What is Analytical Geomatry?

Answer 1

Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry, is the study of geometry using the principles of algebra. Usually the Cartesian coordinate system is applied to manipulate equations for planes, lines, curves, and circles, often in two and sometimes in three dimensions of measurement. As taught in school books, analytic geometry can be explained more simply: it is concerned with defining geometrical shapes in a numerical way, and extracting numerical information from that representation. The numerical output, however, might also be a vector or a shape. Some consider that the introduction of analytic geometry was the beginning of modern mathematics.

René Descartes is popularly regarded as having introduced the foundation for the methods of analytic geometry in 1637 in the appendix titled Geometry of the titled Discourse on the Method of Rightly Conducting the Reason in the Search for Truth in the Sciences, commonly referred to as Discourse on Method. This work, written in his native language (French), and its philosophical principles, provided the foundation for calculus in Europe.

The relationship between algebra and geometry is nontrivial. There exist proofs of this relationship based on the theory of geometry and notions of geometric length.

Answer 2

ANALYTIC GEOMETRY [analytic geometry] branch of geometry in which points are represented with respect to a coordinate system, such as Cartesian coordinates , and in which the approach to geometric problems is primarily algebraic. Its most common application is in the representation of equations involving two or three variables as curves in two or three dimensions or surfaces in three dimensions. For example, the linear equation ax + by + c =0 represents a straight line in the xy -plane, and the linear equation ax + by + cz + d =0 represents a plane in space, where a, b, c, and d are constant numbers (coefficients). In this way a geometric problem can be translated into an algebraic problem and the methods of algebra brought to bear on its solution. Conversely, the solution of a problem in algebra, such as finding the roots of an equation or system of equations, can be estimated or sometimes given exactly by geometric means, e.g., plotting curves and surfaces and determining points of intersection.

In plane analytic geometry a line is frequently described in terms of its slope, which expresses its inclination to the coordinate axes; technically, the slope m of a straight line is the (trigonometric) tangent of the angle it makes with the x -axis. If the line is parallel to the x -axis, its slope is zero. Two or more lines with equal slopes are parallel to one another. In general, the slope of the line through the points ( x1 , y1 ) and ( x2 , y2 ) is given by m = ( y2 - y1 ) / ( x2 - x1 ). The conic sections are treated in analytic geometry as the curves corresponding to the general quadratic equation ax2 + bxy + cy2 + dx + ey + f =0, where a, b, … , f are constants and a, b, and c are not all zero.

In solid analytic geometry the orientation of a straight line is given not by one slope but by its direction cosines, λ, μ, and ν, the cosines of the angles the line makes with the x-, y-, and z -axes, respectively; these satisfy the relationship λ 2 +μ 2 +ν 2 = 1. In the same way that the conic sections are studied in two dimensions, the 17 quadric surfaces, e.g., the ellipsoid, paraboloid, and elliptic paraboloid, are studied in solid analytic geometry in terms of the general equation ax2 + by2 + cz2 + dxy + exz + fyz + px + qy + rz + s =0.

The methods of analytic geometry have been generalized to four or more dimensions and have been combined with other branches of geometry. Analytic geometry was introduced by René Descartes in 1637 and was of fundamental importance in the development of the calculus by Sir Isaac Newton and G. W. Leibniz in the late 17th cent. More recently it has served as the basis for the modern development and exploitation of algebraic geometry .

Answer 3

s(-a million,a million)t(-3,6)u(5,-2) and midpoints between st is X=-2, 2.5 midpoint between su is Y=2, -a million.5 Your straightforward information is incorrect and inconsistent. Please superb it. As i will see it relatively is: s(-a million,a million), t(-3,6), u(5,-2) and midpoints bteween st is X=-2, 3.5 midpoint between su is Y=2, -0.5 we are able to certainly see after correction that XY isn't equivalent to TU as XY = squarert[4^2 + 4^2] = 4*squarert(2) and TU = squarert[8^2 + 8^2] = 8*squarert(2) -------------------- (a million) yet XY = (a million/2)*TU -----------------------------------(1a) yet XY and TU are parallel as [{-0.5-3.5}/{2-(-2)}] = -a million = [{-2-6}/{5-(-3)}] -------------- (2) As X and y are mid factors of ST and SU, Equations (1a) and a pair of teach that the traingles SXY and STU are comparable. as their corresponding facets are in a fastened ratio of two

Answer 4

I thought it was just regular geometry.