Projective geometry

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Lambert conformal conic projection

Let be the longitude, the reference longitude, the latitude, the reference latitude, and and the standard parallels. Then the transformation of spherical coordinates to the plane via the Lambert conformal conic projection is given by(1)(2)where(3)(4)(5)(6)The inverse formulas are(7)(8)where(9)(10)with , , and as defined above.

Bonne projection

The Bonne projection is a map projection that resembles the shape of a heart. Let be the standard parallel, the central meridian, be the latitude, and the longitude on a unit sphere. Then(1)(2)where(3)(4)The illustrations above show Bonne projections for two different standard parallels.The inverse formulas are(5)(6)where(7)The Werner projection is a special case of theBonne projection.

Sinusoidal projection

The sinusoidal projection is an equal-area projectiongiven by the transformation(1)(2)The inverse formulas are(3)(4)

Gnomonic projection

The gnomonic projection is a nonconformal map projection obtained by projecting points (or ) on the surface of sphere from a sphere's center to point in a plane that is tangent to a point (Coxeter 1969, p. 93). In the above figure, is the south pole, but can in general be any point on the sphere. Since this projection obviously sends antipodal points and to the same point in the plane, it can only be used to project one hemisphere at a time. In a gnomonic projection, great circles are mapped to straight lines. The gnomonic projection represents the image formed by a spherical lens, and is sometimes known as the rectilinear projection.In the projection above, the point is taken to have latitude and longitude and hence lies on the equator. The transformation equations for the plane tangent at the point having latitude and longitude for a projection with central longitude and central latitude are given by(1)(2)and is the angular distance of the point..

Azimuthal equidistant projection

An azimuthal projection which is neither equal-area nor conformal. Let and be the latitude and longitude of the center of the projection, then the transformation equations are given by(1)(2)Here,(3)and(4)where is the angular distance from the center. The inverse formulas are(5)and(6)with the angular distance from the center given by(7)

Polyconic projection

A class of map projections in which the parallels are represented by a system of non-concentric circular arcs with centers lying on the straight line representing the central meridian (Lee 1944). The term was first applied by Hunt, and later extended by Tissot (1881).(1)(2)where(3)The inverse formulas are(4)and is determined from(5)starting with the initial vale and defining(6)(7)

Peters projection

The Peters projection is a cylindrical equal-area projection that de-emphasizes the exaggeration of areas at high latitudes by shifting the standard latitude to (or sometimes or ; Dana).

Orthographic projection

The orthographic projection is a projection from infinity that preserves neitherarea nor angle. It is given by(1)(2)where is the latitude, is the longitude, and and are reference longitudes and latitudes, respectively.The inverse transformations are(3)(4)where(5)(6)and the two-argument form of the inverse tangentfunction is best used for this computation.

Orthogonal projection

A projection of a figure by parallel rays. In such a projection, tangencies are preserved. Parallel lines project to parallel lines. The ratio of lengths of parallel segments is preserved, as is the ratio of areas.Any triangle can be positioned such that its shadow under an orthogonal projection is equilateral. Also, the triangle medians of a triangle project to the triangle medians of the image triangle. Ellipses project to ellipses, and any ellipse can be projected to form a circle. The center of an ellipse projects to the center of the image ellipse. The triangle centroid of a triangle projects to the triangle centroid of its image. Under an orthogonal transformation, the Steiner inellipse can be transformed into a circle inscribed in an equilateral triangle.Spheroids project to ellipses (or circles in the degenerate case).In an orthogonal projection, any vector can be written , soand the projection matrix is a symmetric matrix iff the vector..

Eckert vi projection

The equations are(1)(2)where is the solution to(3)This can be solved iteratively using Newton's method with to obtain(4)The inverse formulas are(5)(6)where(7)

Authalic latitude

A parametric latitude which gives a sphere equal surface area relative to an ellipsoid. The authalic latitude is defined by(1)where(2)and is evaluated at the north pole (). Let be the radius of the sphere having the same surface area as the ellipsoid, then(3)The series for is(4)The inverse formula is found from(5)where(6)and . This can be written in series form as(7)

Eckert iv projection

The equations are(1)(2)where is the solution to(3)This can be solved iteratively using Newton's method with to obtain(4)The inverse formulas are(5)(6)where(7)

Mollweide projection

The Mollweide projection is a map projection also called the elliptical projection or homolographic equal-area projection. The forward transformation is(1)(2)where is given by(3)Newton's method can then be used to compute iteratively from(4)where(5)(Snyder 1987, p. 251) or, better yet,(6)can be used as a first guess.The inverse formulas are(7)(8)where(9)

Cylindrical projection

A cylindrical projection of points on a unit sphere centered at consists of extending the line for each point until it intersects a cylinder tangent to the sphere at its equator at a corresponding point . If the sphere is tangent to the cylinder at longitude , then a point on the sphere with latitude and longitude is mapped to a point on the cylinder with height . Unwrapping and flattening out the cylinder then gives the Cartesian coordinates(1)(2)The cylindrical projection of the Earth is illustrated above.This form of the projection, however, is seldom used in practice, and the term "cylindrical projection" is used instead to refer to any projection in which lines of longitude are mapped to equally spaced parallel lines and lines of latitude (parallels) are mapped to parallel lines with arbitrary mathematically spaced separations (Snyder 1987, p. 5). For example, the common Mercator projection uses the complicated transformation(3)instead..

Miller equidistant projection

Several cylindrical equidistant projections were devised by R. Miller. Miller's projections have standard parallels of (giving minimal overall scale distortion), (giving minimal scale distortion over continents), and (Miller 1949).

Cylindrical equidistant projection

The map projection having transformation equations(1)(2)and the inverse formulas are(3)(4)The following table gives special cases of the cylindrical equidistant projection. projection nameequirectangular projectionMiller equidistant projectionMiller equidistant projectionMiller equidistant projection

Miller cylindrical projection

A map projection given by the following transformation,(1)(2)(3)Here, and are the plane coordinates of a projected point, is the longitude of a point on the globe, is central longitude used for the projection, and is the latitude of the point on the globe.The inverse formulas are(4)(5)(6)

Airy projection

A map projection. The inverse equations for are computed by iteration. Let the angle of the projection plane be . Define(1)For proper convergence, let and compute the initial point by checking(2)As long as , take and iterate again. The first value for which is then the starting point. Then compute(3)until the change in between evaluations is smaller than the acceptable tolerance. The (inverse) equations are then given by(4)(5)

Perspectrix

The line joining the three collinear points of intersection of the extensions of corresponding sides in perspective triangles, also called the perspective axis or homology axis.The following table summarizes the perspectrices for various pairs of named triangles. Pairs in which one triangle is inscribed in another have the line at infinity as their perspectrix and are omitted in the list below.triangle 1triangle 2perspectrixKimberlinglineanticomplementary trianglecircum-medial trianglethird power pointde Longchamps lineanticomplementary trianglecircum-orthic triangle***anticomplementary trianglecontact triangle***anticomplementary triangleD-triangle***anticomplementary triangleextouch triangle***anticomplementary trianglefirst Brocard triangle***anticomplementary trianglefirst Neuberg triangle***anticomplementary trianglefirst Yff triangle***anticomplementary triangleFuhrmann..

Fundamental theorem of projective geometry

Any collineation from to , where is a three-dimensional vector space, is associated with a semilinear map from to .

Perspective triangles

Two triangles and are said to be perspective, or sometimes homologic, from a line if the extensions of their three pairs of corresponding sides meet in collinear points , , and . The line joining these points is called the perspectrix.Two triangles are perspective from a point if their three pairs of corresponding polygon vertices are joined by lines which meet in a point of concurrence . This point is called the perspector, perspective center, homology center, or pole.Desargues' theorem guarantees that if two triangles are perspective from a point, they are perspective from a line (called the perspectrix). Triangles in perspective are sometimes said to be homologous or copolar.

Perspective

Perspective is the art and mathematics of realistically depicting three-dimensional objects in a two-dimensional plane, sometimes called centric or natural perspective to distinguish it from bicentric perspective. The study of the projection of objects in a plane is called projective geometry. The principles of perspective drawing were elucidated by the Florentine architect F. Brunelleschi (1377-1446). These rules are summarized by Dixon (1991): 1. The horizon appears as a line. 2. Straight lines in space appear as straight lines in the image. 3. Sets of parallel lines meet at a vanishingpoint. 4. Lines parallel to the picture plane appear paralleland therefore have no vanishing point. There is a graphical method for selecting vanishing points so that a cubeor box appears to have the correct dimensions (Dixon 1991)...

Symplectic map

Informally, a symplectic map is a map which preserves the sum of areas projected onto the set of planes. It is the generalization of an area-preserving map.Formally, a symplectic map is a real-linear map that preserves a symplectic form , i.e., for whichfor all , . Every symplectic map on a complex Hilbert space may be written as , where is unitary, is positive, and is an anti-linear involution (i.e., complex conjugation).

Sondat's theorem

The perspectrix of a pair of paralogic triangles and bisects the line joining the two orthocenters and (Johnson 1929, p. 259).

Shephard's problem

Measurements of a centered convex body in Euclidean n-space (for ) show that its brightness function (the volume of each projection) is smaller than that of another such body. Is it true that its volume is also smaller? C. M. Petty and R. Schneider showed in 1967 that the answer is yes if the body with the larger brightness function is a projection body, but no in general for every .

Affine transformation

An affine transformation is any transformation that preserves collinearity (i.e., all points lying on a line initially still lie on a line after transformation) and ratios of distances (e.g., the midpoint of a line segment remains the midpoint after transformation). In this sense, affine indicates a special class of projective transformations that do not move any objects from the affine space to the plane at infinity or conversely. An affine transformation is also called an affinity.Geometric contraction, expansion, dilation, reflection, rotation, shear, similarity transformations, spiral similarities, and translation are all affine transformations, as are their combinations. In general, an affine transformation is a composition of rotations, translations, dilations, and shears.While an affine transformation preserves proportions on lines, it does not necessarily preserve angles or lengths. Any triangle can be transformed..

Projective geometry

The branch of geometry dealing with the properties and invariants of geometric figures under projection. In older literature, projective geometry is sometimes called "higher geometry," "geometry of position," or "descriptive geometry" (Cremona 1960, pp. v-vi).The most amazing result arising in projective geometry is the duality principle, which states that a duality exists between theorems such as Pascal's theorem and Brianchon's theorem which allows one to be instantly transformed into the other. More generally, all the propositions in projective geometry occur in dual pairs, which have the property that, starting from either proposition of a pair, the other can be immediately inferred by interchanging the parts played by the words "point" and "line."The axioms of projective geometry are: 1. If and are distinct points on a plane, there is at least one line containing..

Affine function

Affine functions represent vector-valued functions of the formThe coefficients can be scalars or dense or sparse matrices. The constant term is a scalar or a column vector.In geometry, an affine transformation or affine map (from the Latin, affinis, "connected with") between two vector spaces consists of a linear transformation followed by a translation. In a geometric setting, these are precisely the functions that map straight lines to straight lines.

Projection

A projection is the transformation of points and lines in one plane onto another plane by connecting corresponding points on the two planes with parallel lines. This can be visualized as shining a (point) light source (located at infinity) through a translucent sheet of paper and making an image of whatever is drawn on it on a second sheet of paper. The branch of geometry dealing with the properties and invariants of geometric figures under projection is called projective geometry.The projection of a vector onto a vector is given bywhere is the dot product, and the length of this projection isGeneral projections are considered by Foley and VanDam (1983).The average projected area over all orientations of any ellipsoid is 1/4 the total surface area. This theorem also holds for any convex solid...

Pohlke's theorem

The principal theorem of axonometry, first published without proof by Pohlke in 1860. It states that three segments of arbitrary length , , and which are drawn in a plane from a point under arbitrary angles form a parallel projection of three equal segments , , and from the origin of three perpendicular coordinate axes. However, only one of the segments or one of the angles may vanish.

Conic projection

A conic projection of points on a unit sphere centered at consists of extending the line for each point until it intersects a cone with apex which tangent to the sphere along a circle passing through a point in a point . For a cone with apex a height above , the angle from the z-axis at which the cone is tangent is given by(1)and the radius of the circle of tangency and height above at which it is located are given by(2)(3)Letting be the colatitude of a point on a sphere, the length of the vector along is(4)The left figure above shows the result of re-projecting onto a plane perpendicular to the z-axis (equivalent to looking at the cone from above the apex), while the figure on the right shows the cone cut along the solid line and flattened out. The equations transforming a point on a sphere to a point on the flattened cone are(5)(6)This form of the projection, however, is seldom used in practice, and the term "conic projection" is used instead to refer..

Mercator projection

The Mercator projection is a map projection that was widely used for navigation since loxodromes are straight lines (although great circles are curved). The following equations place the x-axis of the projection on the equator and the y-axis at longitude , where is the longitude and is the latitude.(1)(2)(3)(4)(5)(6)The inverse formulas are(7)(8)(9)(10)where is the Gudermannian.An oblique form of the Mercator projection is illustrated above. It has equations(11)(12)(13)where(14)(15)(16)The inverse formulas are(17)(18)There is also a transverse form of the Mercator projection, illustrated above (Deetz and Adams 1934, Snyder 1987). It is given by the equations(19)(20)(21)(22)(23)where(24)(25)Finally, the "universal transverse Mercator projection" is a map projection which maps the sphere into 60 zones of each, with each zone mapped by a transverse Mercator projection with central meridian in the center of the..

Conic equidistant projection

A map projection with transformation equations(1)(2)where(3)(4)(5)(6)(7)The inverse formulas are given by(8)(9)where(10)(11)

Map projection

A projection which maps a sphere (or spheroid) onto a plane. Map projections are generally classified into groups according to common properties (cylindrical vs. conical, conformal vs. area-preserving, , etc.), although such schemes are generally not mutually exclusive. Early compilers of classification schemes include Tissot (1881), Close (1913), and Lee (1944). However, the categories given in Snyder (1987) remain the most commonly used today, and Lee's terms authalic and aphylactic are not commonly encountered.No projection can be simultaneously conformaland area-preserving.

Conformal projection

A map projection which is a conformal mapping, i.e., one for which local (infinitesimal) angles on a sphere are mapped to the same angles in the projection. On maps of an entire sphere, however, there are usually singular points at which local angles are distorted.The term conformal was applied to map projections by Gauss in 1825, and eventually supplanted the alternative terms "orthomorphic" (Lee 1944; Snyder 1987, p. 4) and "autogonal" (Tissot 1881, Lee 1944).No projection can be both equal-area and conform, and projections which are neither equal-area nor conformal are sometimes called aphylactic (Lee 1944; Snyder 1987, p. 4).

Vertical perspective projection

The vertical perspective projection is a map projection that corresponds to the appearance of a globe when directly viewed from some distance away with the -axis of the viewer aligned parallel to the positive -axis of the globe. It is given by the transformation equations(1)(2)where is the distance of the point of perspective in units of sphere radii and(3)(4)Note that points corresponding to are on the back side of the globe and so should be suppressed when making the projection.

Conformal latitude

Conformal latitude is defined by(1)(2)(3)The inverse is obtained by iterating the equation(4)using as the first trial. A series form is(5)The conformal latitude was called the isometriclatitude by Adams (1921), but this term is now used to refer to a different quantity.

Van der grinten projection

The van der Grinten projection is a map projectiongiven by the transformation(1)(2)where(3)(4)(5)(6)(7)The inverse formulas are(8)(9)where(10)(11)(12)(13)(14)(15)(16)(17)(18)

Longitude

The azimuthal coordinate on the surface of a sphere ( in spherical coordinates) or on a spheroid (in prolate or oblate spheroidal coordinates). Longitude is defined such that . Lines of constant longitude are generally called meridians. The other angular coordinate on the surface of a sphere is called the latitude.The shortest distance between any two points on a sphere is the so-called great circle distance, which can be directly computed from the latitude and longitudes of two points.

Cassini projection

A map projection defined by(1)(2)The inverse formulas are(3)(4)where(5)

Stereographic projection

A map projection obtained by projecting points on the surface of sphere from the sphere's north pole to point in a plane tangent to the south pole (Coxeter 1969, p. 93). In such a projection, great circles are mapped to circles, and loxodromes become logarithmic spirals.Stereographic projections have a very simple algebraic form that results immediately from similarity of triangles. In the above figures, let the stereographic sphere have radius , and the -axis positioned as shown. Then a variety of different transformation formulas are possible depending on the relative positions of the projection plane and -axis.The transformation equations for a sphere of radius are given by(1)(2)where is the central longitude, is the central latitude, and(3)The inverse formulas for latitude and longitude are then given by(4)(5)where(6)(7)and the two-argument form of the inverse tangentfunction is best used for this computation.For an oblate..

Special affine curvature

Special affine curvature, also called as the equi-affine or affine curvature, is a type of curvature for a plane curve that remains unchanged under a special affine transformation.For a plane curve parametrized by , the special affine curvature is given by(1)(2)(Blaschke 1923, Guggenheimer 1977), where the prime indicates differentiation with respect to t. This reduces for a curve to(3)(4)(Blaschke 1923, Shirokov 1988), where the prime here indicated differentiation with respect to .The following table summarizes the special affine curvatures for a number of curves.curveparametrizationcatenarycircleellipsehyperbolaparabola0

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