Basic Geometrical Ideas

Curves

Looking back at the ancient times, the use of curves has always been persistent. They were visualized as art and decoration; however, their exact geometric definition was decoded later. Mathematicians from the past used to stay very curious related to curved lines. In present-day mathematics, the curved lines are utilized for graphical representation of functions. Hence, we are now going to gain useful knowledge about curves together with their related types and algebraic version.

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Definition of Curve

A “curvy line”, “curved line” or just “curve” is a line which is not straight. In simple terms, it can be any line which is somewhat bent. Ideally, based on mathematics, a curve needs to be continuous and smooth.

It is important to know that, curves hold different definitions as per different disciplines of mathematics. This is primarily due to the diverse field of study in which they are used. Further, there are some exclusive instances in which a curve’s definition may vary from the traditional one.

A curve can be stated as the set of points which look like a straight line falling between two adjacent points. Moreover, a curve is highlighted as a topological space which is locally homeomorphic to a line. You can call a parabola as the ideal example for a curve.

Types of Curves

Let us now try to understand the different types of curves which have varying definitions and properties.

Simple Curve

As we know, a curve is a line that is not straight. Hence, Simple Curve can be termed as a curve that doesn’t cross itself.

Closed Curve

A curve in which the starting point and ending point match is known as a closed curve. Such type of curves creates a path which may start from any point and conclude at the same point. For your information, a closed curve doesn’t necessarily have to be a curve. For example, a square is a form of a closed curve. That is, you can’t get in, nor can get out of it. It can be called as a design with no endpoints and stays on a flat surface.

Simple Closed Curve

A simple closed curve is a connected curve which doesn’t cross itself and concludes at the same point from which it began. For example circles, polygons and ellipses.

Algebraic and Transcendental Curve

Usually, the curved lines are classified into two forms, that is, algebraic curves and transcendental curves. Let us try to understand what they actually are.

Algebraic Curve

Algebraic curves can be defined as a plane curve in which the set of points are situated on the Euclidean plane. Further, it is represented in the form of a polynomial. Moreover, the set of points for an algebraic curve do fulfil a polynomial equation.

It can be written as: C = {(x, y) ∈∈ R2: P(x, y) = 0}. Additionally, the degree of a curve can be termed as the degree of the polynomial it eventually denotes.

Transcendental Curve

In a simple manner, the curve which doesn’t represent an algebraic form is called as a transcendental curve. Such curves can have an infinite number of intersecting points together with a straight line. Moreover, they can also hold an infinite number of intersection points. A transcendental curve isn’t a polynomial based on x and y.

Question For You

Q. Provide examples for Algebraic Curves?

Answer. A circle x2+y2=r2 can be termed as an algebraic curve. Also, parabola y2 = 4ax is a good example of an algebraic curve.

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