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Thread: polar coordinates vs polar equations - terminology

  1. #1
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    polar coordinates vs polar equations - terminology

    The general definition of a "coordinate system" says that the coordinates of a point specify a unique point. It doesn't say that a given point has only one set of coordinates. However, it simplifies matters if a point has unique coordinates. When a precise definition of "polar coordinates" is given, which is the case - do we allow a point to have many different polar coordinates or do we define them in such a way that the polar coordinates of a point are unique? Or is there some technical phrase like "standard polar coordinates" that indicates that such a definition is made?

    A "polar equation" like r = cos(theta) is satisifed by pairs of numbers (r, theta) where r is negative. Do we say that the pairs of numbers that satisfy a polar equation "are" polar coordinates? Or do we only say that they are pairs of numbers can be converted to polar coordinates? (by finding a representation where r is positive).

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    In polar coordinates, a point can be represented by a positive radius or a negative radius, r.
    Imagine the point (1,1) in the xy plane. In polar coordinates(r, theta), it is (1.414, Pi/4) or (-1.414, 5Pi/4) or (1.414, -7PI/4) or (-1.414, -3Pi/4).
    Using theta makes it so that there are infinite possible representations of the same point. Just add 2*Pi to the angle.

    I try to stick to positive r and positive theta.

    I don't understand the second part. An ordered pair can be a solution on the polar curve. What is "Do we say that the pairs of numbers that satisfy a polar equation "are" polar coordinates?"? Isn't the solution to r=cos(theta) simply a circle centered at (1/2, 0) with radius 1/2 in polar coordinates?

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    Quote Originally Posted by ShinAce View Post
    In polar coordinates, a point can be represented by a positive radius or a negative radius, r.
    Imagine the point (1,1) in the xy plane. In polar coordinates(r, theta), it is (1.414, Pi/4) or (-1.414, 5Pi/4) or (1.414, -7PI/4) or (-1.414, -3Pi/4).
    Using theta makes it so that there are infinite possible representations of the same point. Just add 2*Pi to the angle.
    I don't know if your example is intended as a justification of using a negative radius. Whether a point can be represented by a negative radius in polar coordinates isn't (only) a matter of whether one can use a negative radius and angle to compute the cartesian coordinates of a point. It is a matter of what conventions are adopted in the formal definition of "polar coordinates".

    I don't understand the second part.
    If you take for granted that a point can have many different representations in polar coordinates, you wouldn't! The question is whether one can take this for granted. If not then we have to speak carefully about what pairs of numbers (r,theta) are legitimate "polar coordinates".

    As another example, many sources say "the" formula for computing "the" value of the polar coordinate r for a point whose cartesian coordinates are (x,y) is r = sqrt( x^2 + y^2) ..... If a point can have many different polar coordinates, then these sources should say "a" formula for computing "an" r value of a point whose cartesian coordinates are (x,y) is ... .etc.

    I'm asking about a technicality of a definition - not about "what we usually do" or "the most common practice is..". (I don't have any axe to grind one way or another about how polar coordinates are defined. I just find it amusing when mathematics, which is suppose to deal in precise language, gets written up imprecisely.)

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    Quote Originally Posted by tashirosgt View Post
    I don't know if your example is intended as a justification of using a negative radius. Whether a point can be represented by a negative radius in polar coordinates isn't (only) a matter of whether one can use a negative radius and angle to compute the cartesian coordinates of a point. It is a matter of what conventions are adopted in the formal definition of "polar coordinates".



    If you take for granted that a point can have many different representations in polar coordinates, you wouldn't! The question is whether one can take this for granted. If not then we have to speak carefully about what pairs of numbers (r,theta) are legitimate "polar coordinates".

    As another example, many sources say "the" formula for computing "the" value of the polar coordinate r for a point whose cartesian coordinates are (x,y) is r = sqrt( x^2 + y^2) ..... If a point can have many different polar coordinates, then these sources should say "a" formula for computing "an" r value of a point whose cartesian coordinates are (x,y) is ... .etc.

    I'm asking about a technicality of a definition - not about "what we usually do" or "the most common practice is..". (I don't have any axe to grind one way or another about how polar coordinates are defined. I just find it amusing when mathematics, which is suppose to deal in precise language, gets written up imprecisely.)
    In polar coordinates, it is generally understood that the radius argument is positive and the angle argument is between 0 and 2 pi. We generally say that (r,theta) is the location of p (for example) 'in polar coordinates' or 'in polar form'. Remember, in more general problems, especially physics problems, coordinate systems are imposed upon a space for convenience, and 'location' isn't really an intrinsic property of a point. For example, two points in a space may be arbitrarily close given one notion of distance and arbitrarily far given another.

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    Ill add that a more convenient notation is to use De Moivre's thm, then any polar coordinate can be written as a number.

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    And also, when r=0 any theta will still indicate the same point.

    This non-uniqueness of polar-coordinates is intrinsic unless you make very well-defined canonization rules which is one of the reasons why they're really not good for comparing point equality.
    As this is something very seldom needed for the applications where polar coordinates makes sense, it's not a problem unless you really want it to be.

    So basically tashirosgt: why do you think this non-uniqueness might be a problem?
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    Polar coordinates were introduced by Newton, who did not have a computer.

    My calc textbook does not say (r,theta) are THE polar coordinates. It just says that the pair are called polar coordinates of a point P. It mentions that positive angles are ccw from the pole. It also mentions that we can extend the meaning to include negative radius. (-r, theta)=(r, theta+pi)

    Unless you're dealing with computer programming that limits your use of polar coordinates, there is no problem. Kind of like how a calculator never gives a negative answer for a square root, which we understand is an incomplete answer.

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    I think the issue here is reduceability.
    Is saying I am 2/4 of the way there different than saying 1/2 of the way there?
    No; it is a unique value that is just expressed in a different way.

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    Quote Originally Posted by HenrikOlsen View Post
    So basically tashirosgt: why do you think this non-uniqueness might be a problem?
    It isn't a problem that involves explaining any aspect of Nature. It's a matter of terminology and legalistic interpretation. The terminology and language is just something human beings have created. Many bautforum members are concerned with such problems (There are often threads on the origin of words and expressions, grammar, spelling etc.) , so this is an excellent place to ask.

    Shinace's report of his calculus text shows it has been very careful with its language (avoiding any statement which refers to "the" polar coordinates of a point.). But it has given less than a precise mathematical definition. To say "we can extend the meaning" may be good informal instruction, but a mathematical definition should either extend the meaning or not extend the meaning.

    To me, an example of straightforward definition would be:

    The pair of real numbers (r,theta) is a polar coordinate for a point p iff p has the cartesian coordinates (r cos(theta), r sin(theta)) (where "iff" is the standard abbreviation for "if and only if").

    If you want to refer to the special case where r is non-negative and theta is in , say, (-pi, pi], you could have terminology like "the standard polar coordinates" or "the principal polar coordinates". I don't know whether this is done by any authoritative sources.

    The usual online sources one consults avoid giving a precise definition of polar coordinates. An exception is the Wolfram MathWorld page, which unambiguously begins "The polar coordinates..." and defines r in such a manner that it must be non-negative. Of course, one could argue that this page refers to the Wolfram software, Mathematica. However, I think the site is intended as a general reference for mathematics.

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    The wiki page looks good. It even has a section talking about the uniqueness of polar coordinates.

    During our entire childhood, we are taught to recognize xyz(ijk basis vectors). However, we could use an infinite number of bases. There's nothing unique about the cartesian system. There are other fully orthogonal bases with 3 vectors which are cartesian systems but don't have coordinates as (x,y,z). Consider the basis vectors (1,-1,0) , (1,1,0), (0,0,1).

    It seems you've taken to thinking that cartesian is unique and any translation should also be unique. As long as the ordered pair gets me to the same point, I'm happy.

    It took a long time to come up with a good definition of the limit. The idea comes first, then the precise definition follows. It's nice to simply define and derive, but that's not how it always works out.

    As an aside, I once blasted my group for submitting a university report with not a single legitimate reference. Out of the 15 references, 3 were mine and all 3 came from books. The other 12 were websites. How many websites are going to tell me where they got their info from?

    If you want a precise definition, don't use wolfram. You lose all credibility when you do.

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    Quote Originally Posted by tashirosgt View Post
    To me, an example of straightforward definition would be:

    The pair of real numbers (r,theta) is a polar coordinate for a point p iff p has the cartesian coordinates (r cos(theta), r sin(theta)) (where "iff" is the standard abbreviation for "if and only if").
    To me that's not straightforward as you forgot "assuming an euclidean plane geometry".

    One very good place to use polar coordinates is on the surface of a sphere, where your straightforward definition is meaningless but the polar coordinates nevertheless are perfectly logical and easily understood.
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    Quote Originally Posted by HenrikOlsen View Post
    To me that's not straightforward as you forgot "assuming an euclidean plane geometry".
    That's unnecessary for defining a coordinate system. If I defined a distance function, that would get into geometry. All I need in the definition is the fact that a point has unique cartesian coordinates. The point with cartesian coodinates need not be a point "in space". The coordinates could represent two temperatures or other quantities.

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    Quote Originally Posted by ShinAce View Post

    It seems you've taken to thinking that cartesian is unique and any translation should also be unique.
    Coordinates for a point in a given cartesian coodinate system are unique. You're talking about the possibility of different cartesian coordinate systems. If we want to talk about different coordinate systems, then (20, pi/4) and (6, pi/2) could be polar coordinates for the same point since we'd have the choice of where to put the origin and how to orient the reference line for measuring the angle.

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    Quote Originally Posted by tashirosgt View Post
    Coordinates for a point in a given cartesian coodinate system are unique. You're talking about the possibility of different cartesian coordinate systems. If we want to talk about different coordinate systems, then (20, pi/4) and (6, pi/2) could be polar coordinates for the same point since we'd have the choice of where to put the origin and how to orient the reference line for measuring the angle.
    Be careful. The point is unique, yes. The coordinates for the point are not unique. In other words, given coordinates, you have a unique point. But given a unique point, there is not necessarily unique coordinates.

    A implies B does not mean B implies A.

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    Quote Originally Posted by ShinAce View Post
    Be careful. The point is unique, yes. The coordinates for the point are not unique. In other words, given coordinates, you have a unique point. But given a unique point, there is not necessarily unique coordinates.
    For the general definition of a coordinate system, I already pointed that out in my original post. There do exist coordinate systems where the coordinates for a given point are unique. The cartesian system is such a system.

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    Quote Originally Posted by tashirosgt View Post
    That's unnecessary for defining a coordinate system. If I defined a distance function, that would get into geometry. All I need in the definition is the fact that a point has unique cartesian coordinates. The point with cartesian coodinates need not be a point "in space". The coordinates could represent two temperatures or other quantities.
    And in that case a geometric mapping to polar would be absurd.

    I though a good common assumption when talking coordinates and coordinate transforms is that the none of the coordinate systems are insane.

    There are point sets that doesn't map well to cartesian coordinates.
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    Polar coordinates need not be unique, but it's sometimes useful to impose restrictions upon them that make them unique. Like in calculators, or when making a change of variables in an integral. But different restrictions may be convenient in different problems. This is contextual, and there isn't really much point in getting very nitpicky about it. There are literally infinite ways to make polar coordinates unique.

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    If I give you the cartesian coordinates (sqrt(5), sqrt(16)), in which quadrant is the point?
    sqrt(16) can be reduced to +/-4. While sqrt(5) becomes +/-sqrt(5).

    It looks unique, but it's not. The point could be in any of the 4 quadrants. So to make cartesian unique, we would have to write (+sqrt(5), -sqrt(16)), which I have yet to see people do. If there's no + in front of the sqrt, we assume it's positive. However, we started with something that wasn't unique and assumed it was. That's the convention and there's nothing wrong with it.

    It's really just a computer problem. Computers see uppercase and lowercase letters as different values. People do not. So in computers we often convert everything to a certain case and then work on it.

    The same applies with polar coordinates. Some operations may apply no matter what the coordinates given are. Sometimes we would have to convert to positive r and 0 < theta < 2Pi. For all real values of r and theta, you have polar coordinates which can be mapped to a single point in cartesian coordinates.

    How do you find polar coordinates for given cartesian coordinates? You use pythagoras to say that the radius squared = x squared + y squared. But then r could be positive or negative. Then you use the fact that tan(theta)=y/x to find theta. Again, you get two angles with the same value of y/x . Even our system to convert cartesian to polar coordinates requires you to do a bit of work. But cartesian coordinates are unique, so why don't we go around teaching people that you always use positive r and then you use the signs of x and y to deduce the proper angle? Because that's overkill. Use positive r and figure out which of the two angles is the right one. Why add steps that aren't necessary?

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    Quote Originally Posted by ShinAce View Post
    sqrt(16) can be reduced to +/-4. While sqrt(5) becomes +/-sqrt(5).
    That is not correct. The sqrt() function is not multi-valued if you are talking about the one defined on the real numbers. If you are talking about complex analysis and "multi-valued functions", then you haven't given the cartesian coordinates of a point.

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    Quote Originally Posted by ShinAce View Post
    If I give you the cartesian coordinates (sqrt(5), sqrt(16)), in which quadrant is the point?
    sqrt(16) can be reduced to +/-4. While sqrt(5) becomes +/-sqrt(5).
    You are confusing for the equation . is defined as the positive solution to , so i.e. first quadrant with no ambiguity.
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