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January 29 Information
Cartesian curve
Is there a curve equation in cartesian system that can trace any set of points.
Consider the set consisting of a single point (a,b). This is a degenerate circle and has the equation (x−a)²+(y−b)²=0.
Now consider the set consisting of exactly two points, (a,b) and (c,d). Since an equation like pq=0 is solved if either p=0 or q=0, we can combine two equations like the first example and write the equation ((x−a)²+(y−b)²)((x−c)²+(y−d)²)=0 for this new set.
Now consider a set consisting of an infinite number of points chosen at random in the plane. Since they are random it is not possible to write a simple equation that covers them all, the way an equation like x+2y=42 could trace the infinite number of points in a straight line. The only way to get an equation would be to combine a separate component for each point, in the manner shown above. And I don't think this fits any definition that most peoplecitation needed would use for an equation. So I think the answer to the original question is no.
As long as the set of points is non-empty and finite (say with n elements), you can always chose a cartesian coordinate system K' that will make the points into a subset of a function (i.e. that will not have two different y values for a single x value). In that case, you can always find an
interpolating polynomial of degree n-1 that goes through all the n points. I'm fairly sure that a marginally better mathematician than me can easily find a shift/rotate operation that will convert this polynomial equation back into an equation in the original coordinate system K. --
Stephan Schulz (
talk)
13:17, 29 January 2017 (UTC)reply
Say a is the first element. If you mean that each element after the first one is 1/2 the sum of all previous elements, then what you have is a, 3a/2, 9a/4, 27a/16, 81a/32..., which is a
geometric progression with the common ratio 3/2?. --
76.71.6.254 (
talk)
08:33, 30 January 2017 (UTC)reply
Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is a
transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the
current reference desk pages.
January 29 Information
Cartesian curve
Is there a curve equation in cartesian system that can trace any set of points.
Consider the set consisting of a single point (a,b). This is a degenerate circle and has the equation (x−a)²+(y−b)²=0.
Now consider the set consisting of exactly two points, (a,b) and (c,d). Since an equation like pq=0 is solved if either p=0 or q=0, we can combine two equations like the first example and write the equation ((x−a)²+(y−b)²)((x−c)²+(y−d)²)=0 for this new set.
Now consider a set consisting of an infinite number of points chosen at random in the plane. Since they are random it is not possible to write a simple equation that covers them all, the way an equation like x+2y=42 could trace the infinite number of points in a straight line. The only way to get an equation would be to combine a separate component for each point, in the manner shown above. And I don't think this fits any definition that most peoplecitation needed would use for an equation. So I think the answer to the original question is no.
As long as the set of points is non-empty and finite (say with n elements), you can always chose a cartesian coordinate system K' that will make the points into a subset of a function (i.e. that will not have two different y values for a single x value). In that case, you can always find an
interpolating polynomial of degree n-1 that goes through all the n points. I'm fairly sure that a marginally better mathematician than me can easily find a shift/rotate operation that will convert this polynomial equation back into an equation in the original coordinate system K. --
Stephan Schulz (
talk)
13:17, 29 January 2017 (UTC)reply
Say a is the first element. If you mean that each element after the first one is 1/2 the sum of all previous elements, then what you have is a, 3a/2, 9a/4, 27a/16, 81a/32..., which is a
geometric progression with the common ratio 3/2?. --
76.71.6.254 (
talk)
08:33, 30 January 2017 (UTC)reply