The mathematician formerly known as DayLateDon

14h
comment Geometric intuition for derivatives of basic trig functions
Related: This is my three-dimensional "trigonograph" of the derivatives of sine and cosine. As for the other functions ... I have no solution to offer here, but a thought: I've always found it interesting that, just as sine and cosine are derivatives of each other (sign changes notwithstanding), secant and tangent are as well ... except for an "extra" factor of secant: $$\frac{d}{d\theta}\color{red}{\sec\theta} = \color{blue}{\tan\theta}\cdot\sec\theta \qquad \frac{d}{d\theta}\color{blue}{\tan\theta} = \color{red}{\sec\theta}\cdot\sec\theta$$
23h
comment For points A, B, does there there a billiard such that any trajectory from A will reflect twice and then reach B?
Not an answer, but a thought: Take $A$ and $B$ each to be the focus of a matching parabolic "cap" (where each "cap" is deep enough to contain its focus), and glue the caps together to make an ellipse-like shape. Trajectories through $A$ that first hit $A$'s cap will reflect to become parallel to $\overline{AB}$ until hitting $B$'s cap, at which point they reflect to hit $B$. Of course, a trajectory through $A$ that first hits $B$'s cap fails miserably to meet the goal, but hey ... I achieved a 50% success rate on the first try! :) (@dtldarek ... "Great minds", eh?)
1d
comment Circumference of hyperbolic circle is $2\pi \sinh r$
Do you want an integral proof? Using the Poincaré model metric tensor $ds = \frac{2 \sqrt{dx^2+dy^2}}{1-x^2-y^2}$, and parameterizing the origin-centered circle with Euclidean radius $R$ by $x = R\cos t$, $y = R\sin t$, the hyperbolic circumference of the circle is $$\int_{0}^{2\pi}\frac{2 R dt}{1-R^2}=\frac{4\pi R}{1-R^2} \qquad(\star)$$ Since Euclidean distance $R$ from the origin corresponds to hyperbolic distance $r=\log\frac{1+R}{1-R}$, we have $R = \frac{\exp r - 1}{\exp r + 1}$. Substituting into $(\star)$ gives $2\pi\sinh r$.
1d
comment Are circles and lines in two-space one-dimensional?
Related: "Why the unit circle in $R^2$ has one dimension?", "Confused about dimension of circle", "Why is a circle one-dimensional?".
2d
revised Circumference of hyperbolic circle is $2\pi \sinh r$
added 9 characters in body; edited title
2d
comment Circumference of hyperbolic circle is $2\pi \sinh r$
I don't see how the model matters. Here's a proof that doesn't reference a particular model.
2d
comment Where can I find a good drawing software?
Satimage's "Smile" leverages the Mac's AppleScript language to draw images programmatically. The interface is poor, and the documentation difficult to navigate, but Smile has some built-in geometric primitives and labeling options (including TeX support) that could be helpful. I've used it to create diagrams such as those shown in this answer. See also some of Satimage's sample images­.
2d
awarded Electorate
May
24
comment An equation involving ratios in a triangle.
Good. Write up your solution as an answer, and I'll up-vote it. :)
May
24
comment An equation involving ratios in a triangle.
Multiplying the numerator and denominator of the right-hand side by $r/2$ (where $r$ is the inradius), the fraction becomes $|\triangle ABC|/|\triangle IBC|$, which reduces to the ratio of the altitudes of those triangles. The altitudes make convenient similar triangles with $\overline{AD}$ and $\overline{ID}$.
May
24
comment What is reflection across parabola?
Note that parabolas of the form $y=h x^2$ and ellipses of the form $x^2 + 2 y^2 = 2 k^2$ comprise an orthogonal family, with which we can assign unique $hk$-coordinates to every point in the plane (taking $k$ negative in the left half-plane). If there's any justice, these families are preserved under parabolic reflection: ellipses invariant (as point-sets), and parabolas permuted. That is to say, reflection takes $(h,k)$ to some $(h^\prime, k)$. The question then becomes: How are $h$ and $h^\prime$ related? (Surely, $h=1$ iff $h^\prime=1$, and $h=\pm \infty$ iff $h^\prime=\mp \infty$.)
May
24
revised I think I see mysterious lines inside triangles—how to prove their existence?
added 26 characters in body
May
24
comment I think I see mysterious lines inside triangles—how to prove their existence?
@Alexey: I would've beaten you if I didn't take so long editing. ;) I should probably just delete this answer. (Well ... hmmm ... someone just up-voted me. I guess there's perceived value in this near-duplicate, so I'll leave it. :)
May
24
revised I think I see mysterious lines inside triangles—how to prove their existence?
added 59 characters in body
May
24
answered I think I see mysterious lines inside triangles—how to prove their existence?
May
23
comment What is the name of this geometric shape?
I don't believe there's an official name for this shape, so the trick is to find a concise descriptor. "Concave bisymmetric hexagon" would (I think) cover #1, but there's no elegant counterpart for #2. Of course, you're free to call these things whatever you like. I might be inclined to play off of the resemblance to an anvil.
May
23
answered Is there another way to solve this Trigo in series?
May
21
comment How to prove that a straight line is an infinite set of points?
When you fill-in the "...", I'll remove my comment. :)
May
21
comment How to prove that a straight line is an infinite set of points?
See my comment under @lixu's answer.
May
21
comment How to prove that a straight line is an infinite set of points?
Of course, this assumes a correspondence between geometric lines and algebraic linear equations of real-number variables. There are geometries with no such correspondence. As mentioned by myself and others, one must be clear about the axioms system being used.
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