of surfaces having orthogonal intersections is Cartesian coordinates (Moon and Spencer 1988, p. 1). Including degenerate cases, there are 11 sets of quadratic surfaces having orthogonal coordinates. Furthermore, Laplace's equation and the Helmholtz differential equation are separable in all of these coordinate systems (Moon and Spencer 1988, p. 1).Planar..
The Euclidean plane parametrized by coordinates, so that each point is located based on its position with respect to two perpendicular lines, called coordinate axes. They are two copies of the real line, and the zero point lies at their intersection, called the origin. The coordinate axes are usually called the x-axis and y-axis, depicted above. Point is associated with the coordinates corresponding to its orthogonal projections onto the -axis and the -axis respectively.
A system of curvilinear coordinates for which several different notations are commonly used. In this work is used, whereas Arfken (1970) uses and Moon and Spencer (1988) use . The toroidal coordinates are defined by(1)(2)(3)where is the hyperbolic sine and is the hyperbolic cosine. The coordinates satisfy , , and .Surfaces of constant are given by the toroids(4)surfaces of constant by the spherical bowls(5)spheres centered at with radii (6)and surfaces of constant by(7)The scale factors are(8)(9)(10)The Laplacian is(11)The Helmholtz differential equation is not separable in toroidal coordinates, but Laplace's equation is.
Coordinates useful for plotting projective three-dimensional curves of the form which are defined by(1)(2)(3)(4)
One of the eight regions of space defined by the eight possible combinations of signs for , , and .
Cartesian coordinates are rectilinear two- or three-dimensional coordinates (and therefore a special case of curvilinear coordinates) which are also called rectangular coordinates. The two axes of two-dimensional Cartesian coordinates, conventionally denoted the x- and y-axes (a notation due to Descartes), are chosen to be linear and mutually perpendicular. Typically, the -axis is thought of as the "left and right" or horizontal axis while the -axis is thought of as the "up and down" or vertical axis. In two dimensions, the coordinates and may lie anywhere in the interval , and an ordered pair in two-dimensional Cartesian coordinates is often called a point or a 2-vector.The three-dimensional Cartesian coordinate system is a natural extension of the two-dimensional version formed by the addition of a third "in and out" axis mutually perpendicular to the - and -axes defined above. This new axis is conventionally..
A plane coordinate system whose axes are not perpendicular. The -coordinate of a point is the abscissa of its projection onto the -axis in the direction of the -axis, and the -coordinate is similarly determined.Unlike in the more general case of affine coordinates, in oblique coordinates, the unit length is the same on both axes. If the Cartesian equation of a curve in a Cartesian coordinate system is applied to an oblique coordinate system, the result will be a distorted curve. For example, a circle will be transformed into an ellipse.
A system of curvilinear coordinates in which two sets of coordinate surfaces are obtained by revolving the curves of the elliptic cylindrical coordinates about the y-axis which is relabeled the z-axis. The third set of coordinates consists of planes passing through this axis.(1)(2)(3)where , , and . Arfken (1970) uses instead of . The scale factors are(4)(5)(6)The Laplacian is(7)An alternate form useful for "two-center" problems is defined by(8)(9)(10)(11)where , , and . In these coordinates,(12)(13)(14)(Abramowitz and Stegun 1972). The scale factors are(15)(16)(17)and the Laplacian is(18)The Helmholtz differential equationis separable.
A coordinate system defined by the coordinate transformation(1)(2)(3)with and . Surfaces of constant are given by the cardioids of revolution intersecting the positive half of the -axis(4)surfaces of constant by the cardioids of revolution intersecting the negative half of the -axis(5)and surfaces of constant by the half-planes(6)The metric coefficients are(7)(8)(9)
Synergetics coordinates are a set of triangular coordinates in their plane (or their generalization to tetrahedral coordinates in space, or the analogs in higher dimensions). In the plane, coordinates are measured along three axes , , and , with the -axis oriented downward and the and axes oriented at angles to each other as illustrated above (left figure). Interpreting , , and as points on the sides of an equilateral triangle obtained by parallel-displacing from the origin three pairs of lines oriented at angles with respect to one another, the coordinates can be interpreted as specifying a given equilateral triangle (right figure).A nice property of these coordinates is that the vertices of the triangle obtained by parallel-displacing by are given by , , and (see above figure), so that the sums of the coordinates of the vertices are always zero. This property also holds when the coordinates are generalized to three and higher dimensions.The..
An authalic latitude which is directly proportional to the spacing of parallels of latitude from the equator on an ellipsoidal Mercator projection. It is defined by(1)where the symbol is sometimes used instead of . The isometric latitude is related to the conformal latitude by(2)The inverse is found by iterating(3)with the first trial as(4)
A system of curvilinear coordinates variously denoted (Arfken 1970) or (Moon and Spencer 1988). Using the notation of Arfken, the bispherical coordinates are defined by(1)(2)(3)Surfaces of constant are given by the spheres(4)surfaces of constant by apple surfaces () or lemon surfaces ()(5)and surface of constant by the half-planes(6)The scale factors are(7)(8)(9)The Laplacian is given by(10)In bispherical coordinates, Laplace's equation is separable (Moon and Spencer 1988), but the Helmholtz differential equation is not.
A plot of a function expressed in spherical coordinates, with radius as a function of angles and . Polar plots can be drawn using SphericalPlot3D[r, phi, phimin, phimax, theta, thetamin, thetamax]. The plots above are spherical plots of the equations and , where denotes the real part and the imaginary part. The spherical plot of a constant is a sphere of radius .
A system of coordinates obtained by inversion of the prolate spheroids and two-sheeted hyperboloids in prolate spheroidal coordinates. The inverse prolate spheroidal coordinates are given by the transformation equations(1)(2)(3)with , , and . Surfaces of constant are given by the cyclides of rotation(4)surfaces of constant by the cyclides of rotation(5)and surfaces of constant by the half-planes(6)The metric coefficients are given by(7)(8)(9)
A set of curvilinear coordinates definedby(1)(2)(3)where , , and . There are several notational conventions, and whereas is used in this work, Arfken (1970) prefers . The following identities show that curves of constant and are circles in -space.(4)(5)The scale factors are(6)(7)(8)The Laplacian is(9)Laplace's equation is not separable in bipolar cylindrical coordinates, but it is in two-dimensional bipolar coordinates.
Spherical coordinates, also called spherical polar coordinates (Walton 1967, Arfken 1985), are a system of curvilinear coordinates that are natural for describing positions on a sphere or spheroid. Define to be the azimuthal angle in the -plane from the x-axis with (denoted when referred to as the longitude), to be the polar angle (also known as the zenith angle and colatitude, with where is the latitude) from the positive z-axis with , and to be distance (radius) from a point to the origin. This is the convention commonly used in mathematics.In this work, following the mathematics convention, the symbols for the radial, azimuth, and zenith angle coordinates are taken as , , and , respectively. Note that this definition provides a logical extension of the usual polar coordinates notation, with remaining the angle in the -plane and becoming the angle out of that plane. The sole exception to this convention in this work is in spherical harmonics,..
A system of coordinates obtained by inversion of the oblate spheroids and one-sheeted hyperboloids in oblate spheroidal coordinates. The inverse oblate spheroidal coordinates are given by the transformation equations(1)(2)(3)where , , and . Surfaces of constant are given by the cyclides of rotation(4)surfaces of constant by the cyclides of rotation(5)and surfaces of constant by the half-planes(6)The metric coefficients are given by(7)(8)(9)
Bipolar coordinates are a two-dimensional system of coordinates. There are two commonly defined types of bipolar coordinates, the first of which is defined by(1)(2)where , . The following identities show that curves of constant and are circles in -space.(3)(4)The scale factors are(5)(6)The Laplacian is(7)Laplace's equation is separable.Two-center bipolar coordinates are two coordinates giving the distances from two fixed centers and , sometimes denoted and . For two-center bipolar coordinates with centers at ,(8)(9)Combining (8) and (9) gives(10)Solving for Cartesian coordinates and gives(11)(12)Solving for polar coordinates gives(13)(14)
A -hyperoctant is one of the regions of space defined by the possible combinations of signs . The 2-hyperoctant is known as a quadrant and the 3-hyperoctant is known as an octant.
A coordinate system which is similar to bispherical coordinates but having fourth-degree surfaces instead of second-degree surfaces for constant . The coordinates are given by the transformation equations(1)(2)(3)where(4), , , and , , and are Jacobi elliptic functions. Surfaces of constant are given by the bicyclides(5)surfaces of constant by the cyclides of rotation(6)and surfaces of constant by the half-planes(7)
Homogeneous coordinates of a finite point in the plane are any three numbers for which(1)(2)Coordinates ( for which(3)describe the point at infinity in the direction of slope .In homogeneous coordinates, the equation of a line(4)is given by(5)Two points expressed using homogeneous coordinates and are identical iff(6)Two lines expressed using homogeneous coordinates(7)(8)are identical iff(9)The intersection of the two lines above is given by(10)(11)(12)
Barycentric coordinates are triples of numbers corresponding to masses placed at the vertices of a reference triangle . These masses then determine a point , which is the geometric centroid of the three masses and is identified with coordinates . The vertices of the triangle are given by , , and . Barycentric coordinates were discovered by Möbius in 1827 (Coxeter 1969, p. 217; Fauvel et al. 1993).To find the barycentric coordinates for an arbitrary point , find and from the point at the intersection of the line with the side , and then determine as the mass at that will balance a mass at , thus making the centroid (left figure). Furthermore, the areas of the triangles , , and are proportional to the barycentric coordinates , , and of (right figure; Coxeter 1969, p. 217).Barycentric coordinates are homogeneous, so(1)for .Barycentric coordinates normalized so that they become the actual areas of the subtriangles are called homogeneous..
Homogeneous barycentric coordinates are barycentric coordinates normalized such that they become the actual areas of the subtriangles. Barycentric coordinates normalized so that(1)so that the coordinates give the areas of the subtriangles normalized by the area of the original triangle are called areal coordinates (Coxeter 1969, p. 218). Barycentric and areal coordinates can provide particularly elegant proofs of geometric theorems such as Routh's theorem, Ceva's theorem, and Menelaus' theorem (Coxeter 1969, pp. 219-221).The homogeneous barycentric coordinates corresponding to exact trilinear coordinates are , where(2)(3)(4)The homogeneous barycentric coordinates for some common triangle centers are summarized in the following table, where is the circumradius of the reference triangle.triangle centerhomogeneous barycentric coordinatescircumcenter incenter orthocenter symmedian point triangle..
An -dimensional subspace of an -dimensional vector space can be specified by an matrix whose rows are the coordinates of a basis of . The set of all minors of this matrix are then called the Grassmann (or sometimes Plücker; Stofli 1991) coordinates of , where is a binomial coefficient. Hodge and Pedoe (1952) give a thorough treatment of Grassmann coordinates.
An axis is a line with respect to which a curve or figure is drawn, measured, rotated, etc. The most common axes encountered are commonly the mutually perpendicular Cartesian axes in the plane or in space.The plural of "axis" is "axes," pronounced "ax-ees."The term is also used to refer to a line through a sheafof planes (Woods 1961; Altshiller-Court 1979, p. 12).
An authalic latitude given by(1)The series expansion is(2)where(3)
Barycentric coordinates normalized so that they become the areas of the triangles , , and , where is the point whose coordinates have been specified, normalized by the area of the original triangle . This is equivalent to application of the normalization relation(Coxeter 1969, p. 218).
An auxiliary latitude which gives a sphere having correct distances along the meridians. It is denoted (or ) and is given by(1) is evaluated for at the north pole (), and is given by(2)(3)A series for is(4)and a series for is(5)where(6)The inverse formula is(7)
Gaussian curvature, sometimes also called total curvature (Kreyszig 1991, p. 131), is an intrinsic property of a space independent of the coordinate system used to describe it. The Gaussian curvature of a regular surface in at a point is formally defined as(1)where is the shape operator and det denotes the determinant.If is a regular patch, then the Gaussian curvature is given by(2)where , , and are coefficients of the first fundamental form and , , and are coefficients of the second fundamental form (Gray 1997, p. 377). The Gaussian curvature can be given entirely in terms of the first fundamental form(3)and the metric discriminant(4)by(5)where are Christoffel symbols of the first kind. Equivalently,(6)where(7)(8)Writing this out,(9)The Gaussian curvature is also given by(10)(Gray 1997, p. 380), as well as(11)where is the permutation symbol, is the unit normal vector and is the unit tangent vector. The Gaussian..
The study of the geometry of figures by algebraic representation and manipulation of equations describing their positions, configurations, and separations. Analytic geometry is also called coordinate geometry since the objects are described as -tuples of points (where in the plane and 3 in space) in some coordinate system.
The term "real line" has a number of different meanings in mathematics.Most commonly, "real line" is used to mean real axis, i.e., a line with a fixed scale so that every real number corresponds to a unique point on the line. The generalization of the real line to two dimensions is called the complex plane.The term "real line" is also used to distinguish an ordinary line from a so-called imaginary line which can arise in algebraic geometry.Renteln and Dundes (2005) give the following (bad) mathematical jokes about the real line:Q: What is green and homeomorphic to the open unit interval?A: The real lime.
One of the four regions of the plane defined by the four possible combinations of signs , , , and for .
The coordinates are the asymptotic angle of confocal hyperbolic cylinders symmetrical about the x-axis. The coordinates are confocal elliptic cylinders centered on the origin.(1)(2)(3)where , , and . They are related to Cartesian coordinates by(4)(5)The scale factors are(6)(7)(8)(9)(10)(11)(12)The matrices of Christoffel symbolsof the second kind in the sense of Misner et al. (1973) are given by(13)(14)(15)The Jacobian is(16)The Laplacian is(17)Let(18)(19)(20)Then the new scale factors are(21)(22)(23)The Helmholtz differential equationis separable in elliptic cylindrical coordinates.
A system of curvilinear coordinates in which two sets of coordinate surfaces are obtained by revolving the curves of the elliptic cylindrical coordinates about the x-axis, which is relabeled the z-axis. The third set of coordinates consists of planes passing through this axis.(1)(2)(3)where , , and . Note that several conventions are in common use; Arfken (1970) uses instead of , and Moon and Spencer (1988, p. 28) use .In this coordinate system, the scale factors are(4)(5)(6)The Laplacian is (7)(8)An alternate form useful for "two-center" problems is defined by(9)(10)(11)where , , and (Abramowitz and Stegun 1972). In these coordinates,(12)(13)(14)In terms of the distances from the two foci,(15)(16)(17)The scale factors are(18)(19)(20)and the Laplacian is(21)The Helmholtz differential equationis separable in prolate spheroidal coordinates...
The adjective "affine" indicates everything that is related to the geometry of affine spaces. A coordinate system for the -dimensional affine space is determined by any basis of vectors, which are not necessarily orthonormal. Therefore, the resulting axes are not necessarily mutually perpendicular nor have the same unit measure. In this sense, affine is a generalization of Cartesian or Euclidean.An example of an affine property is the average area of a random triangle chosen inside a given triangle (i.e., triangle triangle picking). Because this problem is affine, the ratio of the average area to the original triangle is a constant independent of the actual triangle chosen. Another example of an affine property is the areas (relative to the original triangle) of the regions created by connecting the side -multisectors of a triangle with lines drawn to the opposite vertices (i.e., Marion's theorem).An example of a property that..
A plot of a function expressed in polar coordinates, with radius as a function of angle . Polar plots can be drawn in the Wolfram Language using PolarPlot[r, t, tmin, tmax]. The plot above is a polar plot of the polar equation , giving a cardioid.Polar plots of give curves known as roses, while polar plots of produce what's known as Archimedes' spiral, a special case of the Archimedean spiral corresponding to . Other specially-named Archimedean spirals include the lituus when , the hyperbolic spiral when , and Fermat's spiral when . Note that lines and circles are easily-expressed in polar coordinates as(1)and(2)for the circle with center and radius , respectively. Note that equation () is merely a particular instance of the equation(3)defining a conic section of eccentricity and semilatus rectum . In particular, the circle is the conic of eccentricity , while yields a general ellipse, a parabola, and a hyperbola.The plotting of a complex number..
A ternary diagram is a triangular diagram which displays the proportion of three variables that sum to a constant and which does so using barycentric coordinates. The coordinate axes of such a diagram are shown in the figure above, where each of the x-, y-, and z-axes are scaled so that , and where the grid lines denote the values , . In most instances, ternary plots are drawn on equilateral triangles as in the figure above, though it is not uncommon for certain scenarios to be better graphed on right triangular diagrams as well (West 2013).Ternary diagrams are sometimes called ternary plots, triangle plots, ternary graphs, simplex plots, and de Finetti diagrams, though the latter term is usually reserved for a specific family of ternary diagrams commonly studied in population genetics. Such diagrams are encountered often in the study of phase equilibria and appear somewhat often throughout a number of physical sciences.pointcoordinates For..