As we learned about the Euclidean geometry and its five basic axioms in class, some terms like “straight line”, “circle”, and “right angle” kept jumping in my mind. I thought I had a picture of them. for example, a straight line is as straight as the rope with a ball attached and hang in the air, and a right angle is shown like a corner of a rectangular table. However, as a math major student, such a simple cognition of them is not enough, I hope to have some more mathematical concept to express them.
- The Cartesian coordinate system
2.1 The invention of Cartesian coordinates
In the 17th century, René Descartes (Latinized name: Cartesius), a well-known mathematician and philosopher to today’s people all around the world, published his work La Géométrie , in which he made a breakthrough. More concretely, Descartes uses two straight lines that are perpendicular to each other as axes x, y, and uses these axes to measure the positions of any points in a plane.
2.2 The rule of representing a point in Cartesian coordinates
One point in Cartesian coordinates has two parameters: one is the x parameter, the other is the y parameter. To measure the x parameter, we need to draw a straight line y’ parallel to the y axis(we will discuss the definition of parallel in Cartesian coordinates later) that through the point, and then set the x parameter of that point as the number of the intersection of y’ and x axis, for its y parameter, draw a line x’ parallel to the x axis through the point and take the number on the intersection of x’ and y-axis as this point’s y parameter.
2.3 To express a straight line in Cartesian coordinates
A straight line in Euclidean geometry is a straight object with negligible width and depth. So, it is an idealization of such objects in Euclidean geometry. However, in Cartesian coordinates, a line has a strict definition, a straight line is the set of points that satisfies a certain equation. And the line equation usually can be written as:
A*x + B*y + C = 0,
The A, B, and C are the coefficients of x, y, and constant. Moreover, the -A/B is the slope of the straight line, -C/B is the y-intercept of this line, which means the intersection of the y-axes and the line.
So, all above is how we express a straight line in Cartesian coordinates.
- To express the five postulates in the Cartesian coordinate system
In Cartesian coordinates, to express a line we only need one point and a direction. Suppose we have two points a=(A, C) and b=(B, D). By doing a subtraction of the two points, we can get a vector (B – A, D – C). We only need this vector to provides a direction, which is (B – A)/(D – C). So this unique straight line can be expressed as
(x – A)*(B – A)/(D – C) = y – C;
2.”To produce [extend] a finite straight line continuously in a straight line.”
Any line in Cartesian coordinates is a straight line(infinite). It can be limited by a range for x or y, such as
A*x + B*y + C = 0,( a < x < b or c < y < d)
- “To describe a circlewith any centerand distance [radius].”
To describe a circle in the Cartesian system, we only need the center’s x and y coordinates (x0, y0), and a distance as the radius r, it is
(x – x0)2 + (y – y0)2 = r2,
So, above is a typical circle in the Cartesian coordinates.
- “That all right anglesare equal to one another.”
A right angle in the Cartesian system is always equal to the angle between x and y axes, for x, y axes in the Cartesian system is perpendicular to each others.
- “That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.”
For the parallel postulate, it is far more easier to be expressed in Cartesian coordinates, suppose we already know a line as
A*x + B*y + C = 0,
And we have a point (x1, y1) out of the line, have the point and a direction of the other line, the slope is – A/B, and the other line can be described as
-A/B*(x – x1) = y – y1;
And it is easy to know these two lines are parallel, because they have same slope and do not share one point. And by the property of Cartesian coordinates, this is the only line that parallel with the first one.
In Euclidean geometry, some concept are hard to imagine or describe, while Cartesian coordinate make it possible and easy to express, such a great combination of geometry and algebra!