Plane curves in maths are those which are plotted using two axes only. For instance, all conics, lines etc. kind of shapes come in category of plane shapes. Points lying on these curves are formed using a pair of x and y values such as (x, y). Thus these shapes are also called as 2 – Dimensional shapes. A Plane Curve can be open or closed depending on geometry of the curve. These curves are obtained by plotting their respective coordinates in rectangular or Cartesian plane using two axes: x – axis and y axis. The equations of plane curves can be of form:
1. Y = MX + c
2. X^{2} /A^{2} + Y^{2} /B^{2} = 1
3. X^{2} /A^{2} + Y^{2} /B^{2} = - 1
4. (X – H)^{2} + (Y – K)^{2} = R^{2}
5. Y^{2} = 4AX
In contrast with 2 – dimensional figures we have 3 dimensional figures that would require 3 axes to represent them and so they have representation of coordinates using three values: x, y and z values.
Plane curves are therefore possible to be drawn on plain paper, which is not the case with 3 – Dimensional figures like sphere, cylinder, cone, pyramid etc.
We can also say that plane shapes are a kind of function equations that have one independent variable and one dependent variable. On differentiating the plane curves with respect to independent variable of equation, we get the rate of change of function or slope of curve at that particular point. Double differentiation of plane curves will give maxima or minima for the curve. When integration of these curves is done, area of curve is evaluated. Like this other calculus operations are also possible with plane curves.
Plane curve is a curve that lies completely on the plane, we can find the length of a plane curve using integration method. Suppose y = f(x) is a curve on [p, q], p and q are continuous derivative of 'f'. So
length of this curve is defined by:
L = ∫a b √(1 + (f’(x))^{2})dx ------equation (1).
Let’s find out the length of given curve...Read More