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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
using a
coordinate system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the position of the points or other geometric elements on a manifold such as Euclidean space. The coordinates are ...
. This contrasts with
synthetic geometry Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is geometry without the use of coordinates. It relies on the axiomatic method for proving all results from a few basic properties initially called postulates ...
. Analytic geometry is used in
physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
and
engineering Engineering is the practice of using natural science, mathematics, and the engineering design process to Problem solving#Engineering, solve problems within technology, increase efficiency and productivity, and improve Systems engineering, s ...
, and also in
aviation Aviation includes the activities surrounding mechanical flight and the aircraft industry. ''Aircraft'' include fixed-wing and rotary-wing types, morphable wings, wing-less lifting bodies, as well as lighter-than-air aircraft such as h ...
,
rocketry Rocketry may refer to: Science and technology * The design and construction of rockets ** The hobbyist or (semi-)professional use of model rockets * Aerospace engineering Aerospace engineering is the primary field of engineering concerned wit ...
,
space science Space is a three-dimensional continuum containing positions and directions. In classical physics, physical space is often conceived in three linear dimensions. Modern physicists usually consider it, with time, to be part of a boundless ...
, and
spaceflight Spaceflight (or space flight) is an application of astronautics to fly objects, usually spacecraft, into or through outer space, either with or without humans on board. Most spaceflight is uncrewed and conducted mainly with spacecraft such ...
. It is the foundation of most modern fields of geometry, including algebraic, differential,
discrete Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory * Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit * Discrete group, ...
and computational geometry. Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and circles, often in two and sometimes three dimensions. Geometrically, one studies the Euclidean plane (two dimensions) and Euclidean space. As taught in school books, analytic geometry can be explained more simply: it is concerned with defining and representing geometric shapes in a numerical way and extracting numerical information from shapes' numerical definitions and representations. That the algebra of the real numbers can be employed to yield results about the linear continuum of geometry relies on the
Cantor–Dedekind axiom In mathematical logic, the Cantor–Dedekind axiom is the thesis that the real numbers are order-isomorphic to the linear continuum of geometry. In other words, the axiom states that there is a one-to-one correspondence between real numbers and po ...
.


History


Ancient Greece

The
Greek Greek may refer to: Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group *Greek language, a branch of the Indo-European language family **Proto-Greek language, the assumed last common ancestor of all kno ...
mathematician
Menaechmus Menaechmus (, c. 380 – c. 320 BC) was an ancient Greek mathematician, list of geometers, geometer and philosopher born in Alopeconnesus or Prokonnesos in the Thracian Chersonese, who was known for his friendship with the renowned philosopher P ...
solved problems and proved theorems by using a method that had a strong resemblance to the use of coordinates and it has sometimes been maintained that he had introduced analytic geometry.
Apollonius of Perga Apollonius of Perga ( ; ) was an ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the earlier contributions of Euclid and Archimedes on the topic, he brought them to the state prior to the invention o ...
, in '' On Determinate Section'', dealt with problems in a manner that may be called an analytic geometry of one dimension; with the question of finding points on a line that were in a ratio to the others. Apollonius in the ''Conics'' further developed a method that is so similar to analytic geometry that his work is sometimes thought to have anticipated the work of Descartes by some 1800 years. His application of reference lines, a diameter and a tangent is essentially no different from our modern use of a coordinate frame, where the distances measured along the diameter from the point of tangency are the abscissas, and the segments parallel to the tangent and intercepted between the axis and the curve are the ordinates. He further developed relations between the abscissas and the corresponding ordinates that are equivalent to rhetorical equations (expressed in words) of curves. However, although Apollonius came close to developing analytic geometry, he did not manage to do so since he did not take into account negative magnitudes and in every case the coordinate system was superimposed upon a given curve ''a posteriori'' instead of ''a priori''. That is, equations were determined by curves, but curves were not determined by equations. Coordinates, variables, and equations were subsidiary notions applied to a specific geometric situation.


Persia

The 11th-century Persian mathematician
Omar Khayyam Ghiyāth al-Dīn Abū al-Fatḥ ʿUmar ibn Ibrāhīm Nīshābūrī (18 May 1048 – 4 December 1131) (Persian language, Persian: غیاث الدین ابوالفتح عمر بن ابراهیم خیام نیشابورﻯ), commonly known as Omar ...
saw a strong relationship between geometry and algebra and was moving in the right direction when he helped close the gap between numerical and geometric algebra with his geometric solution of the general
cubic equation In algebra, a cubic equation in one variable is an equation of the form ax^3+bx^2+cx+d=0 in which is not zero. The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of th ...
s, but the decisive step came later with Descartes. Omar Khayyam is credited with identifying the foundations of
algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
, and his book ''Treatise on Demonstrations of Problems of Algebra'' (1070), which laid down the principles of analytic geometry, is part of the body of Persian mathematics that was eventually transmitted to Europe. Because of his thoroughgoing geometrical approach to algebraic equations, Khayyam can be considered a precursor to Descartes in the invention of analytic geometry.Cooper, G. (2003). Journal of the American Oriental Society,123(1), 248-249.


Western Europe

Analytic geometry was independently invented by
René Descartes René Descartes ( , ; ; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and Modern science, science. Mathematics was paramou ...
and
Pierre de Fermat Pierre de Fermat (; ; 17 August 1601 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his d ...
, although Descartes is sometimes given sole credit. ''Cartesian geometry'', the alternative term used for analytic geometry, is named after Descartes. Descartes made significant progress with the methods in an essay titled ''
La Géométrie ''La Géométrie'' () was published in 1637 as an appendix to ''Discours de la méthode'' ('' Discourse on the Method''), written by René Descartes. In the ''Discourse'', Descartes presents his method for obtaining clarity on any subject. ''La ...
(Geometry)'', one of the three accompanying essays (appendices) published in 1637 together with his ''Discourse on the Method for Rightly Directing One's Reason and Searching for Truth in the Sciences'', commonly referred to as ''
Discourse on Method ''Discourse on the Method of Rightly Conducting One's Reason and of Seeking Truth in the Sciences'' () is a philosophical and autobiographical treatise published by René Descartes in 1637. It is best known as the source of the famous quotation ...
''. ''La Geometrie'', written in his native French tongue, and its philosophical principles, provided a foundation for
calculus Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the ...
in Europe. Initially the work was not well received, due, in part, to the many gaps in arguments and complicated equations. Only after the translation into
Latin Latin ( or ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken by the Latins (Italic tribe), Latins in Latium (now known as Lazio), the lower Tiber area aroun ...
and the addition of commentary by van Schooten in 1649 (and further work thereafter) did Descartes's masterpiece receive due recognition.
Pierre de Fermat Pierre de Fermat (; ; 17 August 1601 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his d ...
also pioneered the development of analytic geometry. Although not published in his lifetime, a manuscript form of ''Ad locos planos et solidos isagoge'' (Introduction to Plane and Solid Loci) was circulating in Paris in 1637, just prior to the publication of Descartes' ''Discourse''. Clearly written and well received, the ''Introduction'' also laid the groundwork for analytical geometry. The key difference between Fermat's and Descartes' treatments is a matter of viewpoint: Fermat always started with an algebraic equation and then described the geometric curve that satisfied it, whereas Descartes started with geometric curves and produced their equations as one of several properties of the curves. As a consequence of this approach, Descartes had to deal with more complicated equations and he had to develop the methods to work with polynomial equations of higher degree. It was
Leonhard Euler Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
who first applied the coordinate method in a systematic study of space curves and surfaces.


Coordinates

In analytic geometry, the plane is given a coordinate system, by which every point has a pair of
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
coordinates. Similarly,
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
is given coordinates where every point has three coordinates. The value of the coordinates depends on the choice of the initial point of origin. There are a variety of coordinate systems used, but the most common are the following: Stewart, James (2008). ''Calculus: Early Transcendentals'', 6th ed., Brooks Cole Cengage Learning.


Cartesian coordinates (in a plane or space)

The most common coordinate system to use is the
Cartesian coordinate system In geometry, a Cartesian coordinate system (, ) in a plane (geometry), plane is a coordinate system that specifies each point (geometry), point uniquely by a pair of real numbers called ''coordinates'', which are the positive and negative number ...
, where each point has an ''x''-coordinate representing its horizontal position, and a ''y''-coordinate representing its vertical position. These are typically written as an
ordered pair In mathematics, an ordered pair, denoted (''a'', ''b''), is a pair of objects in which their order is significant. The ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a''), unless ''a'' = ''b''. In contrast, the '' unord ...
(''x'', ''y''). This system can also be used for three-dimensional geometry, where every point in
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
is represented by an ordered triple of coordinates (''x'', ''y'', ''z'').


Polar coordinates (in a plane)

In
polar coordinates In mathematics, the polar coordinate system specifies a given point (mathematics), point in a plane (mathematics), plane by using a distance and an angle as its two coordinate system, coordinates. These are *the point's distance from a reference ...
, every point of the plane is represented by its distance ''r'' from the origin and its
angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...
''θ'', with ''θ'' normally measured counterclockwise from the positive ''x''-axis. Using this notation, points are typically written as an ordered pair (''r'', ''θ''). One may transform back and forth between two-dimensional Cartesian and polar coordinates by using these formulae: x = r\, \cos\theta,\, y = r\, \sin\theta; \, r = \sqrt,\, \theta = \arctan(y/x). This system may be generalized to three-dimensional space through the use of cylindrical or spherical coordinates.


Cylindrical coordinates (in a space)

In
cylindrical coordinates A cylinder () has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infinite ...
, every point of space is represented by its height ''z'', its
radius In classical geometry, a radius (: radii or radiuses) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The radius of a regular polygon is th ...
''r'' from the ''z''-axis and the
angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...
''θ'' its projection on the ''xy''-plane makes with respect to the horizontal axis.


Spherical coordinates (in a space)

In spherical coordinates, every point in space is represented by its distance ''ρ'' from the origin, the
angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...
''θ'' its projection on the ''xy''-plane makes with respect to the horizontal axis, and the angle ''φ'' that it makes with respect to the ''z''-axis. The names of the angles are often reversed in physics.


Equations and curves

In analytic geometry, any
equation In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for ...
involving the coordinates specifies a
subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
of the plane, namely the solution set for the equation, or locus. For example, the equation ''y'' = ''x'' corresponds to the set of all the points on the plane whose ''x''-coordinate and ''y''-coordinate are equal. These points form a line, and ''y'' = ''x'' is said to be the equation for this line. In general, linear equations involving ''x'' and ''y'' specify lines,
quadratic equation In mathematics, a quadratic equation () is an equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where the variable (mathematics), variable represents an unknown number, and , , and represent known numbers, where . (If and ...
s specify
conic section A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, tho ...
s, and more complicated equations describe more complicated figures.Percey Franklyn Smith, Arthur Sullivan Gale (1905)''Introduction to Analytic Geometry'', Athaeneum Press Usually, a single equation corresponds to a
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
on the plane. This is not always the case: the trivial equation ''x'' = ''x'' specifies the entire plane, and the equation ''x''2 + ''y''2 = 0 specifies only the single point (0, 0). In three dimensions, a single equation usually gives a
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
, and a curve must be specified as the
intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...
of two surfaces (see below), or as a system of
parametric equation In mathematics, a parametric equation expresses several quantities, such as the coordinates of a point (mathematics), point, as Function (mathematics), functions of one or several variable (mathematics), variables called parameters. In the case ...
s. The equation ''x''2 + ''y''2 = ''r''2 is the equation for any circle centered at the origin (0, 0) with a radius of r.


Lines and planes

Lines in a Cartesian plane, or more generally, in affine coordinates, can be described algebraically by ''linear'' equations. In two dimensions, the equation for non-vertical lines is often given in the '' slope-intercept form'': y = mx + b where: * ''m'' is the
slope In mathematics, the slope or gradient of a Line (mathematics), line is a number that describes the direction (geometry), direction of the line on a plane (geometry), plane. Often denoted by the letter ''m'', slope is calculated as the ratio of t ...
or
gradient In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The g ...
of the line. * ''b'' is the y-intercept of the line. * ''x'' is the
independent variable A variable is considered dependent if it depends on (or is hypothesized to depend on) an independent variable. Dependent variables are studied under the supposition or demand that they depend, by some law or rule (e.g., by a mathematical function ...
of the function ''y'' = ''f''(''x''). In a manner analogous to the way lines in a two-dimensional space are described using a point-slope form for their equations, planes in a three dimensional space have a natural description using a point in the plane and a vector orthogonal to it (the
normal vector In geometry, a normal is an object (e.g. a line, ray, or vector) that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the infinite straight line perpendicular to the tangent line to the cu ...
) to indicate its "inclination". Specifically, let \mathbf_0 be the position vector of some point P_0 = (x_0,y_0,z_0), and let \mathbf = (a,b,c) be a nonzero vector. The plane determined by this point and vector consists of those points P, with position vector \mathbf, such that the vector drawn from P_0 to P is perpendicular to \mathbf. Recalling that two vectors are perpendicular if and only if their dot product is zero, it follows that the desired plane can be described as the set of all points \mathbf such that \mathbf \cdot (\mathbf-\mathbf_0) =0. (The dot here means a
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a Scalar (mathematics), scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. N ...
, not scalar multiplication.) Expanded this becomes a (x-x_0)+ b(y-y_0)+ c(z-z_0)=0, This is just a
linear equation In mathematics, a linear equation is an equation that may be put in the form a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coeffici ...
: ax + by + cz + d = 0, \text d = -(ax_0 + by_0 + cz_0). Conversely, it is easily shown that if ''a'', ''b'', ''c'' and ''d'' are constants and ''a'', ''b'', and ''c'' are not all zero, then the graph of the equation ax + by + cz + d = 0, This familiar equation for a plane is called the ''general form'' of the equation of the plane. In three dimensions, lines can ''not'' be described by a single linear equation, so they are frequently described by
parametric equation In mathematics, a parametric equation expresses several quantities, such as the coordinates of a point (mathematics), point, as Function (mathematics), functions of one or several variable (mathematics), variables called parameters. In the case ...
s: x = x_0 + at y = y_0 + bt z = z_0 + ct where: * ''x'', ''y'', and ''z'' are all functions of the independent variable ''t'' which ranges over the real numbers. * (''x''0, ''y''0, ''z''0) is any point on the line. * ''a'', ''b'', and ''c'' are related to the slope of the line, such that the
vector Vector most often refers to: * Euclidean vector, a quantity with a magnitude and a direction * Disease vector, an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematics a ...
(''a'', ''b'', ''c'') is parallel to the line.


Conic sections

In the
Cartesian coordinate system In geometry, a Cartesian coordinate system (, ) in a plane (geometry), plane is a coordinate system that specifies each point (geometry), point uniquely by a pair of real numbers called ''coordinates'', which are the positive and negative number ...
, the
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...
of a
quadratic equation In mathematics, a quadratic equation () is an equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where the variable (mathematics), variable represents an unknown number, and , , and represent known numbers, where . (If and ...
in two variables is always a conic section – though it may be degenerate, and all conic sections arise in this way. The equation will be of the form Ax^2 + Bxy + Cy^2 +Dx + Ey + F = 0\textA, B, C\text As scaling all six constants yields the same locus of zeros, one can consider conics as points in the five-dimensional
projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
\mathbf^5. The conic sections described by this equation can be classified using the
discriminant In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the zero of a function, roots without computing them. More precisely, it is a polynomial function of the coef ...
B^2 - 4AC . If the conic is non-degenerate, then: * if B^2 - 4AC < 0 , the equation represents an
ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
; ** if A = C and B = 0 , the equation represents a
circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
, which is a special case of an ellipse; * if B^2 - 4AC = 0 , the equation represents a
parabola In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exactl ...
; * if B^2 - 4AC > 0 , the equation represents a
hyperbola In mathematics, a hyperbola is a type of smooth function, smooth plane curve, curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected component ( ...
; ** if we also have A + C = 0 , the equation represents a rectangular hyperbola.


Quadric surfaces

A quadric, or quadric surface, is a ''2''-dimensional
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
in 3-dimensional space defined as the locus of zeros of a
quadratic polynomial In mathematics, a quadratic function of a single variable is a function of the form :f(x)=ax^2+bx+c,\quad a \ne 0, where is its variable, and , , and are coefficients. The expression , especially when treated as an object in itself rather tha ...
. In coordinates , the general quadric is defined by the
algebraic equation In mathematics, an algebraic equation or polynomial equation is an equation of the form P = 0, where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For example, x^5-3x+1=0 is an algebraic equati ...
Silvio Lev
Quadrics
in "Geometry Formulas and Facts", excerpted from 30th Edition of ''CRC Standard Mathematical Tables and Formulas'',
CRC Press The CRC Press, LLC is an American publishing group that specializes in producing technical books. Many of their books relate to engineering, science and mathematics. Their scope also includes books on business, forensics and information technol ...
, from
The Geometry Center The Geometry Center was a mathematics research and education center at the University of Minnesota. It was established by the National Science Foundation in the late 1980s and closed in 1998. The focus of the center's work was the use of computer ...
at
University of Minnesota The University of Minnesota Twin Cities (historically known as University of Minnesota) is a public university, public Land-grant university, land-grant research university in the Minneapolis–Saint Paul, Twin Cities of Minneapolis and Saint ...
\sum_^ x_i Q_ x_j + \sum_^ P_i x_i + R = 0. Quadric surfaces include
ellipsoid An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional Scaling (geometry), scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface;  that is, a Surface (mathemat ...
s (including the
sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
),
paraboloid In geometry, a paraboloid is a quadric surface that has exactly one axial symmetry, axis of symmetry and no central symmetry, center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar p ...
s,
hyperboloid In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by def ...
s,
cylinder A cylinder () has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infinite ...
s, cones, and planes.


Distance and angle

In analytic geometry, geometric notions such as
distance Distance is a numerical or occasionally qualitative measurement of how far apart objects, points, people, or ideas are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two co ...
and
angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...
measure are defined using
formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwe ...
s. These definitions are designed to be consistent with the underlying
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
. For example, using
Cartesian coordinates In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...
on the plane, the distance between two points (''x''1, ''y''1) and (''x''2, ''y''2) is defined by the formula d = \sqrt, which can be viewed as a version of the
Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
. Similarly, the angle that a line makes with the horizontal can be defined by the formula \theta = \arctan(m), where ''m'' is the
slope In mathematics, the slope or gradient of a Line (mathematics), line is a number that describes the direction (geometry), direction of the line on a plane (geometry), plane. Often denoted by the letter ''m'', slope is calculated as the ratio of t ...
of the line. In three dimensions, distance is given by the generalization of the Pythagorean theorem: d = \sqrt, while the angle between two vectors is given by the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a Scalar (mathematics), scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. N ...
. The dot product of two Euclidean vectors A and B is defined by \mathbf A\cdot\mathbf B \stackrel \left\, \mathbf A\right\, \left\, \mathbf B\right\, \cos\theta, where ''θ'' is the
angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...
between A and B.


Transformations

Transformations are applied to a parent function to turn it into a new function with similar characteristics. The graph of R(x,y) is changed by standard transformations as follows: * Changing x to x-h moves the graph to the right h units. * Changing y to y-k moves the graph up k units. * Changing x to x/b stretches the graph horizontally by a factor of b. (think of the x as being dilated) * Changing y to y/a stretches the graph vertically. * Changing x to x\cos A+ y\sin A and changing y to -x\sin A + y\cos A rotates the graph by an angle A. There are other standard transformation not typically studied in elementary analytic geometry because the transformations change the shape of objects in ways not usually considered. Skewing is an example of a transformation not usually considered. For more information, consult the Wikipedia article on affine transformations. For example, the parent function y=1/x has a horizontal and a vertical asymptote, and occupies the first and third quadrant, and all of its transformed forms have one horizontal and vertical asymptote, and occupies either the 1st and 3rd or 2nd and 4th quadrant. In general, if y=f(x), then it can be transformed into y=af(b(x-k))+h. In the new transformed function, a is the factor that vertically stretches the function if it is greater than 1 or vertically compresses the function if it is less than 1, and for negative a values, the function is reflected in the x-axis. The b value compresses the graph of the function horizontally if greater than 1 and stretches the function horizontally if less than 1, and like a, reflects the function in the y-axis when it is negative. The k and h values introduce translations, h, vertical, and k horizontal. Positive h and k values mean the function is translated to the positive end of its axis and negative meaning translation towards the negative end. Transformations can be applied to any geometric equation whether or not the equation represents a function. Transformations can be considered as individual transactions or in combinations. Suppose that R(x,y) is a relation in the xy plane. For example, x^2+y^2-1=0 is the relation that describes the unit circle.


Finding intersections of geometric objects

For two geometric objects P and Q represented by the relations P(x,y) and Q(x,y) the intersection is the collection of all points (x,y) which are in both relations.While this discussion is limited to the xy-plane, it can easily be extended to higher dimensions. For example, P might be the circle with radius 1 and center (0,0): P = \ and Q might be the circle with radius 1 and center (1,0): Q = \. The intersection of these two circles is the collection of points which make both equations true. Does the point (0,0) make both equations true? Using (0,0) for (x,y), the equation for Q becomes (0-1)^2+0^2=1 or (-1)^2=1 which is true, so (0,0) is in the relation Q. On the other hand, still using (0,0) for (x,y) the equation for P becomes 0^2+0^2=1 or 0=1 which is false. (0,0) is not in P so it is not in the intersection. The intersection of P and Q can be found by solving the simultaneous equations: x^2+y^2 = 1 (x-1)^2+y^2 = 1. Traditional methods for finding intersections include substitution and elimination. Substitution: Solve the first equation for y in terms of x and then substitute the expression for y into the second equation: x^2+y^2 = 1 y^2=1-x^2. We then substitute this value for y^2 into the other equation and proceed to solve for x: (x-1)^2+(1-x^2)=1 x^2 -2x +1 +1 -x^2 =1 -2x = -1 x=1/2. Next, we place this value of x in either of the original equations and solve for y: (1/2)^2+y^2 = 1 y^2 =3/4 y = \frac. So our intersection has two points: \left(1/2,\frac\right) \;\; \text \;\; \left(1/2,\frac\right). Elimination: Add (or subtract) a multiple of one equation to the other equation so that one of the variables is eliminated. For our current example, if we subtract the first equation from the second we get (x-1)^2-x^2=0. The y^2 in the first equation is subtracted from the y^2 in the second equation leaving no y term. The variable y has been eliminated. We then solve the remaining equation for x, in the same way as in the substitution method: x^2 -2x +1 -x^2 =0 -2x = -1 x=1/2. We then place this value of x in either of the original equations and solve for y: (1/2)^2+y^2 = 1 y^2 = 3/4 y = \frac. So our intersection has two points: \left(1/2,\frac\right) \;\; \text \;\; \left(1/2,\frac\right). For conic sections, as many as 4 points might be in the intersection.


Finding intercepts

One type of intersection which is widely studied is the intersection of a geometric object with the x and y coordinate axes. The intersection of a geometric object and the y-axis is called the y-intercept of the object. The intersection of a geometric object and the x-axis is called the x-intercept of the object. For the line y=mx+b, the parameter b specifies the point where the line crosses the y axis. Depending on the context, either b or the point (0,b) is called the y-intercept.


Geometric axis

Axis in geometry is the perpendicular line to any line, object or a surface. Also for this may be used the common language use as a: normal (perpendicular) line, otherwise in engineering as ''axial line''. In
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, a normal is an object such as a line or vector that is
perpendicular In geometry, two geometric objects are perpendicular if they intersect at right angles, i.e. at an angle of 90 degrees or π/2 radians. The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', � ...
to a given object. For example, in the two-dimensional case, the normal line to a curve at a given point is the line perpendicular to the
tangent line In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...
to the curve at the point. In the three-dimensional case a surface normal, or simply normal, to a
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
at a point ''P'' is a
vector Vector most often refers to: * Euclidean vector, a quantity with a magnitude and a direction * Disease vector, an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematics a ...
that is
perpendicular In geometry, two geometric objects are perpendicular if they intersect at right angles, i.e. at an angle of 90 degrees or π/2 radians. The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', � ...
to the
tangent plane In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...
to that surface at ''P''. The word "normal" is also used as an adjective: a line normal to a plane, the normal component of a
force In physics, a force is an influence that can cause an Physical object, object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the Magnitu ...
, the normal vector, etc. The concept of normality generalizes to
orthogonality In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendicular'' is more specifically ...
.


Spherical and nonlinear planes and their tangents

Tangent is the linear approximation of a spherical or other curved or twisted line of a function.


Tangent lines and planes

In
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, the tangent line (or simply tangent) to a plane
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
at a given point is the straight line that "just touches" the curve at that point. Informally, it is a line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where ''f'' is the
derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
of ''f''. A similar definition applies to
space curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
s and curves in ''n''-dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
. As it passes through the point where the tangent line and the curve meet, called the point of tangency, the tangent line is "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point. Similarly, the tangent plane to a
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
at a given point is the plane that "just touches" the surface at that point. The concept of a tangent is one of the most fundamental notions in
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
and has been extensively generalized; see
Tangent space In mathematics, the tangent space of a manifold is a generalization of to curves in two-dimensional space and to surfaces in three-dimensional space in higher dimensions. In the context of physics the tangent space to a manifold at a point can be ...
.


See also

* Applied geometry *
Cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...
*
Rotation of axes In mathematics, a rotation of axes in two dimensions is a mapping from an ''xy''-Cartesian coordinate system to an ''x′y′''-Cartesian coordinate system in which the origin is kept fixed and the ''x′'' and ''y′'' axes ar ...
* Translation of axes *
Vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...


Notes


References


Books

* * * John Casey (1885
Analytic Geometry of the Point, Line, Circle, and Conic Sections
link from
Internet Archive The Internet Archive is an American 501(c)(3) organization, non-profit organization founded in 1996 by Brewster Kahle that runs a digital library website, archive.org. It provides free access to collections of digitized media including web ...
. * * Mikhail Postnikov (1982
Lectures in Geometry Semester I Analytic Geometry
via Internet Archive *


Articles

* * * * *


External links



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