Curved Line

Introduction

A curved line is simply that line which is not straight but has bent. We can see curves in our day to day lives. In fact, most of the things of nature are some or the other kinds of curves. A curve is defined as a curved line rather than a straight line. Ideally, straight lines have zero curvature, but curves have non-zero curvature, and are continuous and smooth. Curves are prominent shapes that we see all around us. You can find curves in works of art, ornaments, or everyday objects. Curves are shapes that appear all around us. Initially, the trunk should be curved or straight. In modern mathematical terminology, curves are curved and straight lines are straight lines, specifically to distinguish them.

Curves are one of the important subjects of mathematics, so they are often used to represent features graphically.

Curves

A "curved line" or simply "curve" is not a straight line. Curves are found everywhere around us. Curves can be found all around us in art, decoration and everyday life.

A curve is more curved than a straight line. In the best of circumstances, it should be simple and non-stop. In other words, a curve is a set of elements similar to a straight line, between two points. It is known that a straight line has zero curvature. As a result, a curve can be called a curve if its curvature is greater than zero.

Arcs

In mathematics, an "arc" is a smooth curve connecting two endpoints. In general, an arc is part of a circle. It's basically part of a circle. In mathematics, arc means part of a curve or part of a circle. The straight lines joining the ends of the arc are called the chords of the circle. Arcs are called portions or segments of the circumference. A straight line that must be drawn connecting the two ends of an arc is called a chord. If the arc length is exactly half the circle, it is recognized as a half arc.

Closed and Open Curves

An open curve is a curve with a definite start and end point. Example: An open curve is an instance of a parabola. A curve that has the same start and end points is called a closed curve. Example: A circle is an ideal example of a closed curve.

A curve is said to be open if its endpoints do not currently intersect. In an open curve, the endpoints never meet. The parabola is a perfect example of an open curve.

A curve is considered closed if the start point is equal to the end point.

A circle or an eclipse is a perfect example of a closed curve.

Curved Shapes

Two-dimensional curvilinear shapes consist of circles, ellipses, parabolas, hyperbolas, and arcs, sectors, and segments. 3D curved shapes are the geometrical figures like spheres, cylinders and cones.

Probably the most common two-dimensional curved shape is the circle.

In order to work with circles (and various curvilinear shapes) in geometry, it is important to know the main properties of circles.

• The straight line through the center of the circle is the diameter.

• Half the diameter is the radius.

• The line surrounding the area of the circle is the circumference.

Every element on the circle has exactly the same distance from the center of the circle as any other point on the circle. The sectors and segments are "slices" of the circle.

A sector is shaped like a slice of pizza, curved on one side, and each straight side is the same size as the radius of the previously cut circle or pizza. A pie chart consists of different sectors related to the information displayed in the measurement.

Regions can be of any size, but regions containing half circles (180°) are called semicircles, and regions containing quadrants (90°) are called quadrants.

The phase is the curved phase of the sector, the part that remains when the triangle is removed from the sector. A segment consists of two lines- Arcs (parts of the circumference of a circle) and chords - the straight lines which are responsible for joining two points that lie at the opposite sides of the circle.

Area Under a curve on a Cartesian Plane

Finding the area under a curve on a Cartesian plane is quite simple. We just simply need to know the area of the curve, let in our case be f(x). If we need to find the area of this curve between two points, say, a and b, then we simply need to integrate f(x) between a and b. So, area of the curve f(x) between the points a and b will be $\mathrm{\int_a^bf(x).}$

Curves in Daily Life

Curves can be seen almost everywhere in our real life. Starting from our doors and windows, to domes and in many architectural marvels, we can see curves. If we look at the rainbow in the sky, it’s a curve, many of the gates of houses are curved in nature, half moon looks like a curve etc.

Solved Examples

What would be the area of the curve x² in the range (3,9)?

$$\mathrm{The\: area\: will\: be:\int_3^9 x^2 =[\frac{x^3}{3}]_3^9=243-9=234.}$$

What would be the area of the curve x⁸ in the range (0,1)?

The area will be: $\mathrm{\int_0^1 x^8 =[\frac{x^9}{9}]_0^1=\frac{1}{9}-0=\frac{1}{9}.}$

Conclusion

A curved line is simply that line which is not straight but has bent. We can see curves in our day to day lives. In fact, most of the things of nature are some or the other kinds of curves. A curve is defined as a curved line rather than a straight line. Ideally, straight lines have zero curvature, but curves have non-zero curvature, and are continuous and smooth.

FAQs

1. Briefly describe what an arc is.

In mathematics, an "arc" is a smooth curve connecting two endpoints. In general, an arc is part of a circle.

2. Briefly describe what a curve is.

A curved line is simply that line which is not straight but has bent. We can see curves in our day to day lives.

3. What are some of the curves that we see in our day to day lives?

Curves can be seen almost everywhere in our real life. Starting from our doors and windows, to domes and in many architectural marvels, we can see curves. If we look at the rainbow in the sky, it’s a curve, many of the gates of houses are curved in nature, half moon looks like a curve etc.

4. Briefly describe what an open curve is.

An open curve is a curve with a definite start and end point. Example: An open curve is an instance of a parabola.

5. Briefly describe what a closed curve is.

A curve that has the same start and end points is called a closed curve. Example: A circle is an ideal example of a closed curve.