When we start to learn about a curve, we get to know about an arbitrary point (x, y) which traces the path of the curve in accordance with an algebraic equation [which is given].
For example, when we have to define a curve, say, a circle, we try to establish a mathematical expression which generates that circle [i.e., an algebraic equation which the arbitrary point (x, y) traces] We do so on a plane which acts as a reference system for the curve [or for set of all points satisfying the algebraic equation]. This reference system is called as Cartesian Co-ordinate System.
Mutually perpendicular axes of reference X-axis and Y-axis intersect at O – the Origin. Every Curve tracing is done with respect to the origin.
So, in Cartesian Co-ordinate System, we draw a plane figure [locus of a point] which satisfies the given equation. Well, that equation is a relation between x and y.
For a circle of radius ''r'' with center at point (0,0) i.e., origin the equation will be x^2 + y^2 = r [simply found by using distance formula. Hence we try to find a relation between arbitrary point (x,y) and center of the circle, say, (h, k) here the origin, by means of radius ''r''. We do this as we know the property of circle that all points on its circumference are equidistant form center of the circle and that distance is radius.]
Hence we generate the curve.
When it comes to graphing a function, we say there are two sets R and R [R – set of Real numbers], we take the cross products which results in a plane like Cartesian Plane. But is it the same or different? It has X and Y co-ordinates. We can draw a curve here too. We just don't relate the points, we relate them such that y is the unique image of x. A function has its domain and range [set of numbers which obey what the function says, i.e., the points where function is defined.. its real]
What is the difference between the reference planes where we trace a plane figure and plot graph?
When we are asked to plot the graph of circle we discussed above, it turns out to a semicircle lying on one side of X axis. [as we are plotting a function, it should pass the vertical line test!]
But what makes them different? The reference Planes.. are they different?
I know one thing that Graphs are relative and we plot how y changes / depends on x. But in case of tracing a curve we simply do it by a known algebraic equation that describes it.
While doing analysis of a physical situation what do we use? I mean, say. a particle is moving and we have got to represent its motion. A position – time graph says where a particle is / was at a given time. It doesn't say actual path traced by the particle. But to get that, we should trace a curve using analytic geometry i.e., a position v/s position curve.
Am I right.
I have already messed up what I really intended to ask. It is as if I know the answer / concept but can't convince myself / put them together or find link..
If you are getting me, please answer.
Its kinda silly and is worth being reported as a wrong or bad question. I accept that.
Thank you anyways.
Best Answer
Hmm...
I think she is asking clarification between a reference system which is simply used to describe a particle in 2 or 3 dimension [its position, path or displacement] and graph where we plot variation of a quantity with respect to the other.
Well, we made a reference system so that we can strictly define a particular point or event in space. But only the co-ordinates of a point are not sufficient, so the vectors are the tools which define a point, a line or a plane in space or a 2 dimensional system.
You should first study the Cartesian Co-ordinate system to know how points are plotted in a plane. A point is generated by a simple relation between its co-ordinates x and y. It may be random or described by an algebraic equation. So, if there happens to be unique y for x, it will be a function [for all x belonging to the domain of definition]
This is how we know about plotting a curve [which is basically showing which x is related to which y by a rule] It can be a relation or a function. This is where a basic math about this ends.
Then lets talk about Vectors and 3 Dimensional Geometry. There you are, to again learn how to mark a point, know its co-ordinates [x,y,z], plot a curve etc use of vectors makes it all easier.
To put it all together, "There is nothing called ABSOLUTE " when we are measuring anything. So every measurement is done relative to the other. By looking at your question I feel like you are a physics student who has trouble in applying math to physical world. So, let us make things clear one by one.
Like I said, measurement is relative. You can't just say a particle is at ''Rest".. it is at rest with respect to some other quantity. If anything is moving that means it is moving [changing position with respect to time or your position] relative to some quantity. So, there was a need to create a reference system. Thus arose Cartesian Co-ordinate system. It has mutually perpendicular axes X Y and Z. with an origin O with respect to which we can make measurements.
In math we JUST learn how to plot an algebraic equation i.e., a relation [sometimes] and a function. So, using terms like variation / dependence of a quantity makes MORE sense in physics.
We know some math and we effectively try to apply it to the physical quantities or changes we see by establishing a mathematical relation between objects. So, whether your curve is talking about actual path traversed by a particle or variation of its position w.r.t. time is a simple matter of what you are plotting in a Cartesian Plane. like position - time or position - position variation. As the X O X', Y O Y', Z O Z' are just axes of a REFERENCE system RELATIVE to which you can make measurement. So, what you plot tells you whether its a relation or function [in math] or actual path [position - position curve... Well while studying things like position vector or displacement vector there will be elaborate explanation for this. Also study 3 D geometry. How a particle moves or traces its path in space is better understood by this. ] or variation of its position w.r.t time. So is true for the case of simple pendulum. It all depends on what you are measuring relative to the reference system you have. Don't think a graph and Cartesian plane are different. We just want a reference system to make a sensible measurement which will be true regardless of time etc.
"No measurement is ABSOLUTE"
Hope this works for you.