Sets of all Points

The point, line, and plane are the main parts of geometry mathematics; they occupy an important place in geometric mathematics, where its main function is placed in its set and all points. In geometry, a set of points satisfy a certain rule. Set and point occupy a prominent place in geometry as undefined terms. 

They are used to better understand geometry, such as visual representations, where one cannot be presented without a set of points with fixed dimensions. With its help, it is easy to describe the dimensions on the basis of thickness, width, or height, also showing their prime position in the set of points.

Here the position, size, or shape of any geometrical structure can also be better understood on the basis of the set of points. As we can do for circles. We can understand all the points of the set by means of graphs, which can be directed based on the points, which tell about the position of the points. These dots are written with braces, as described here. 

A set {2, 4, 6} represents the set of even numbers, which includes 2, 4, and 6, which helps in better understanding of points and sets of points in mathematics.

Sets of all points and locus

Locus is a term used in geometry to represent a set of points that describe the positions of their geometry as well as their location. Here we will take the example of sunrise to understand the locus. With the help of this information, we can know about the sun’s position and direction. The same thing is done with points in geometry, using locus to find out things like position or location. The main relationship of locus is with the point, and the position and shape of this geometry are likewise determined by this point. When we talk about a circle, we must remember that it is made up of points.

Other examples of geometry, like the locus of points, such as in the ellipse, parabola, and hyperbola, can be comprehended in the same way. This holds true for other shapes, both regular and irregular. Locus is essential not only because of its shape but also because of its angle.

Points are the most important component in the construction of any line; they help in the formation of geometric shapes such as line segments, circles, and curves, as these sets of points help satisfy the condition of any geometrical structure.

The collection of points describes the shape of the structure, as well as its actual location. The link between the set of points and the locus is complex, allowing it to be easily understood as the set of all points equidistant from any given figure, as well as the position of the center point of all figures.  In addition, it provides information about the location of several points.

Sets of all points: Graphical or mathematical representation

We can understand the set of all points or locus by mathematical and graphical representation of the circle. Let the circle be the set of points, and its diameter and radius are all the same for given points. Here the distance of each point along with the radius of the circle will represent.

A circle’s equation is as follows:

( x – h )2 + ( y – k )2 = ( r  )2

Where r is the circle’s radius and h and k are the circle’s center coordinates (x=h, and y=k).

When the origin (0, 0) of a coordinate system is used as the center of a circle, the equation becomes:

( x – 0 )2 + ( y – 0  )2 = ( r  )2

( x )2 + ( y  )2 = ( r  )2

x 2 + y  2 =   r  2

For a circle with a radius of 4 meters with a center at the origin, uses the formula

x 2+ y  2 =   4 2

x 2+ y 2 =  16

We saw here a set of points (x, y) satisfying the formula, which will describe part of the locus of the points. This will be proved by finding a segment of a circle with a radius of 4 m and a center at the point O. (0, 0). Here E, F, G, H, and J will denote the points which are part of the set of points on this circle.

x 2+ y 2 =  16

If we put, x = 0, y = 4

0 2 + 4 2 =  16

16 = 16

Which means (0,4) is part of the locus of the circle.

It helps in understanding the position of the set of points in a better way.

Example

Circle – A circle is mainly a group of points; it is also a group of many points made in a plane area. This circle also has a central point, with the help of which this circle is constructed. Also, the distance of the points in the circle from this point is the same, which we call the radius.

Ellipse – The locus of all points in a plane is an ellipse if the sum of the distances of points from two fixed points in the plane, known as foci, which are circumscribed by the curve, is constant.

Conclusion

Locus is a term used in geometry to represent a set of points that describe the positions of their geometry as well as their location. When we will join all the given points we will get a shape which is called the locus of these points. We also took the example of a circle showing how it is representing locus. Hope you liked the article.