A curve is a line or a shape which is smoothly drawn in a plane having turns or a bent in it. For instance, an ellipse is an example of a curved shape. In Maths, Geometry is a branch that deals with sizes, shapes, and the properties of figures. Geometry can be divided into two types. They are:

1. Plane or Two Dimensional Geometry
2. Solid or Three Dimensional Geometry

Two-dimensional geometry is the study of flat shapes like lines, curves, polygons, etc. that can be drawn on a \(2\,D\) piece of paper. On the other hand, three-dimensional geometry is the study of \(3\,D\) objects such as cubes, cuboids, spheres, cylinders, etc. Here we will discuss a fundamental element of plane geometry closely about curves.

What is Surface in Math?

In Mathematics, a surface is a two-dimensional collection of points (flat surface), a three-dimensional collection of points whose cross-section is a curve (curved surface), or the edges of any three-dimensional solid. In general, a surface is a continuous boundary of an object.

The outside layer of an object has only area but no thickness. It is a two-dimensional boundary that can be curved or flat.

Example: A sphere has a smooth surface.

Definition of Curve in Math

In mathematics, a curve is an abstract term used to describe the path of an uninterruptedly moving point (see continuity). Such a path is usually given by an equation. A closed curve is a path that repeats itself and thus encloses one or more than one region. Simple examples include circles, polygons, and ellipses.

What is a Curve in Math?

A curve can be defined as the constant movement of points in all directions. It means that each and every possible shape can be classified as curves because all the shapes are formed by the movement of the points in one or multi-dimension.

A curve can be an open curve or a closed curve. The figures below show an example of each type of simple curve:

Figure \((1)\) is an open curve, and Figure \((2)\) is a closed curve.

Types of Curves in Math

Curves are categorised into different categories. Let us discuss the different types of curves with examples.

The different types of curves are discussed below with figure for better understanding-

1. Simple Curve: A curve that changes its direction but it does not connect itself. A simple curve changes direction but does not criss-cross itself while changing direction. A simple curve can be open as well as closed.

2. Non-Simple Curve: A type of curve that crosses its path is called a non-simple curve.

3. Open Curve: An open curve does not include any area within itself, and it has two endpoints.  The following figures indicate some of the open curves.

4. Closed Curve: A closed curve do not have any endpoints and encloses an area (or a region). It is formed by joining the starting and endpoints of an open curve together.
Examples of the closed curve are circles, polygons, ellipses etc.

5. Upward Curve: An upward curve is a curve that turns in the upward direction. It is also known as a concave upward. Concave upward is also called “Convex Downward”.

6. Downward Curve. A downward curve is a curve that turns in the downward direction. It is also known as a concave downward. Concave downward is also called “Convex Upward”.

Examples of Curves in Math

Let us identify the different types of curves in the given figure based on their properties. We can see that in Figure \(1\) the curve is a simple closed curve, as an area is bounded by the boundaries of the curve. In Figure \(2,\) the curve is a closed curve, as we observe the smooth surface and continuous edges. Figure \(3\) is an upward curve. Figure \(4\) is a downward curve. Figure \(5\) shows a non-simple closed curve because it crosses itself. And, Figure \(6\) shows a non-simple open curve as it crosses itself and has two endpoints.

Real-life Example of Curves

Curves Around Us: Roads on mountains are curvy.

Curved Line

A curved line is a line that is bent and not straight. If the curvature is not equal to zero, then it is taken as a curve line. Generally, it is smooth and continuous. 

Curved Line Images

We can see many objects and things that are in the shape of a curved line. Some of them include a track of a roller coaster, railway track at the turning points, paths of a road particularly for hilly areas, and many more. Apart from the real-life examples, we can also notice the curve-shaped lines in mathematics, such as graphs of a quadratic polynomial, i.e., ogive curve, parabola, arrows, etc. Images below are curve-shaped lines.

Interior and Exterior of Closed Figures

1. The point that is lying inside the closed curve is called its interior
2. The point that is lying outside the closed curve is called its exterior
3. The point that lie on the curve constitute its boundary

In the given figure, point \(P\) lies in its interior, \(Q\) in its exterior, and \(R\) lies on its boundary. Together the interior of the curve with its boundary is called the region.

Let’s take a real-life example. If Rohan is playing cricket in a playground having a closed boundary rope and Khushi is watching him play from outside, then we say that Rohan is in the interior of the playground and Khushi is in its exterior. The flags on the rope constitute its boundary.

Solved Examples – Curves

Q.1. Demonstrate each one of the following with a diagram roughly if possible.
(a) A closed curve that is not a polygon.
(b) An open curve made up of only line segments.
(c) A curve that is neither closed nor simple. Explain.
(d) A polygon with two sides.
Ans:
(a) A closed figure that is not a polygon is shown below:

(b) An open curve made up entirely of the line segment

(c) A curve that is neither closed nor simple is shown below:

The curve is not simple as it is joining itself. The curve is not closed as its starting point and endpoint are different.

(d) A polygon with two sides is not possible since the minimum number of line segments required to form a closed figure is three.

Q.2. Consider the adjacent figure and answer the following questions

(i) Is it a curve?
(ii) Is it closed?
Ans: (i) Yes, it is a curve.
(ii) Yes, it is closed because start and endpoints are connected

Q.3. Classify the following curve as a closed or open curve.

Ans:
(a) Closed curve
(b) Open curve
(c) Closed curve
(d) Closed curve

Q.4. Draw a polygon of any number of sides and shade its interior. Is every polygon closed?
Ans: Yes, every polygon is closed.
The following figure is a polygon, and its interior is shaded.

Q.5. Is a straight line a curve?
Ans: No. A curve is not a straight line, simultaneously a straight line is not a curve.

Summary

In this article, we learnt that a curve is a shape or a line that is smoothly drawn in a plane having a bent or turns in it. There are different types of curves such as simple curves, non-simple curves, open curves, closed curves, upward and downward curves, etc.

Also, we discussed points that are interior and exterior to a closed figure. We also provided a few solved examples on this topic and a few frequently asked questions on curves. We learnt to identify the given figures based on their properties. A clear understanding of all the topics described in this article will help the students learn better about curves.

Frequently Asked Questions on Curves

Q.1. What is called a curve?
Ans: A curve is a shape or a line that is smoothly drawn in a plane having a bent or turns in it.

Q.2. What is a curve definition suitable for Class 6?
Ans: A curve is a path drawn on a sheet of paper without lifting the pencil and without retracing any point of the path except for single points.

Q.3. Illustrate some examples of a curve.
Ans: The following figures indicate examples of curves:

Q.4. What are the types of curves? Or Discuss types of curves in mathematics.
Ans: The following are the types of curves:
1. Simple Curve
2. Non-Simple Curve
3. Open Curve
4. Closed
5. Upward Curve
6. Downward Curve

Q.5. What is the primary difference between a curve and a line?
Ans: Curves are ideally lines having different shapes while a line has a straight shape. A straight line can be shaped into a curve as per requirement. In the Euclidean theory, a straight line has a short distance which is ideally between two points. This idea only appears in Euclidean geometry.
Moreover, a straight line is a short line joining any two points and moving into a single direction. But a curve that is curved in shape doesn’t move in a pre-defined direction. They can be bent in a circle or a sine wave pattern for fulfilling requirements.