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We're talking about Bezier curves in the context of CAD and graphics. In this case, I believe there is no reason to assume that these curves will have a more sp
by sfpotter 2y ago
We're talking about Bezier curves in the context of CAD and graphics. In this case, I believe there is no reason to assume that these curves will have a more special form than sqrt(arbitrary quartic). Do you think that they do have a more special form, and that this form will simplify nicely? Or are you suggesting that sqrt(abrbitrary quartic) might simplify more?
- Dylan16807 2y agoThe issue is that "sqrt(arbitrary quartic)" is already more specialized than "elliptic integral". So we can't just talk about elliptic integrals in general, we need proof that this specialization doesn't give rise to closed forms.
- bubblyworld 2y agoIf I may, your confusion seems to come from the meaning of "elliptic curves in general have no closed form". You seem to interpret that as "given any elliptic integral A, there is no closed form for the solution of A". This is false (there are many counterexamples). What it actually means is "there is no single closed-form that produces the solution of any given elliptic integral". The quantifiers are the other way around. This is why I brought up the halting problem. There's a similar confusion that often comes up, where people think that it means there's no way to determine if any given program halts. But this is false - the program "let X=5*6" trivially halts, for instance. What it actually means is that there's no single program that can uniformly determine whether any given input program halts. It's exactly the same situation.
- sfpotter 2y agoNo, I'm not getting this confused at all. I understand the point. What I'm saying is that because the general elliptic integral corresponding to sqrt(quartic) doesn't have a closed form, and because this (or maybe a slightly more specific form) is what's of interest in this context (CAD), saying something about the closed form of specific sqrt(quartic) elliptic integrals isn't very interesting as far as I can see.
- bubblyworld 2y agoOkay, I'm not really sure what you're talking about, apologies. I don't have anything else to add.
- sfpotter 2y agoYou're getting hung up on a logical point which, while correct, isn't particularly salient to the context in which this conversation is occurring (CAD and graphics). The original article, while mathematically interesting, is also not particularly salient within that context.
- bubblyworld 2y agoDon't tell people how to take their drugs =) I'm personally more interested in the maths.
- sfpotter 2y agoTake whatever drugs you like. You just seem a little bewildered why people aren't more fired up about what you have to say about the halting problem. Just trying to suggest that it may be because this isn't the right venue for it. I wish you many happy mathematical returns. :-)
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- kevinventullo 2y agothe general elliptic integral corresponding to sqrt(quartic) doesn't have a closed form I see, so you’re claiming that there is no closed form for the integral of sqrt(quartic). That is a different statement that neither directly implies nor is implied by the statement in the Wikipedia article. Maybe it is true and has been proven though, I don’t know!