Sunday, 8 September 2013

Introduction of Analytical Geometory

Analytic geometry, or analytical geometry, has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties. This article focuses on the classical and elementary meaning.
In classical mathematics, analytic geometry, also known as coordinate geometry, or Cartesian geometry, is the study of geometry using a coordinate system and the principles of algebra and analysis. This contrasts with the synthetic approach of Euclidean geometry, which treats certain geometric notions as primitive, and uses deductive reasoning based on axioms and theorems to derive truth. Analytic geometry is widely used in physics and engineering, and is the foundation of most modern fields of geometry, including algebraic, differential, discrete, and computational geometry.
Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and squares, often in two and sometimes in three dimensions. Geometrically, one studies the Euclidean plane (2 dimensions) and Euclidean space (3 dimensions). As taught in school books, analytic geometry can be explained more simply: it is concerned with defining and representing geometrical shapes in a numerical way and extracting numerical information from shapes' numerical definitions and representations. The numerical output, however, might also be a vector or a shape. 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.


Basic principles


Coordinates

In analytic geometry, the plane is given a coordinate system, by which every point has a pair of real number coordinates. The most common coordinate system to use is the Cartesian coordinate system, 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 (xy). This system can also be used for three-dimensional geometry, where every point in Euclidean space is represented by an ordered triple of coordinates (xyz).
Other coordinate systems are possible. On the plane the most common alternative is polar coordinates, where every point is represented by its radius r from the origin and its angle θ. In three dimensions, common alternative coordinate systems include cylindrical coordinates and spherical coordinates.

Equations of curves

In analytic geometry, any equation involving the coordinates specifies a subset of the plane, namely the solution set for the equation. 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 equations specify conic sections, and more complicated equations describe more complicated figures.
Usually, a single equation corresponds to a curve on the plane. This is not always the case: the trivial equation x = x specifies the entire plane, and the equation x2 + y2 = 0 specifies only the single point (0, 0). In three dimensions, a single equation usually gives a surface, and a curve must be specified as the intersection of two surfaces (see below), or as a system of parametric equations. The equation x2 + y2 = r2 is the equation for any circle with a radius of r.

Distance and angle

In analytic geometry, geometric notions such as distance and angle measure are defined using formulas. These definitions are designed to be consistent with the underlying Euclidean geometry. For example, using Cartesian coordinates on the plane, the distance between two points (x1y1) and (x2y2) is defined by the formula
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2},\!
which can be viewed as a version of the Pythagorean theorem. Similarly, the angle that a line makes with the horizontal can be defined by the formula
\theta = \arctan(m)\!
where m is the slope of the line.


Transformations

Transformations are applied to parent functions to turn it into a new function with similar characteristics. 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
x2 + y2 -1= 0
is the relation that describes the unit circle. 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 xcosA+ ysinA and changing y to -xsinA + ycosA 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.

Intersections

While this discussion is limited to the xy-plane, it can easily be extended to higher dimensions. 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. For example, P might be the circle with radius 1 and center (0,0): P = {(x,y) | x2+y2=1} and Q might be the circle with radius 1 and center (1,0): Q = {(x,y) | (x-1)2+y2=1}. 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+02=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+02=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:
x2+y2 = 1
(x-1)2+y2 = 1
Traditional methods 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.
x2+y2 = 1
y2=1-x2 We then substitute this value for y2 into the other equation:
(x-1)2+(1-x2)=1 and proceed to solve for x:
x2 -2x +1 +1 -x2 =1
-2x = -1
x=½
We next place this value of x in either of the original equations and solve for y:
½2+y2 = 1
y2 = ¾
y = \frac{\pm \sqrt{3}}{2}
So that our intersection has two points:
 \left(1/2,\frac{+ \sqrt{3}}{2}\right) \;\;  \mathrm{and} \;\;  \left(1/2,\frac{-\sqrt{3}}{2}\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-x2=0 The y2 in the first equation is subtracted from the y2 in the second equation leaving no y term. y has been eliminated. We then solve the remaining equation for x, in the same way as in the substitution method. x2 -2x +1 +1 -x2 =1 -2x = -1 x=½ We next place this value of x in either of the original equations and solve for y: ½2+y2 = 1
y2 = ¾
y = \frac{\pm \sqrt{3}}{2}
So that our intersection has two points:
 \left(1/2,\frac{+ \sqrt{3}}{2}\right) \;\;  \mathrm{and} \;\; \left(1/2,\frac{-\sqrt{3}}{2}\right)
For conic sections, as many as 4 points might be in the intersection.

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.

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