# Cartesian and polar terminology

• I
• DrGreg
In summary: It is not what you want. So you have decided that you can't show the plots. In that case you can't get help and I am out of the conversation.I have given you my opinion. It is not what you want. So you have decided that you can't show the plots. In that case you can't get help and I am out of the conversation.In summary, the conversation discusses the different ways of plotting a scalar quantity V as a function of angle θ. The options include a Cartesian plot, a polar plot, and a parametric plot. The difficulty lies in finding clear and unambiguous terminology to describe these plots and their respective coordinates. The suggested terminology includes directional voltage response for

#### DrGreg

Gold Member

I have a scalar quantity ##V## (let's call it a voltage for concreteness) that is a function of angle ##\theta##. There are two obvious ways to plot it, as a Cartesian plot (see A above) or as a polar plot (see B). I can also express the polar plot in terms of Cartesian coordinates ##V_x = V \, \cos \theta## and ##V_y = V \, \sin \theta## (see C). A particular mathematical process that I have to document involves calculating ##V_x## and ##V_y## as an intermediate step to providing a final output of ##V## and ##\theta##.

(The plots above are illustrative and do not show the function that I have to document in my real-world problem.)

In my document, there is a part where I can describe all this in detail using equations and plots, but there are other parts of the document where I have to summarise the steps of the process in words only. I want to find words to describe A, B, C in just a few words per plot, but still clearly and unambiguously. I have no hesitation in describing B as a "polar plot", but A and C are more problematic. Both could be described as "Cartesian plots" but that isn't enough to distinguish them. Indeed my co-author has unintentionally described both A and C identically, including the phrase "in Cartesian coordinates". I don't really like to describe ##(V,\theta)## as Cartesian coordinates even in the context of Plot A.

As I see it, A and B show the same coordinates but different plots, whereas B and C show the same plot with different coordinates.

Part of my difficulty is that I'm not sure whether the words "polar" and "Cartesian" (or "rectangular" if you prefer) are properties of the coordinates, or of the plots, or the combination of both.

For clarification, ##\theta## does represent a geometrical angle in the real world, but ##V## does not represent a distance.

My suggested terminology is

A = directional voltage response as a Cartesian plot

B = directional voltage response as a polar plot

C = ??

##(V,\theta)## = polar coordinates (or polar voltage coordinates??)

##(V_x,V_y)## = ??
Any thoughts?

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I would call,
A: Plot of V as a function of θ.
B: Polar plot of V.
C: Parametric plot of V.

Such are the names that Mathematica uses for these plots, so there must be some common understanding that that's what they are.

phinds and mfb
kuruman said:
I would call,
A: Plot of V as a function of θ.
B: Polar plot of V.
C: Parametric plot of V.

Such are the names that Mathematica uses for these plots, so there must be some common understanding that that's what they are.

I tried to simplify the statement of my problem to exclude unnecessary detail. But I now realize I've oversimplified it.

Actually I'm not plotting ##V## against ##\theta##. I'm really plotting a third variable ##Q## which is a function of both ##V## and ##\theta##. It's plotted by colouring each pixel on the plot by a colour that represents the value of ##Q##.

DrGreg said:
Actually I'm not plotting VV against θ\theta. I'm really plotting a third variable Q which is a function of both V and θ. It's plotted by colouring each pixel on the plot by a colour that represents the value of Q.
In that case, I would call that a Contour Plot (another term used by Mathematica) where a given color represents a constant value of Q.

kuruman said:
In that case, I would call that a Contour Plot (another term used by Mathematica) where a given color represents a constant value of Q.
But all three are contour plots of ##Q## (as a function of ##V## and ##\theta## in A and B, and of ##V_x## and ##V_y## in C). But I need to find descriptions in words only, no algebraic symbols.

A, B and C in your example, post #1, are not contour plots but show single contours. A contour plot has many contours labeled by numbers as, say isobars on a weather map or different colors as you described in post #3 for values of quantity Q. Perhaps it might be expedient to post the actual plots instead of describing them.

kuruman said:
A, B and C in your example, post #1, are not contour plots but show single contours. A contour plot has many contours labeled by numbers as, say isobars on a weather map or different colors as you described in post #3 for values of quantity Q. Perhaps it might be expedient to post the actual plots instead of describing them.
Consider that the curve I drew in my three plots shows one of many contours. Imagine a family of similar "parallel" curves above/around it. I can't show the actual plots.

DrGreg said:
Consider that the curve I drew in my three plots shows one of many contours. Imagine a family of similar "parallel" curves above/around it. I can't show the actual plots.
I have given you my opinion.

## 1. What is the difference between Cartesian and polar coordinates?

Cartesian coordinates use two perpendicular axes, x and y, to represent a point's distance from the origin in a straight line. Polar coordinates use a point's distance from the origin and its angle from a reference line (usually the positive x-axis) to represent its location.

## 2. How do you convert from Cartesian to polar coordinates?

To convert from Cartesian coordinates (x,y) to polar coordinates (r,θ):
1. Calculate r using the Pythagorean theorem: r = √(x² + y²)
2. Calculate θ using the inverse tangent function: θ = tan⁻¹(y/x)

## 3. What is the meaning of the origin in Cartesian coordinates?

The origin (0,0) in Cartesian coordinates is the point where the x and y axes intersect. It is considered the starting point for measuring the distance and direction of other points on the coordinate plane.

## 4. How do you plot a point using polar coordinates?

To plot a point using polar coordinates (r,θ):
1. Start at the origin (0,0).
2. Move r units away from the origin in the direction of θ (angle) from the reference line.
3. Mark the point where the line intersects with a circle of radius r centered at the origin.

## 5. What are some examples of real-world applications of Cartesian and polar coordinates?

Cartesian coordinates are commonly used in maps, graphs, and computer graphics. Polar coordinates are often used in physics, engineering, and navigation systems such as GPS. They are also used in polar graphs to represent equations and in astronomy to locate stars and other celestial objects.