Is radius a misnomer in a polar equation?

[nomadreid]

Is "radius" a misnomer in a polar equation?

Often I see the description of "r" in a polar equation r = r(theta) as being "radius", but "radius" is a length, and here you can have a negative r. Hence "radius" is a misnomer, as far as I can tell. Perhaps it would be better described with some term like "signed radius" or some vector terminology?

 

[Stephen Tashi]

here you can have a negative r

More than making the connotation of "radius" offensive, wouldn't that invalidate the standard formulae for converting between polar and cartesian coordinates?

 

[mathman]

In polar coordinates r is always non-negative.

 

[nomadreid]

First, Stephen Tashi: I am not sure having negative r's would invalidate the formulae for converting from polar to Cartesian, even if it would play havoc with the intuitive formulation. [For example, the conversion to parametric proceeds quite smoothly with negative values of r(theta): for example, r=cos(theta) translates into (x=cos^2(theta), y = sin(2*theta)/2, which is the same graph as r = cos(theta). Onward to Cartesian just requires care with the inverses.]. However, in the reverse direction, from Cartesian to polar, one would end up, using the standard conversions, with non-negative r's, and various cases with tangent, which will not necessarily be the most elegant formulation, but would still be valid. I think, without having worked it out. I would be love to be corrected, though, if you have worked this out, as it would also bother me.
To mathman: when polar coordinates are given as isolated points, then r is always non-negative; however, when describing a curve, one uses such formulae as r=r(theta) =cos(theta), not r = |cos(theta)|. This latter gives a different graph to the former; the latter gives a "bunny hop" graph, while the latter gives a circle, the standard graph for this polar equation. Hence, it appears that r(theta) can be negative. Alternatively, one could have an awkward definition, saying that r= cos(theta) really means the collection of points {(|cos(theta)|, alpha) : alpha = theta when cos(theta) is non-negative, & alpha = theta + pi when cos(theta) is negative}, but I have never seen such a definition. However, I would be happy to have my mistake here pointed out to me.

 

[Hurkyl]

There's nothing wrong with negative r; it's still clear what point in the plane an [itex](r,\theta)[/itex] pair refers to.

This isn't much different than the fact [itex](r, \theta)[/itex] and [itex](r, \theta + 2 \pi)[/itex] refer to the same point.

 

[nomadreid]

Thanks, Hurkyl. So, negative r's are OK. Back to my original point, that the r being referred to as "radius" is misleading, since "radius" is defined as a positive quantity.

Your point about (r,θ) = (r,θ+2π) brings up an error in my previous post (in my "awkward definition"), in that "α=-θ" should have been "α=θ+π". I have therefore edited that. Thanks.

Any suggestions for a better name for r?

 

[HallsofIvy]

No, "radius" or "radial length" are perfectly good.

 

[Stephen Tashi]

for example, r=cos(theta) translates into (x=cos^2(theta), y = sin(2*theta)/2, which is the same graph as r = cos(theta).

You are arguing that it is possible to modify the standard formulas, not that it is possible to use the standard formulas as they are.

while the latter gives a circle, the standard graph for this polar equation.

The graph of a circle in standard polar coordinates is r = c where c > 0 is the constant radius of the circle. There is no [itex] \theta [/itex] in the equation.

There's nothing wrong with negative r; it's still clear what point in the plane an [itex](r,\theta)[/itex] pair refers to.

There's nothing ambiguous about a negative r, but a negative r is not used in standard polar coordinates - at least as I read the Wolfram Alpha page on the subject. Of course, it is possible to define many different coordinate systems. The question of whether we can use a negative r in standard polar coordinates is a legalistic issue, it's a matter of definition and I don't claim that Wolfram Alpha is ultimate authority. (Another interesting legalistic issue of what range of angles is permitted in standard polar coordinates. Wolfram says to "use the two argument arctan function" to compute the angle.)

Maybe the people who do the "chart and atlas" thing for the coordinates of manifolds can point out an authoritative source.

 

[Office_Shredder]

Stephen, negative values of r aren't typically used when expressing the polar coordinates of a point, but often come up when writing down a polar equation

r=sin(theta) for example gives many negative values of r. So we just say that negative values of r are on the other side of the origin because that is what gives us graphs that look good and useful to us

 

[Stephen Tashi]

negative values of r aren't typically used when expressing the polar coordinates of a point, but often come up when writing down a polar equation

But are the ordered pairs produced by a "polar equation" necessarily "polar coordinates"? Or are they merely numbers that can be converted to a polar cordinates?

As I said, this is a legalistic technicality that depends on the strict definition of "polar coordinates". It's interesting how the usual web sources are slightly ambiguous about the relation between "polar equations" and "polar coordinates".

A more general technicality is what are the general requirements for a "coordinate system". In particular, do we wish the same point to be described by more than one set of coordinates? Do we allow the the conversion formulae from one coordinate system to another to be functions that don't have inverses?

This is why I suspect that the people who study mainfolds in rigorous fashion are the ones to explain this.