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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 t

by ffhhj 5y ago

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.

mbustamanter 5y ago
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 https://en.wikipedia.org/wiki/Stereographic_projection
JadeNB 5y ago
> 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 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.
macrolocal 5y ago
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-emphasis-on-projective-space-in-algebraic-geometry 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.

by ffhhj 5y ago

mbustamanter 5y ago

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 https://en.wikipedia.org/wiki/Stereographic_projection

JadeNB 5y ago

> 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 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.

macrolocal 5y ago

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-emphasis-on-projective-space-in-algebraic-geometry 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.