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Not a math guy, but I like working with computer graphics. Is there some theory in which the tangent curve has its +/- infinities connected? I like to imagine these are connected in 3D as a circle or spiral, and what we see in a 2D graph is the projection that makes the curve look broken.
What you're intuiting might be projective space, which is secretly fundamental to the study of plane curves!

https://math.stackexchange.com/questions/85394/why-the-empha...

It's sometimes inconvenient to have all lines to intersect at a single infinite point, eg. for Bezout's theorem, so mathematicians instead add a point for every +- direction.

> Is there some theory in which the tangent curve has its +/- infinities connected?

I'm sure there are much simpler ways in which this can be done, but one need only regard the tangent function as valued in the one-point compactification (https://en.wikipedia.org/wiki/One-point_compactification) of `\mathbb R`, which is a circle. In this sense, the graph of the tangent function is a subset of the cylinder `\mathbb R \times S^1`; and the usual graph comes from making a slice through the "`\infty` section" `\mathbb R \times \{\infty\}` and unrolling.

If I understand correctly, you're probably referring to a stereographic projection [0], and I see that the upper tip would 'close' the inverse image of the graph of 1/x on the sphere, something which doesn't happen if you take the closure of the graph with respect to the usual topology on R^2. This is not my field, there may even be some fancy topological concepts such as connectedness or arc-connectedness related but I don't know.

[0] https://en.wikipedia.org/wiki/Stereographic_projection

Can someone explain what I’m looking at in the crazy plot [1] at the top of Sine Curve? It’s just captioned with the expression that generated it:

  ContourPlot[Sin[x Sin[y]] == Cos[y Cos[x]], …]
Simply pretty or is there some significance to it?

[1]: http://xahlee.info/SpecialPlaneCurves_dir/Sinusoid_dir/sinus...

Simply pretty, I haven't seen an application of that particular expression elsewhere, so I think he was tinkering. Goofing around with curves in Desmos is actually something I do once in a while, while eating breakfast or something.
Somewhat disappointed in the Bezier section, which seems to want to incorrectly teach people that Bezier curves have four points, "the end", rather than teaching them that they can be defined with any number of points, and that an n-point Bezier represents a section of an nth order polygon, with point-on-curve calculation based on summing n terms with binomial coefficients.
Yes, I thought so too. But I think the site reasonably and generally fulfills the objective of a 'dictionary'. I was going to reply to your comment with the best introduction to Bézier curves I've read (and I'm a math guy), before realizing that it was you who wrote it. You helped me understand some dry numerical analysis text I had been assigned since your primer is interactive and so 'concrete', thanks.
Always happy to hear the primer was able to help make something more easy to understand =)
> I was going to reply to your comment with the best introduction to Bézier curves I've read (and I'm a math guy), before realizing that it was you who wrote it.

Would you link it anyway, please, for the benefit of those of us who didn't write it? :-)

Here you go. https://pomax.github.io/bezierinfo/