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Plane Curve

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In Geometry, we deal with many types of shapes and figures like lines, circles, ellipses, curve and many more. They all come under the family of curves. In geometry, there are two types of curve:
1. Open curve: In open curve, we consider those curves whose end points never meet together.
2. Close curve: Close curve are types of curve whose closing points are connected or joined.
Equation of curve which is given as:
y = x2, here ‘y’ denotes the y – coordinates value and ‘x’ denotes the x – coordinate values. We can also write equation of curve using parameter of curves and these equations are called parametric equation of curve.
Let’s talk about curves in geometry. Basically curves are of two types:
1. Analytic Curve: - These are easy curves so that they can be managed by an equation such as ellipse, straight line and helices.
2. Interpolated Curve: - A ll B - spline curves are considered as interpolated curve.
As we know curve can be use in two -dimensional (plane curves) curve and three-dimensional (space).
A curve in two-dimensional is also called as “plane curve” that lies in a single plane. It may be closed or open. It is a curve in a real Euclidean plane R2. A smooth plane curve can be represented by an equation ƒ(x, y) = 0, here ƒ: R2 -> R. It is a smooth function. Some of plane curves are inside curves which are known as interior curves and curves which are outside the curve are known as exterior curves. In spite of these curves, some are on edges (means on boundary) which are known boundary curves.
Examples:
Straight line ax + by = c
Parabola y – x2 = 0
Circle x2 + y2 = r2 (here r is the radius of the Circle)
Ellipse x2 / (a2 + y2) / b2 = 1.

Length of a Plane Curve

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A curve is simply a straight line with a slight difference that it contains a curve in its formation. Curve may be less or more. Parabola, hyperbola, ellipse, circle and many more, all are formed by curve. Any projection, like projection of a ball from a particular Point also contains a curve. The point at which direction of motion change, is the point of a curve. Plane curve is simply defined as a change in Straight Line with a slight difference. It is a one dimensional curve which lies in a single plane and its partial Derivatives are never zero. They are categorized as open curve and close curve.
Open curves are defined as the curves which are open from one side and are not compact. For example Parabola, hyperbola, straight line, etc. and the closed curve are like, circle, ellipse, etc.
Length is defined as the measure of any figure. It can be determined by different formulas specified for different figures. Length of a line is determined with help of distance formula which is written as:
d = (x2-x1)2 + (y2-y1)2
In order to find the length of a Plane Curve we need to break the curve into two segments. Its total length is represented as:
L = ∑n k=1(∆x n)2 + (∆y n)2
Thus length of a plane curve is determined by the Set of two points and a set of point at curve. This point of curve divides it into two segments and then the length is determined by the above formula. Limits define The Range up to which submission or Integration is to be performed for finding the length.
The order of limits while solving the equation must be quiet specified. That is, x limit will be substituted in place of variable x and y limit must be put in place of variable y.