Introduction to Analytical Geometry

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Sub Topics Define Inclination of a Line Analytical Geometry Definition History of Analytical Geometry

The concept of Analytical Geometry was given by Mr. Percey F. Smith and Mr. Arthur Sullivan Gale. Mr. Percey F. Smith was a professor of mathematics in Sheffield scientific school, Yale University and Arthur Sullivan Gale was an assistant professor of mathematics in the University of Rochester. Now, we will see introduction to analytical geometry. Analytic geometry and Algebra both are similar, they are used to design the geometric objects in the mathematics, and these geometric objects are points, straight line, and Circle etc. In the plane of geometry the points are represented as Ordered Pair and in case of Straight Line it is represented as Set of points which satisfy the linear equation and the analytic geometry which satisfies linear equation are known as linear algebra. There are some different names for analytic geometry like Cartesian geometry is also known as analytic geometry. This plane analytic geometry is based on the coordinate system and the principal of algebra and analysis. Now, we will see how to find the distance and angle in analytic geometry. Suppose we have coordinates m1, m2 and n1, n2 for the plane of geometry: Then we will see how to find the distance of plane of analytic geometry by using above coordinates: d = √ (m2 - m1)2 + (n2 - n1)2; Where ‘d’ is distance of analytic geometry. We use pythagoras theorem for finding the distance of analytic geometry; Now we will find the angle of plane of geometry: The angle of geometry is: ∅ = arctan (p); Where ‘p’ is the Slope of line; Using this formula we can find the distance of plane of analytic geometric. This is all about analytical geometry introduction.

A line can be plotted on a graph, and when we plot any line of the graph, it is inclined at certain angle in a coordinate plane. This plotting of the line on the Cartesian plane and the angle formed by the line on the axis is called inclination of the line and is represented by the angle Ө. Here, we define Inclination of a Line. By the term Inclination of a line, we Mean how much a line is inclined on a particular axis.
The inclination of an angle helps us to find the Slope of any line which is represented by tan Ө. By the angle of inclination we mean the measure of angle Ө, that the line or the part of the line forms with the positive direction of the x – axis and it is measured in anti clock wise direction. So, we say that the value of Ө is always a positive quantity. Thus, it is clear that the value of Ө is always between 0 degrees to 180 degrees.
We must look at the following observations about the Slope or inclination of the line:
1. The inclination of the line which is parallel to x – axis, or the x – axis itself is always 0 degrees and 180 degrees.
2. The inclination of the line which is parallel to y – axis or y – axis itself is 90 degrees.
Thus we come to the conclusion that any line parallel to the x axis or the x axis itself is called a horizontal line and the line which is parallel to y axis, or the y- axis itself is called the perpendicular of the vertical line. Another type of line is oblique line, which we say is neither a parallel line nor a horizontal line. If we know the inclination of any given line then we can easily find the slope of that line.
Most of us would not be familiar with the term analytical Geometry in the context of the Math. So in this article we will try to define Analytical Geometry in a very precise way to have a good understanding. The analytical geometry is also sometimes called as analytic geometry.
Before we define analytical geometry we should mention that the analytical geometry possesses two distinct types of the meanings in the context of the math. One of those two meanings of the analytical geometry is the modern meaning and the other meaning refers to the classical meaning of the analytical geometry. The definition of analytical geometry in two of its forms can be given as follows.
The definition of analytical geometry in its modern form or we can say that the advanced understanding of the term analytical geometry generally emphasizes on the geometry of such type of the varieties which are analytic.
Now we will define analytical geometry in its classical form but before we do that we should first mention that the meaning of analytical geometry in its classical form is same as the definition of the coordinate geometry or the definition of the Cartesian geometry. In context to classical math, the analytical geometry is defined as the study of the geometry with the help of the system of a coordinate and also by the use of various principles of the analysis and the principles of the Algebra. The analytical geometry has got many applications in the fields of engineering and Physics.
We all study Geometry in the Math and find it quite interesting but as we go further deep in study of geometry we find another type of geometry that is the Analytical Geometry. Before understanding analytical geometry first becomes familiar with history of analytical geometry.
The analytical geometry history starts from the Period when a great mathematician of Greece whose name was Menaechmus started solving some of the problems. With this he start proving some of the theorems by utilizing a method which was very similar to the utilization of the coordinates and even it is sometimes realized that he was the one who was greatly responsible for the introduction of the analytical geometry.
The history of analytical geometry, also includes the contribution of the Apollonius belonging to the Perga who also solved some of the problems in a way that can be referred as the analytical geometry of kind of the one dimension and the question which they used to solve normally was determining the points which exist on a line and that happened to be in the form of a Ratio with the others.
The further development of the analytical geometry was also credited to the Apollonius who in the Conics again gave a process to solve the problems which was very much like the analytical geometry. He gave his contribution in the fields of the lines of the reference also gave the uses of Tangent and Diameter which we very normally use in the modern math in the system of coordinate in which we find the distance along any diameter starting from the Point which is called the Point of Tangency.