6 ms·

Visual Dictionary of Special Plane Curves

31 points by mbustamanter 5y ago

TheRealPomax 5y ago
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.
- mbustamanter 5y ago
  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.
  - JadeNB 5y ago
    > 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? :-)
    - mbustamanter 5y ago
      Here you go. https://pomax.github.io/bezierinfo/ https://pomax.github.io/bezierinfo/
  - TheRealPomax 5y ago
    Always happy to hear the primer was able to help make something more easy to understand =)
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.
aendruk 5y ago
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/sinusoid.html http://xahlee.info/SpecialPlaneCurves_dir/Sinusoid_dir/sinus...
- mbustamanter 5y ago
  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.

31 points by mbustamanter 5y ago

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.

mbustamanter 5y ago
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.
- JadeNB 5y ago
  > 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? :-)
  - mbustamanter 5y ago
    Here you go. https://pomax.github.io/bezierinfo/ https://pomax.github.io/bezierinfo/
- TheRealPomax 5y ago
  Always happy to hear the primer was able to help make something more easy to understand =)

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.

aendruk 5y ago

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/sinusoid.html http://xahlee.info/SpecialPlaneCurves_dir/Sinusoid_dir/sinus...

mbustamanter 5y ago
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.