COGO includesPlane analytic geometryAnd solid analytic geometry.planeanalysisGeometric passing through planeright angleCoordinate systems, establishing points withreal number- correspondence between pairs, andCurves and EquationsCorrespondence betweenAlgebraMethods Study geometric problems, or use geometric methods to study algebraic problems.[1]
analytic geometry(English:analytic geometry）, also known asCoordinate geometry(English:coordinate geometry）OrCartesian geometry(English:Cartesian geometry）, previously known asCartesian geometry, is a branch of geometry that studies graphics by means of analytic expressions.Analytic geometry usually uses 2DRectangular coordinate system Study various general plane curves such as straight line, circle, conic curve, cycloid, star line, etc., and use 3DSpace rectangular coordinate systemTo studyplane、The BallAnd other general space surfaces, while studying theirequation, and define some graphic concepts andparameter。
history
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Menaechmus, an ancient Greek mathematician, solved and proved problems in a way very similar to the coordinate system now used, so that he was sometimes considered the ancestor of analytic geometry.Apollonius' way of solving problems in On Touching is now called single dimensional analytic geometry;He used straight lines to find the ratio between one point and other points.Apollonius further developed this method in his Theory of Conic Curve. This method is very similar to analytic geometry, which is more than 1800 years earlier than Descartes.He used reference lines, diameters and tangents, which are not fundamentally different from the existing coordinate system, that is, the distance measured along the diameter from the tangent point is the abscissa, while the line segment parallel to the tangent and intersecting with the number axis and curve is the ordinate.He further developed the relationship between abscissa and ordinate, that is, they are equivalent to exaggerated curves.However, Apollonius' work was close to analytic geometry, but it failed to complete it because he did not include negative numbers in the system.Here, the equation is determined by the curve, and the curve is not derived from the equation.Coordinates, variables and equations are just footnotes to some given geometric problems.
Omar Hayam, a mathematician in the Persian Empire in the eleventh century, discovered the close relationship between geometry and algebra, and used algebra and geometry in solving cubic equations, making great progress.But the most critical step was completed by Descartes.
Traditionally, analytic geometry was founded by Ren é Descartes.Descartes' pioneering work is recorded in Geometry（La Geometrie）He was published together with his Methodology in 1637.These efforts were written in French and founded on the philosophyInfinitesimalProvides the foundation.At first, these works were not recognized, partly because of the discontinuity in the discussion and the complexity of the equations.It was not until 1649 that the work was translated into Latin and complimented by van Schooten that it was accepted by the public.
FermatIt also contributes to the development of analytic geometry.His Introduction to Plane and Solid Tracks（Ad Locos Planos et Solidos Isagoge）Although not published before his death, the manuscript appeared in Paris in 1637, just a little earlier than Descartes Methodology.The introduction is clear and well received, providing the groundwork for analytic geometry.The difference between Fermat's method and Descartes' method lies in the starting point.Fermat starts with the algebraic formula and then describes its geometric curve, while Descartes starts with the geometric curve and ends with the equation.As a result, Descartes' method can deal with more complex equations, and develops to use higher order polynomials to solve problems.
development
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Since the 17th century, due tonavigation, astronomy, mechanicsEconomics、military、productionThe development of elementary geometry andelementary algebra The rapid development of, promoted the establishment of coordinate geometry, and was widely used inmathematicsEach branch of.Before the establishment of coordinate geometry, geometry and algebra were two independent branches.The establishment of coordinate geometry truly realizes the combination of geometric methods and algebraic methods for the first time, and unifies form and number. This ismathematicsA major breakthrough in the history of development.As the first decisive step in the development of variable mathematics, the establishment of coordinate geometry is very important forCalculusThe birth of has played an immeasurable role.
fundamental theory
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coordinate
In analytic geometry, the plane gives a coordinate system, that is, each point has a corresponding pair of real coordinates.The most common isCartesian coordinate system, where each point hasx-The coordinates correspond to the horizontal position, andy-The coordinates correspond to the vertical position.These are often written asOrdered pair(x,y)。This system can also be used in 3D geometry. Every point in space isMultivariate groupRendering（x,y,z)。
In analytic geometry, any equation contains a definite surfacesubsetThat is, the solution set of the equation.For example, equationy=xOn the plane, it corresponds to allx-Coordinates equal toy-The solution set of coordinates.These points converge into a straight line,y=xIs called the straight line of this equation.In summary, in the linear equationxandyDefine the line, the conic curve is defined by the quadratic equation of one variable, and the more complex equation describes the more complex image.
Usually, a simple equation corresponds to a curve in the plane.But this is not necessarily the case: equationx=xCorresponding to the whole plane, equationxtwo+ytwo=0 only corresponds to (0, 0) one point.In 3D space, an equation usually corresponds to a surface, and a curve often represents the intersection of two surfaces, or a parametric equation.equationxtwo+ytwo=rtwoRepresents all circles with radius r and center on (0,0).
Distance and angle
In analytic geometry, geometric concepts such as distance and angle areformulaTo express.These definitions and theEuclidean geometryThe underlying theme is consistent.For example, using a planeCartesian coordinate systemWhen, two points A（xone,yone)，B(xtwo,ytwo）Distance betweend(Also writing | AB |) is defined as
The above can be considered as aPythagorean theoremForm of.Similarly, the angle between a straight line and a horizontal line can be defined as
Changes can change the parent equation into a new equation, but maintain the original characteristics.For example, the parent equation
There are horizontal and vertical asymptotes, which can be located in the first and third quadrants. All its deformations have horizontal and vertical asymptotes, which appear in the first or third, second or fourth quadrants.In general, if y=f (x), it can become y=af [b (x-k)]+h.The new deformation equation, if the factor a is greater than 1, the vertical tension equation;If it is less than 1, we compress the equation.If the value of a is negative, then the equation is reflected on the x-axis.If the value of b is greater than 1, the horizontal compression equation will be used; if the value of b is less than 1, the tensile equation will be used.As with a, if it is negative, it will be reflected on the y-axis.The values of k and h are translation, h is vertical, and k is horizontal.Positive values of h and k mean that the equation moves in the positive direction of the number axis, and negative values mean that it moves in the negative direction of the number axis.
Variations can be applied to any geometric equation, whether or not the equation represents an equation.Change can be considered as individual processing or combination processing.
intersection
Although this discussion is limited to the plane xOy, it can easily be derived into higher dimensional space.Two geometric objects P and Q refer to P (x, y) and Q (x, y), and their intersection is the set of all points (x, y).
intercept
One kind of intersection that has been widely studied is the intersection of geometric objects and x and y coordinate axes.
The intersection of a geometric object and the y-axis is called the y of the object-intercept。The intersection with the x-axis is called the x-intercept of the object.
For the line y=mx+b, the parameter b defines where the line intersects the y-axis.Therefore, b or point (0, b) is called y-intercept.
Outer product finding a vector perpendicular to two known vectors (and their space volumes)
Intersection Problem
Modern analytic geometry
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Analytic cluster(analytical variety) is defined as severalanalytic functionCommon solution set of.Algebraic varieties similar to real numbers and complex numbers.Any complex manifold is an analytic family.Because resolution clusters may have singularities, not all resolution clusters are complex.
Analytic geometry is generally equivalent to real and complex algebraic geometry. In his book Algebraic Geometry and Analytic Geometry, Jean Pierre Searle（Géometrie Algébrique et Géométrie Analytique）This view is expounded.However, the two fields still have their uniqueness, and the way of proof is very different. Algebraic geometry also includes the finite characteristics of geometry.
