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The Graph w/ my problem

I do most of my graphing using Desmos, and also use graphing as a visual check for some of the math that I'm a little more wary of when I don't have access to someone to double check my work. I was doing something when I noticed a discrepancy which boils down to the title question. Above is a picture of Desmos's graph of those two equations. If it's just a problem with the calculator, I'll soon report it as a bug and be very relived that math isn't broken.

pooja somani
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Aaron Quitta
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    I can't see your graphs, but I'm guessing you didn't tell Desmos (however one does that) that the first one should be in polar coordinates. Is there a place to select "polar" as opposed to "cartesian"? – Ted Shifrin May 21 '18 at 02:42
  • There is no need for that, I'll attach an image instead of the graph, or you can try it yourself just by typing the 2 equations in. – Aaron Quitta May 21 '18 at 02:46
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    When I opened the link to your graph, it was behaving a little buggy. After switching between rectangular grid and polar grid and turning graphs on and off, it eventually righted itself. In particular using just 1 and 3 plots correctly for me. Also went to Default Zoom. Bad cache maybe. – sharding4 May 21 '18 at 02:46
  • @TedShifrin That isn't the issue. It appears (to me) to be the result of a loss-of-precision or an issue with rounding. Basically, the two curves are very, very close to each other, but off by a small amount (maybe a millionth of a unit?). – Xander Henderson May 21 '18 at 02:46
  • I remove myself from this discussion. – Ted Shifrin May 21 '18 at 02:48
  • @sharding4 Very interesting. I'll see if that helps me. EDIT: Doesn't seem to be a local problem as far as I can tell, I can replicate it on other browsers/devices. I'm going to report it as a bug to the developers. – Aaron Quitta May 21 '18 at 02:50
  • I'm fairly sure it's just a precision error. Zoomed out the curves seem to coincide, as should be. – rubikscube09 May 21 '18 at 03:06

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The commenters guessing it was a precision error were correct. As far as I can tell, it's an unfortunate consequence of the way Desmos graphs polar equations. Desmos graphs nearly all Cartesian equations with smooth curves, but it uses minuscule line segments to approximate the graphs of polar equations.

For a rough comparison, you can graph $r=\sin(9\theta)$ and $y=0.77\sin(9x+2)$ (in green and purple below, respectively), which have somewhat similar curvature at their relative extrema, and zoom in on the approximate intersection of two extrema in the first quadrant, at about $(0.64,0.765)$ to see how differently Desmos plots both curves (notice also that when zoomed out, they both look smooth). I suspect you were seeing a similar kind of effect when you zoomed in very far on your graph.

enter image description here

Robert Howard
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