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January 29

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 people^{[citation needed]} would use for an equation. So I think the answer to the original question is no.

--76.71.6.254 (talk) 07:57, 29 January 2017 (UTC)[reply]

Your question's rather vague. But for any reasonable formulation, the answer is either "Yes, a space filling curve" or "No".--2406:E006:C32:1:4CF1:9436:B805:9A8A (talk) 11:51, 29 January 2017 (UTC)[reply]

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]

Just rotation will do, and rotations are easy (do we have an article titled rotation matrix?). --JBL (talk) 14:37, 29 January 2017 (UTC)[reply]

name/formulas series

How do you call a series where each element is 50% of all previous elements? — Preceding unsigned comment added by 31.177.96.51 (talk) 22:57, 29 January 2017 (UTC)[reply]

Do you mean half of the immediately preceding element, or half of the sum of all previous elements, or what? (And why?) —Tamfang (talk) 03:43, 30 January 2017 (UTC)[reply]

The first element is not. Bo Jacoby (talk) 07:49, 30 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]

Surely a, a/2, 3a/4, 9a/8? -- SGBailey (talk) 17:14, 30 January 2017 (UTC)[reply]

Arrgh, sorry about that. So it's actually a geometric progression only after the first element, as Dmcq says below. --76.71.6.254 (talk) 04:58, 31 January 2017 (UTC)[reply]

If a₀ = a and a_n (n > 0) is half of the sum of the previous n elements then shouldn't that be a, a/2, 3a/4, 9a/8 etc. ? Gandalf61 (talk) 09:29, 30 January 2017 (UTC)[reply]

Well a geometric progression except for the very first element. Dmcq (talk) 22:29, 30 January 2017 (UTC)[reply]