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CONSTRUCTION AND READING OF THE CARTESIAN DIAGRAM “1st PART”
In this lesson we will learn how one traces the lines graduated to be able to represent in a suitable way, the sizes given.
In the representations most used in technique, the frame of reference is consisted two perpendicular lines, one horizontal and the other vertical, called axes of the system, or more simply, axes.
Figure 1 shows the two axes separately drawn in order to highlight well conventions which it is advisable to establish for each one of them.
Let us consider initially the horizontal axis corresponding to the graduated right-hand side. We notice that the axis of the figure 1-a, it comprises two classifications indeed: one which leaves the origin and goes towards the line, and the other, which leaves the same origin and goes towards the left; between in addition to, the numbers which are on the left origin, all are preceded by the sign “-” (less). Each number, as well on the right as on the left of the origin, corresponds to one of the subdivisions and those follow to the same distance one of the other.
The numbers and the subdivisions of the right-hand side are called X-coordinates: positive X-coordinates on the right of the origin, and negative X-coordinates on the left.
Let us suppose that the horizontal axis represents the course of time as on the graphics of figure 1.
Origin 0 indicates the moment when we start to count. The positive X-coordinate 1, indicates that as from the initial moment, a unit of time ran out, namely one second, or an hour, or a day (that depends on the unit which we chose).
X-coordinate 2 indicates that as from the initial moment two units of time (2 seconds, 2 hours or 2 days ran out. X-coordinate 3 indicates three units of time after the initial moment; X-coordinate 4 indicates four units of time after the initial moment and so on.
The negative X-coordinates indicate also units of time, but instead of representing the time which passed as from the selected initial moment, they represent the time which preceded this initial moment. Thus, the negative X-coordinate - 1 indicates that there remains a unit of time to arrive at the initial moment, the X-coordinate - 3 indicates three units of time before the initial moment and so on de about times increasingly more remote before the initial moment.
By keeping in mind the significance of the numbers preceded by the sign - (- 4, - 3, - 2, - 1), we observe that while going towards the line, i.e., that while passing from a great number to smaller, the distance as from initial time falls; one can think that the arrow which indicates the increase in times is not in agreement with the representation of times on the left of the origin.
This deduction would be exact if we limit ourselves to look at the classification of the negative X-coordinates, without taking into account the progression of time. If we start to count times, on the basis of a negative X-coordinate, for example - 4 seconds, and, if we want to maintain this account in agreement with the real succession of the seconds, we must express ourselves as follows:
less four seconds, less three seconds, less two seconds, less one second, time zero, one second (after time zero), two seconds, three seconds, four seconds…
As it is seen, the counting which follows the flow of time always goes from the left towards the line, in agreement with the arrow placed at the end of the axis, either that one counts on the negative X-coordinates, or, with stronger reason than one counts on the positive X-coordinates.
This point being cleared up, now let us consider the vertical axis traced on the figure 1-b.
We can repeat all the considerations made for the horizontal axis in connection with the origin, of classifications and the subdivisions. The only difference consists in the name given to the subdivisions and the corresponding numbers which are called ordered: positive ordinates above the origin and negative ordinates below this same origin.
The vertical axis must also represent a given physical size; for example, it can represent the respective temperature and its values where the vertical axis is consisted the scale of the thermometer. By comparing the vertical axis with the scale of the thermometer, we can easily guess the significance of the arrow placed at the higher end of the axis: as in the example of times for the horizontal axis, the arrow indicates the direction of the ascending values.
In the particular case considered, it indicates in which direction the increases in the temperature occur.
An increase in temperature precisely consists, starting from a negative ordered value, to go towards the ascending values of the positive ordinates. This displacement thus goes upwards, as the arrow indicates it, as well as the mercury column of the thermometer goes up when the temperature increases.
The significance of the axes being established, let us see now how they are laid out on the sheet so as to form a frame of reference adapted to the establishment and the reading of the graphs.
To exploit a frame of reference, it is enough to be advisable:
position of each axis in the plan of the sheet,
how, on the basis of a point of the plan, to determine its X-coordinate and ordinate,
how, on the basis of the X-coordinate and of the ordinate, to locate a point in the plan.
Among the possible solutions, in an infinite number, let us adopt that of figure 2, which, under much of aspects, simplest and is most frequently used in electronics.
To compose this frame of reference, it is necessary perpendicularly to cross two axes and in such way that their respective points of origins coincide. Moreover, one lays down the following rule to pass from the points of the plan, with the corresponding values which one can read on the axes:
being given a point, one considers two lines passing by this point; one perpendicular to the x-axis and the other perpendicular with the y-axis.
The line intersection perpendicular to the x-axis with this axis indicates the value of the X-coordinate; the line intersection perpendicular to the y-axis with this axis indicates the value of the ordinate.
While following the rule that we have just stated, one can easily establish the values of X-coordinates and ordinates of the points A, B, C, D, indicated on figure 2.
Not A : the perpendicular with the x-axis indicates on this axis value 2 and the perpendicular with the y-axis indicates on this axis value 2. Consequently, point A corresponds to the value of X-coordinate 2 and the value of ordinate 2.
Not B : while following the same method that for point A, one finds that the point B corresponds to the value of X-coordinate - 3 and with the value of ordered 3.
Not C : it corresponds to the value of X-coordinate - 1 and with the value of ordinate - 5.
Not D : it corresponds to the value of X-coordinate 4 and the value of ordinate - 2.
The graphs based on the preceding rules are called the Cartesian diagrams.
A Cartesian diagram divides the plan of the sheet into four parts, called quadrants, separated between them by the two axes which cross.
It is interesting to notice that the points included/understood in the first quadrant (figure 2), have positive X-coordinates and ordinates; the points of the second quadrant have negative X-coordinates and positive ordinates; the points of the third quadrant have negative X-coordinates and ordinates; finally, the point of the fourth quadrant have negative ordinates and positive X-coordinates.
By combining the positive and negative X-coordinates judiciously, with the positive and negative ordinates, one can represent each point of the plan, by means of a couple of values (X-coordinate and ordinate) called coordinated point.
Usually, to indicate a point of a chart, one writes the values of the co-ordinates between bracket (after the capital letter constituting the name of the point) while always putting in first, the value of the X-coordinate and as a second the value of the ordinate.
For example, to indicate the points of figure 2, one writes in summary:
It will always be necessary to have for the spirit this shortened writing each time a point of the graph must be indicated by its co-ordinates or, conversely, when knowing the co-ordinates, one will have to locate a point in the plan.
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