Saturday, February 14, 2015

The Algebraic and parametric equation of conic section and its Application in Linear Algebra and Singular Value Decomposition and Image or Data compression.



Today i will start from the definition of circle .The circle is the locus of a point P(x,y) which moves in such a way that it's distance from fixed point called center is constant that constant is called the radius of circle Using this definition we can derive equation of circle  using Distance formula from coordinate Geometry.

                   d=sqrt[(X2-X1)^2 + (Y2-Y1)^2]
A circle is a simple shape in Euclidean geometry. It is the set of all points in a plane that are at a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves so that its distance from a given point is constant. The distance between any of the points and the centre is called the radius.
A circle is a simple closed curve which divides the plane into two regions: an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is the former and the latter is called a disk.
A circle may also be defined as a special ellipse in which the two foci are coincident and the eccentricity is 0, or the two-dimensional shape enclosing the most area per unit perimeter, using calculus of variations.

Cartesian coordinates

n an x–y Cartesian coordinate system, the circle with centre coordinates (a, b) and radius r is the set of all points (x, y) such that

\left(x - a \right)^2 + \left( y - b \right)^2=r^2.  
 Is the equation of a circle whose origin is shifted by X=a and Y=b from O(0,0)
This equation, known as the Equation of the Circle, follows from the Pythagorean theorem applied to any point on the circle: as shown in the diagram to the right, the radius is the hypotenuse of a right-angled triangle whose other sides are of length x − a and y − b. If the circle is centred at the origin (0, 0), then the equation simplifies to
x^2 + y^2 = r^2.\!\  
The equation can be written in parametric form using the trigonometric functions sine and cosine as
x = a+r\,\cos t,\,
y = b+r\,\sin t\,
where t is a parametric variable in the range 0 to 2π, interpreted geometrically as the angle that the ray from (a, b) to (x, y) makes with the x-axis.
An alternative parametrisation of the circle is:
x = a + r \frac{2t}{1+t^2}.\,
y = b + r \frac{1-t^2}{1+t^2}\,
In this parametrisation, the ratio of t to r can be interpreted geometrically as the stereographic projection of the line passing through the centre parallel to the x-axis .

Parametric equation

In mathematics, parametric equations of a curve express the coordinates of the points of the curve as functions of a variable, called a parameter. For example,
\begin{align} x&=\cos t\\ y&=\sin t \end{align}
are parametric equations for the unit circle, where t is the parameter. Together, these equations are called a parametric representation of the curve.
A common example occurs in kinematics, where the trajectory of a point is usually represented by a parametric equation with time as the parameter.

Ellipse

It is the locus of a point which moves in such a way that the sum of it's distance from two focus is a constant which is equal to 2a.

An ellipse in canonical position (center at origin, major axis along the X-axis) with semi-axes a and b can be represented parametrically as
x(t)=a\,\cos t
y(t)=b\,\sin t. 
The equation of an ellipse whose major and minor axes coincide with the Cartesian axes is      \left(\frac{x}{a}\right)^2 + \left(\frac{y}{b}\right)^2 = 1. This can be explained as follows:
If we let
{x} = {a}\cos\theta.
{y} = {b}\sin\theta.
Then plotting x and y values for all angles of θ between 0 and 2π results in an ellipse (e.g. at θ = 0, x = a, y = 0 and at θ = π/2, y = b, x = 0).
Squaring both equations gives:

{x}^2 = {a}^2\cos^2\theta.





{y}^2 = {b}^2\sin^2\theta.
Dividing these two equations by a2 and b2 respectively gives:
\frac{{x}^2}{{a}^2} = \cos^2\theta.\frac{{y}^2}{{b}^2} = \sin^2\theta.


  • Adding these two equations together gives:
    • \frac{{x}^2}{{a}^2} + \frac{{y}^2}{{b}^2} = \cos^2\theta + \sin^2\theta.



    \frac{{x}^2}{{a}^2} + \frac{{y}^2}{{b}^2} = 1.


    This means any noncircular ellipse is a compressed (or stretched) circle. If a circle is treated like an ellipse, then the area of the ellipse would be proportional to the length of either axis (i.e. doubling the length of an axis in a circular ellipse would create an ellipse with double the area of the original circle).
    An ellipse in general position can be expressed as

    x(t)=X_c + a\,\cos t\,\cos \varphi - b\,\sin t\,\sin\varphi
    y(t)=Y_c + a\,\cos t\,\sin \varphi + b\,\sin t\,\cos\varphi
    as the parameter t varies from 0 to 2π. Here (X_c,Y_c) is the center of the ellipse, and \varphi is the angle between the X-axis and the major axis of the ellipse

      
    Focus
    The distance from the center C to either focus is f = ae, which can be expressed in terms of the major and minor radii:

    f = \sqrt{a^2-b^2}.
    The sum of the distances from any point P = P(x,y) on the ellipse to those two foci is constant and equal to the major axis

    PF_1 + PF_2 = \sqrt{(x+f)^2+y^2} + \sqrt{(x-f)^2+y^2} = 2a.

    Eccentricity

    The eccentricity of the ellipse (commonly denoted as either e or \varepsilon) is
    e=\varepsilon=\sqrt{\frac{a^2-b^2}{a^2}} =\sqrt{1-\left(\frac{b}{a}\right)^2} =f/a
     

     

     




     
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