What is a Curve?
A curve is a continuous line that flows smoothly and without abrupt turns. A curve can be identified easily by observing if it bends and modifies its course at least once. The examples of geometric shapes in which curves can be observed are circles, semi-circles, spheres, and so on. We can also observe curves in real-life as well, like the moon or a tennis ball.
What is a Curved Line?
A curved line is one that is crooked and not straight. It should ideally be continuous and smooth. To put it another way, a curve is described as a collection of points that resemble a straight line that passes through two adjacent locations. We are aware that the straight line has zero curvature. Therefore, we can refer to a line as being curved if its curvature is greater than zero. The various types of curved lines are depicted along with images for clarity in the next section.
Various Curves
Different curves are categorised according to a few characteristics. Let’s examine the types of curves.
Open Curve
An open curve is a type of curve which does not enclose the area within its two endpoints. Some examples of open curves are as shown in the image.
Closed Curve
A closed curve encloses a given region or area. This type of curve is formed by joining the two endpoints of an open curve, hence a closed curve looks like it does not have endpoints. The best examples of closed curves are circles, ellipses, and so on.
Simple Curve
A curve that changes its direction and it does not intersect with itself is referred to as a simple curve. A simple curve can be either open or closed.
Non-Simple Curve
A particular kind of curve that crosses itself is known as a non-simple curve.
Upward Curve
A curve that points in the upward direction is called an upward curve.
Downward Curve
A curve that points downwards is called a downward curve.
Different Curved Shapes in Geometry
Given below are a few curved shapes in geometry.
Circle:
The circle is a two-dimensional shape and the entire outer surface is curved.
Semi-circle:
The semi-circle, as the name suggests is half a circle, is another two-dimensional shape that has a straight line and a curved portion as well.
Sphere:
A sphere is a three dimensional shape which has a completely curved surface.
Cone:
A cone is a three-dimensional shape which has a pointed surface and a circular base which is a curved surface.
Cylinder:
A cylinder is a three-dimensional shape that has two circular surfaces which have curves.
Solved Curve Examples
Example 1: Identify whether the curves in the following images are closed or open curves.
Image 1:
Image 2:
Solution:
Image one is an open curve and the second image is a closed curve.
Example 2: Identify whether the following images are simple closed closed curves or non simple closed curves.
Image 1:
Image 2:
Solution:
Image 1 is a non-simple closed curve and the second image is a simple closed curve.
Example 3: A wire is bent to form three semicircles as shown below, how long is the newly shaped wire? Justify your answer.
Solution:
Length of the wire = 50 + 50 + 50 = 150m
The length of the bent wire is 150 m which is the same as the length of the wire when it was straight. This is because no matter how the wire is bent, the length of the wire stays constant.
A continuous line that runs smoothly and devoid of sharp twists is referred to as a curve. By looking to see if it bends and modifies its trajectory at least once, a curve can be clearly spotted.
A curve which changes its direction and does not intersect with itself is referred to as a simple curve. If this curve is open it is an open simple curve, and if the curve is closed, it is a closed simple curve.
The curve that changes direction while intersecting itself is a non-simple curve.If this curve is not closed it is an open non-simple curve and if the curve is closed, it is a closed non-simple curve.
An upward curve is also known as a concave upward. A concave upward curve is also called a ‘convex downward’.
A downward curve is also known as a concave downward. A concave downward is also called a ‘convex upward’.