Geometry is a branch of mathematics concerned with questions of shape, relative position of figures, the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a practical way for dealing with lengths and volumes. Geometry began to see elements of formal mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment, Euclid's Elements, set a standard for many centuries to follow. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC. Islamic scientists expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid analytic footing by mathematicians such as René Descartes and Pierre de Fermat. Since and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, describing spaces that lie beyond the normal range of human experience.
While geometry has evolved throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, planes, surfaces and curves, as well as the more advanced notions of manifolds and topology or metric. Geometry has applications to many fields, including art, physics, as well as to other branches of mathematics. Contemporary geometry has many subfields: Euclidean geometry is geometry in its classical sense; the mandatory educational curriculum of the majority of nations includes the study of points, planes, triangles, similarity, solid figures and analytic geometry. Euclidean geometry has applications in computer science and various branches of modern mathematics. Differential geometry uses techniques of linear algebra to study problems in geometry, it has applications in physics, including in general relativity. Topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this means dealing with large-scale properties of spaces, such as connectedness and compactness.
Convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues using techniques of real analysis. It has close connections to convex analysis and functional analysis and important applications in number theory. Algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques, it has applications including cryptography and string theory. Discrete geometry is concerned with questions of relative position of simple geometric objects, such as points and circles, it shares many principles with combinatorics. Computational geometry deals with algorithms and their implementations for manipulating geometrical objects. Although being a young area of geometry, it has many applications in computer vision, image processing, computer-aided design, medical imaging, etc; the earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles and volumes, which were developed to meet some practical need in surveying, construction and various crafts.
The earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum. Clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter's position and motion within time-velocity space; these geometric procedures anticipated the Oxford Calculators, including the mean speed theorem, by 14 centuries. South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks. In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore, he is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. Pythagoras established the Pythagorean School, credited with the first proof of the Pythagorean theorem, though the statement of the theorem has a long history.
Eudoxus developed the method of exhaustion, which allowed the calculation of areas and volumes of curvilinear figures, as well as a theory of ratios that avoided the problem of incommensurable magnitudes, which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose Elements considered the most successful and influential textbook of all time, introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom and proof. Although most of the contents of the Elements were known, Euclid arranged them into a single, coherent logical framework; the Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today. Archimedes of Syracuse used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, gave remarkably accurate approximations of Pi.
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A perimeter is a path that surrounds a two-dimensional shape. The term may be used either for the path itself or its length—it can be thought of as the length of the outline of a shape; the perimeter of a circle or ellipse is called its circumference. Calculating the perimeter has several practical applications. A calculated perimeter is the length of fence required to surround a garden; the perimeter of a wheel describes. The amount of string wound around a spool is related to the spool's perimeter; the perimeter is the distance around a shape. Perimeters for more general shapes can be calculated, as any path, with ∫ 0 L d s, where L is the length of the path and d s is an infinitesimal line element. Both of these must be replaced with by algebraic forms in order to be calculated. If the perimeter is given as a closed piecewise smooth plane curve γ: → R 2 with γ = its length L can be computed as follows: L = ∫ a b x ′ 2 + y ′ 2 d t A generalized notion of perimeter, which includes hypersurfaces bounding volumes in n -dimensional Euclidean spaces, is described by the theory of Caccioppoli sets.
Polygons are fundamental to determining perimeters, not only because they are the simplest shapes but because the perimeters of many shapes are calculated by approximating them with sequences of polygons tending to these shapes. The first mathematician known to have used this kind of reasoning is Archimedes, who approximated the perimeter of a circle by surrounding it with regular polygons; the perimeter of a polygon equals the sum of the lengths of its sides. In particular, the perimeter of a rectangle of width w and length ℓ equals 2 w + 2 ℓ. An equilateral polygon is a polygon. To calculate the perimeter of an equilateral polygon, one must multiply the common length of the sides by the number of sides. A regular polygon may be characterized by the number of its sides and by its circumradius, to say, the constant distance between its centre and each of its vertices; the length of its sides can be calculated using trigonometry. If R is a regular polygon's radius and n is the number of its sides its perimeter is 2 n R sin .
A splitter of a triangle is a cevian that divides the perimeter into two equal lengths, this common length being called the semiperimeter of the triangle. The three splitters of a triangle all intersect each other at the Nagel point of the triangle. A cleaver of a triangle is a segment from the midpoint of a side of a triangle to the opposite side such that the perimeter is divided into two equal lengths; the three cleavers of a triangle all intersect each other at the triangle's Spieker center. The perimeter of a circle called the circumference, is proportional to its diameter and its radius; that is to say, there exists a constant number pi, π, such that if P is the circle's perimeter and D its diameter P = π ⋅ D. In terms of the radius r of the circle, this formula becomes, P = 2 π ⋅ r. To calculate a circle's perimeter, knowledge of its radius or diameter and the number π suffices; the problem is that π is not rational, nor is it algebraic. So, obtaining an accurate approximation of π is important in the calculation.
The computation of the digits of π is relevant to many fields, such as mathematical analysis and computer science. The perimeter and the area are two main measures of geometric figures. Confusing them is a common error, as well as believing that the greater one of them is, the greater the other must be. Indeed, a commonplace observation is that an enlargement of a shape make its area grow as well as its perimeter. For example, if a field is drawn on a 1/10,000 scale map, the actual field perimeter can be calculated multiplying the drawing perimeter by 10,000; the real area is 10,0002 times the area of the shape on the map. There is no relation between the area and the perimeter of an ordinary shape. F
Spherical geometry is the geometry of the two-dimensional surface of a sphere. It is an example of a geometry, not Euclidean. Two practical applications of the principles of spherical geometry are astronomy. In plane geometry, the basic concepts are lines. On a sphere, points are defined in the usual sense; the equivalents of lines are not defined in the usual sense of "straight line" in Euclidean geometry, but in the sense of "the shortest paths between points", which are called geodesics. On a sphere, the geodesics are the great circles. Thus, in spherical geometry, angles are defined between great circles, resulting in a spherical trigonometry that differs from ordinary trigonometry in many respects. Spherical geometry is rather a subset of elliptic geometry. For example, it shares with that geometry the property that a line has no parallels through a given point. Contrast this with Euclidean geometry, in which a line has one parallel through a given point, hyperbolic geometry, in which a line has two parallels and an infinite number of ultraparallels through a given point.
An important geometry related to that of the sphere is that of the real projective plane. Locally, the projective plane has all the properties of spherical geometry, but it has different global properties. In particular, it one-sided. Concepts of spherical geometry may be applied to the oblong sphere, though minor modifications must be implemented on certain formulas. Higher-dimensional spherical geometries exist; the earliest mathematical work of antiquity to come down to our time is On the rotating sphere by Autolycus of Pitane, who lived at the end of the fourth century BC. Spherical trigonometry was studied by early Greek mathematicians such as Theodosius of Bithynia, a Greek astronomer and mathematician who wrote the Sphaerics, a book on the geometry of the sphere, Menelaus of Alexandria, who wrote a book on spherical trigonometry called Sphaerica and developed Menelaus' theorem; the book of unknown arcs of a sphere written by the Islamic mathematician Al-Jayyani is considered to be the first treatise on spherical trigonometry.
The book contains formulae for right-handed triangles, the general law of sines, the solution of a spherical triangle by means of the polar triangle. The book On Triangles by Regiomontanus, written around 1463, is the first pure trigonometrical work in Europe. However, Gerolamo Cardano noted a century that much of its material on spherical trigonometry was taken from the twelfth-century work of the Andalusi scholar Jabir ibn Aflah. Euler published a series of important memoirs on spherical geometry: L. Euler, Principes de la trigonométrie sphérique tirés de la méthode des plus grands et des plus petits, Mémoires de l'Académie des Sciences de Berlin 9, 1755, p. 233–257. XXVII, p. 277–308. L. Euler, Eléments de la trigonométrie sphéroïdique tirés de la méthode des plus grands et des plus petits, Mémoires de l'Académie des Sciences de Berlin 9, 1755, p. 258–293. XXVII, p. 309–339. L. Euler, De curva rectificabili in superficie sphaerica, Novi Commentarii academiae scientiarum Petropolitanae 15, 1771, pp. 195–216.
L. Euler, De mensura angulorum solidorum, Acta academiae scientarum imperialis Petropolitinae 2, 1781, p. 31–54. XXVI, p. 204–223. L. Euler, Problematis cuiusdam Pappi Alexandrini constructio, Acta academiae scientarum imperialis Petropolitinae 4, 1783, p. 91–96. XXVI, p. 237–242. L. Euler, Geometrica et sphaerica quaedam, Mémoires de l'Académie des Sciences de Saint-Pétersbourg 5, 1815, p. 96–114. XXVI, p. 344–358. L. Euler, Trigonometria sphaerica universa, ex primis principiis breviter et dilucide derivata, Acta academiae scientarum imperialis Petropolitinae 3, 1782, p. 72–86. XXVI, p. 224–236. L. Euler, Variae speculationes super area triangulorum sphaericorum, Nova Acta academiae scientarum imperialis Petropolitinae 10, 1797, p. 47–62. XXIX, p. 253–266. With points defined as the points on a sphere and lines as the great circles of that sphere, a spherical geometry has the following properties: Any two lines intersect in two diametrically opposite points, called antipodal points. Any two points that are not antipodal points determine a unique line.
There is a natural unit of length and a natural unit of area. Each line is associated with a pair of antipodal points, called the poles of the line, which are the common intersections of the set of lines perpendicular to the given line; each point is associated with a unique line, called the polar line of the point, the line on the plane through the centre of the sphere and perpendicular to the diameter of the sphere through the given point. As there are two arcs determined by a pair of points, which are not antipodal, on the line they determine, three non-collinear points do not determine a unique triangle. However, if we only consider triangles whose sides are minor arcs of great circles, we have the following properties: The angle sum of a triangle is greater than 180° and less than 270°; the area of a triangle is proportional to the excess of its angle sum over 180
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences and engineering disciplines, as well as in the social sciences. A model may help to explain a system and to study the effects of different components, to make predictions about behaviour. Mathematical models can take many forms, including dynamical systems, statistical models, differential equations, or game theoretic models; these and other types of models can overlap, with a given model involving a variety of abstract structures. In general, mathematical models may include logical models. In many cases, the quality of a scientific field depends on how well the mathematical models developed on the theoretical side agree with results of repeatable experiments. Lack of agreement between theoretical mathematical models and experimental measurements leads to important advances as better theories are developed.
In the physical sciences, a traditional mathematical model contains most of the following elements: Governing equations Supplementary sub-models Defining equations Constitutive equations Assumptions and constraints Initial and boundary conditions Classical constraints and kinematic equations Mathematical models are composed of relationships and variables. Relationships can be described by operators, such as algebraic operators, differential operators, etc. Variables are abstractions of system parameters of interest. Several classification criteria can be used for mathematical models according to their structure: Linear vs. nonlinear: If all the operators in a mathematical model exhibit linearity, the resulting mathematical model is defined as linear. A model is considered to be nonlinear otherwise; the definition of linearity and nonlinearity is dependent on context, linear models may have nonlinear expressions in them. For example, in a statistical linear model, it is assumed that a relationship is linear in the parameters, but it may be nonlinear in the predictor variables.
A differential equation is said to be linear if it can be written with linear differential operators, but it can still have nonlinear expressions in it. In a mathematical programming model, if the objective functions and constraints are represented by linear equations the model is regarded as a linear model. If one or more of the objective functions or constraints are represented with a nonlinear equation the model is known as a nonlinear model. Nonlinearity in simple systems, is associated with phenomena such as chaos and irreversibility. Although there are exceptions, nonlinear systems and models tend to be more difficult to study than linear ones. A common approach to nonlinear problems is linearization, but this can be problematic if one is trying to study aspects such as irreversibility, which are tied to nonlinearity. Static vs. dynamic: A dynamic model accounts for time-dependent changes in the state of the system, while a static model calculates the system in equilibrium, thus is time-invariant.
Dynamic models are represented by differential equations or difference equations. Explicit vs. implicit: If all of the input parameters of the overall model are known, the output parameters can be calculated by a finite series of computations, the model is said to be explicit. But sometimes it is the output parameters which are known, the corresponding inputs must be solved for by an iterative procedure, such as Newton's method or Broyden's method. In such a case the model is said to be implicit. For example, a jet engine's physical properties such as turbine and nozzle throat areas can be explicitly calculated given a design thermodynamic cycle at a specific flight condition and power setting, but the engine's operating cycles at other flight conditions and power settings cannot be explicitly calculated from the constant physical properties. Discrete vs. continuous: A discrete model treats objects as discrete, such as the particles in a molecular model or the states in a statistical model.
Deterministic vs. probabilistic: A deterministic model is one in which every set of variable states is uniquely determined by parameters in the model and by sets of previous states of these variables. Conversely, in a stochastic model—usually called a "statistical model"—randomness is present, variable states are not described by unique values, but rather by probability distributions. Deductive, inductive, or floating: A deductive model is a logical structure based on a theory. An inductive model arises from empirical findings and generalization from them; the floating model rests on neither theory nor observation, but is the invocation of expected structure. Application of mathematics in social sciences outside of economics has been criticized for unfounded models. Application of catastrophe theory in science has been characterized as a floating model. Mathematical models are of great importance in the natural sciences in physics. Physical theories are invariably expressed using mathematic
Center of mass
In physics, the center of mass of a distribution of mass in space is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may be applied to cause a linear acceleration without an angular acceleration. Calculations in mechanics are simplified when formulated with respect to the center of mass, it is a hypothetical point where entire mass of an object may be assumed to be concentrated to visualise its motion. In other words, the center of mass is the particle equivalent of a given object for application of Newton's laws of motion. In the case of a single rigid body, the center of mass is fixed in relation to the body, if the body has uniform density, it will be located at the centroid; the center of mass may be located outside the physical body, as is sometimes the case for hollow or open-shaped objects, such as a horseshoe. In the case of a distribution of separate bodies, such as the planets of the Solar System, the center of mass may not correspond to the position of any individual member of the system.
The center of mass is a useful reference point for calculations in mechanics that involve masses distributed in space, such as the linear and angular momentum of planetary bodies and rigid body dynamics. In orbital mechanics, the equations of motion of planets are formulated as point masses located at the centers of mass; the center of mass frame is an inertial frame in which the center of mass of a system is at rest with respect to the origin of the coordinate system. The concept of "center of mass" in the form of the center of gravity was first introduced by the great ancient Greek physicist and engineer Archimedes of Syracuse, he worked with simplified assumptions about gravity that amount to a uniform field, thus arriving at the mathematical properties of what we now call the center of mass. Archimedes showed that the torque exerted on a lever by weights resting at various points along the lever is the same as what it would be if all of the weights were moved to a single point—their center of mass.
In work on floating bodies he demonstrated that the orientation of a floating object is the one that makes its center of mass as low as possible. He developed mathematical techniques for finding the centers of mass of objects of uniform density of various well-defined shapes. Mathematicians who developed the theory of the center of mass include Pappus of Alexandria, Guido Ubaldi, Francesco Maurolico, Federico Commandino, Simon Stevin, Luca Valerio, Jean-Charles de la Faille, Paul Guldin, John Wallis, Louis Carré, Pierre Varignon, Alexis Clairaut. Newton's second law is reformulated with respect to the center of mass in Euler's first law; the center of mass is the unique point at the center of a distribution of mass in space that has the property that the weighted position vectors relative to this point sum to zero. In analogy to statistics, the center of mass is the mean location of a distribution of mass in space. In the case of a system of particles Pi, i = 1, …, n , each with mass mi that are located in space with coordinates ri, i = 1, …, n , the coordinates R of the center of mass satisfy the condition ∑ i = 1 n m i = 0.
Solving this equation for R yields the formula R = 1 M ∑ i = 1 n m i r i, where M is the sum of the masses of all of the particles. If the mass distribution is continuous with the density ρ within a solid Q the integral of the weighted position coordinates of the points in this volume relative to the center of mass R over the volume V is zero, ∭ Q ρ d V = 0. Solve this equation for the coordinates R to obtain R = 1 M ∭ Q ρ r d V, where M is the total mass in the volume. If a continuous mass distribution has uniform density, which means ρ is constant the center of mass is the same as the centroid of the volume; the coordinates R of the center of mass of a two-particle system, P1 and P2, with masses m1 and m2 is given by R = 1 m 1 + m 2. Let the percentage of the total mass divided between these two particles vary from 100% P1 and 0% P2 through 50% P1 and 50% P2 to 0% P1 and 100% P2 the center of mass R moves along the line from P1 to P2; the percentages of mass at each point can be viewed as projective coordinates of the point R on this line, are termed barycentric coordinates.
Another way of interpreting the process here is the mechanical balancing of moments about an arbitrary point. The numerator gives the total moment, balanced by an equivalent total force at the center of mass; this can be generalized
In mathematics, a conic section is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, the ellipse; the circle is a special case of the ellipse, is of sufficient interest in its own right that it was sometimes called a fourth type of conic section. The conic sections have been studied by the ancient Greek mathematicians with this work culminating around 200 BC, when Apollonius of Perga undertook a systematic study of their properties; the conic sections of the Euclidean plane have various distinguishing properties. Many of these have been used as the basis for a definition of the conic sections. One such property defines a non-circular conic to be the set of those points whose distances to some particular point, called a focus, some particular line, called a directrix, are in a fixed ratio, called the eccentricity; the type of conic is determined by the value of the eccentricity. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2.
This equation may be written in matrix form, some geometric properties can be studied as algebraic conditions. In the Euclidean plane, the conic sections appear to be quite different from one another, but share many properties. By extending the geometry to a projective plane this apparent difference vanishes, the commonality becomes evident. Further extension, by expanding the real coordinates to admit complex coordinates, provides the means to see this unification algebraically; the conic sections have been studied for thousands of years and have provided a rich source of interesting and beautiful results in Euclidean geometry. A conic is the curve obtained as the intersection of a plane, called the cutting plane, with the surface of a double cone, it shall be assumed that the cone is a right circular cone for the purpose of easy description, but this is not required. Planes that pass through the vertex of the cone will intersect the cone in a point, a line or a pair of intersecting lines; these are called degenerate conics and some authors do not consider them to be conics at all.
Unless otherwise stated, "conic" in this article will refer to a non-degenerate conic. There are three types of conics, the ellipse and hyperbola; the circle is a special kind of ellipse, although it had been considered as a fourth type. The circle and the ellipse arise when the intersection of the plane is a closed curve; the circle is obtained when the cutting plane is parallel to the plane of the generating circle of the cone – for a right cone, see diagram, this means that the cutting plane is perpendicular to the symmetry axis of the cone. If the cutting plane is parallel to one generating line of the cone the conic is unbounded and is called a parabola. In the remaining case, the figure is a hyperbola. In this case, the plane will intersect both halves of the cone, producing two separate unbounded curves. A property that the conic sections share is presented as the following definition. A conic section is the locus of all points P whose distance to a fixed point F is a constant multiple of the distance from P to a fixed line L.
For 0 < e < 1 we obtain an ellipse, for e = 1 a parabola, for e > 1 a hyperbola. A circle is not defined by a focus and directrix, in the plane; the eccentricity of a circle is defined to be zero and its focus is the center of the circle, but there is no line in the Euclidean plane, its directrix. An ellipse and a hyperbola each have distinct directrices for each of them; the line joining the foci is called the principal axis and the points of intersection of the conic with the principal axis are called the vertices of the conic. The line segment joining the vertices of a conic is called the major axis called transverse axis in the hyperbola; the midpoint of this line segment is called the center of the conic. Let a denote the distance from the center to a vertex of an ellipse or hyperbola; the distance from the center to a directrix is a/e while the distance from the center to a focus is ae. A parabola does not have a center; the eccentricity of an ellipse can be seen as a measure of how far the ellipse deviates from being circular.
If the angle between the surface of the cone and its axis is β and the angle between the cutting plane and the axis is α, the eccentricity is cos α cos β. A proof that the conic sections given by the focus-directrix property are the same as those given by planes intersecting a cone is facilitated by the use of Dandelin spheres. Various parameters are associated with a conic section. Recall that the principal axis is the line joining the foci of an ellipse or hyperbola, the center in these cases is the midpoint of the line segment joining the foci; some of the other common features and/or. The linear eccentricity is the distance between the focus; the latus rectum is the chord parallel to the directrix and passing through the focus. Its length is denoted by 2ℓ; the semi-latus rectum is half of the length of the latus rec
Projective geometry is a topic in mathematics. It is the study of geometric properties that are invariant with respect to projective transformations; this means that, compared to elementary geometry, projective geometry has a different setting, projective space, a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean space, for a given dimension, that geometric transformations are permitted that transform the extra points to Euclidean points, vice versa. Properties meaningful for projective geometry are respected by this new idea of transformation, more radical in its effects than can be expressed by a transformation matrix and translations; the first issue for geometers is. It is not possible to refer to angles in projective geometry as it is in Euclidean geometry, because angle is an example of a concept not invariant with respect to projective transformations, as is seen in perspective drawing. One source for projective geometry was indeed the theory of perspective.
Another difference from elementary geometry is the way in which parallel lines can be said to meet in a point at infinity, once the concept is translated into projective geometry's terms. Again this notion has an intuitive basis, such as railway tracks meeting at the horizon in a perspective drawing. See projective plane for the basics of projective geometry in two dimensions. While the ideas were available earlier, projective geometry was a development of the 19th century; this included the theory of complex projective space, the coordinates used being complex numbers. Several major types of more abstract mathematics were based on projective geometry, it was a subject with a large number of practitioners for its own sake, as synthetic geometry. Another topic that developed from axiomatic studies of projective geometry is finite geometry; the topic of projective geometry is itself now divided into many research subtopics, two examples of which are projective algebraic geometry and projective differential geometry.
Projective geometry is an elementary non-metrical form of geometry, meaning that it is not based on a concept of distance. In two dimensions it begins with the study of configurations of lines; that there is indeed some geometric interest in this sparse setting was first established by Desargues and others in their exploration of the principles of perspective art. In higher dimensional spaces there are considered hyperplanes, other linear subspaces, which exhibit the principle of duality; the simplest illustration of duality is in the projective plane, where the statements "two distinct points determine a unique line" and "two distinct lines determine a unique point" show the same structure as propositions. Projective geometry can be seen as a geometry of constructions with a straight-edge alone. Since projective geometry excludes compass constructions, there are no circles, no angles, no measurements, no parallels, no concept of intermediacy, it was realised. For example, the different conic sections are all equivalent in projective geometry, some theorems about circles can be considered as special cases of these general theorems.
During the early 19th century the work of Jean-Victor Poncelet, Lazare Carnot and others established projective geometry as an independent field of mathematics. Its rigorous foundations were addressed by Karl von Staudt and perfected by Italians Giuseppe Peano, Mario Pieri, Alessandro Padoa and Gino Fano during the late 19th century. Projective geometry, like affine and Euclidean geometry, can be developed from the Erlangen program of Felix Klein. After much work on the large number of theorems in the subject, the basics of projective geometry became understood; the incidence structure and the cross-ratio are fundamental invariants under projective transformations. Projective geometry can be modeled by the affine plane plus a line "at infinity" and treating that line as "ordinary". An algebraic model for doing projective geometry in the style of analytic geometry is given by homogeneous coordinates. On the other hand, axiomatic studies revealed the existence of non-Desarguesian planes, examples to show that the axioms of incidence can be modelled by structures not accessible to reasoning through homogeneous coordinate systems.
In a foundational sense, projective geometry and ordered geometry are elementary since they involve a minimum of axioms and either can be used as the foundation for affine and Euclidean geometry. Projective geometry is not ``; the first geometrical properties of a projective nature were discovered during the 3rd century by Pappus of Alexandria. Filippo Brunelleschi started investigating the geometry of perspective during 1425. Johannes Kepler and Gérard Desargues independently developed the concept of the "point at infinity". Desarg