1
vote
1answer
21 views

Confusion with Bolyai-Gerwien theorem

The Bolyai-Gerwien theorem states: Given two polygons with the same area, it is possible to cut up one polygon into a finite number of smaller polygonal pieces and from those pieces assemble into ...
7
votes
3answers
140 views

Beginning of Romance

I am a 17 guy from India. The fascination of maths has struck me recently, while I am in standard 12th. But all the resources I have, is some school textbooks. M.L Agrawal's of 11th and 12th. I don't ...
2
votes
1answer
91 views

Prove the Gaussian curvature $K=0$

If two families of a geodesics on a surface intersect at a constant angle $\theta$, prove that the Gaussian curvature of the surface is zero, i.e. $K=0$. Please explain how to show the question. ...
1
vote
1answer
95 views

Prove that the curves of the family $v^3/u^2=k$ are geodesics on a surface

Prove that the curves of the family $v^3/u^2=k$ where $k$ is a constant are geodesics on a surface with the metric $$v^2 \, du^2-2uv \, du+2u^2 \, dv^2$$ where $u,v \gt 0$.
3
votes
1answer
102 views

Showing to be Unit-sphere

First and second fundamental forms are both $du^2 +\cos^2 u dv^2$ I want to show that the surface is a part of the unit sphere. What I did is following; $E=L=1$ $F=M=0$ $N=G=\cos^2 u$ ...
4
votes
1answer
104 views

Geodesic eqautions and length of a curve in geodesic coordinate system.

About geodesic coordinates: Let S be regular surface. $p\in S$ $\gamma$ be unit speed geodesic on $S$ with parameter $v$ and $\gamma (0)=p$ $\tilde \gamma^v$ be unit speed geodesic s.t. ...
0
votes
2answers
148 views

Geodesics on torus

Describe the geodesics on Torus $$\sigma (u,v)= ((a+b \cos u)\cos v, (a+b\cos u)\sin v, b\sin u)$$ First fundamental form for torus is $$b^2 du^2 +(a+b \cos u)^2dv^2$$ Consider unit-speed ...
3
votes
2answers
65 views

Geodesics on spheroid

Describe the geodesics A Spheroid obtained by rotating the ellipse $\frac{x^2}{p^2}+\frac{z^2}{q^2}=1$ around the z-axis where $p, q\gt 0$ Please explain this question explicitly. Thank you:)
3
votes
2answers
132 views

Geodesic equations and christoffel symbols

I want to learn explicitly proof of the proposition 9.2.3. Which books or lecture notes I can find? Please give me a suggestion. Thank you:)
2
votes
1answer
71 views

Affine variety example.

I know how to show the statement. But I cannot find an example (the part I underlined by a yellow pen) please help me for finding an example. Note: For example, can I consider the following ...
1
vote
1answer
79 views

Differential geometry question.

Please explain how to solve this question. Thank you:) And sorry for hand-writing.
4
votes
3answers
206 views

Proving the Obvious

Is it normal that I have the hardest time when I'm trying to prove statements that are blatantly obvious on a visual and/or intuitive level? For instance, how does one go about formally proving the ...
3
votes
6answers
234 views

I want a good dictionary of mathematics/ geometry

I noticed I a made a mistake in some geometrical terminology and wanted to better my life by buying a new dictionary of mathematics or more specialised Geometry. (okay I am just a shopaholic for ...
0
votes
1answer
38 views

Show that the image of Gaussian map of a generalized cone is a curve on $S^2$ and deduce that the cone has zero Gaussian curvature.

Show that the image of Gaussian map of a generalized cone is a curve on $S^2$ and deduce that the cone has zero Gaussian curvature. I dont have enough idea. Please explain the question clearly. ...
-1
votes
1answer
131 views

Compute the geodesic curvature of any sphere on a sphere.

Compute the geodesic curvature of any sphere on a sphere. Again there exists its answer, but not understandable for me. Please explain it explicitly. Thank you so much. (If required, i can post ...
1
vote
1answer
70 views

Second fundamental form question.

Honestly, I dont have any idea for that question I posted. Please can someone help me solving the question. Thank you.
2
votes
1answer
92 views

The second fundamental form and isometry.

What is the effect on the second fundametal form of asurface of applying an isometry of $\Bbb R^3$ ? Or a dilation? I posted its answer. This answer is not understandable for me in general. ...
0
votes
1answer
29 views

Surface patch are taken different for sphere, but their second fund. forms are not completely different.

Shpere First of all, I take the surface patch for sphere $$\sigma(u,v)=(\sin u\cos v, \sin u\sin v, \cos u)$$ And then I calculated its second fundametal form. And I got the following result $$ ...
0
votes
1answer
109 views

Why first fundamental form and second fundamental form are the same?

Surface of Revolution $\gamma(u)=(f(u),0,g(u))$ and $\sigma(u,v)=(f(u)\cos v, f(u)\sin v, g(u))$ Fist of all, I calculated the first fundametal form for surface of revolution. And I obtained ...
0
votes
1answer
99 views

Transition map for Möbius band in differential geometry.

Calculate the transition map $\phi$ between the two surface patches for the möbius band. These two surface patches are the following $U=\{(t,\theta) \ | -1/2\lt t\lt 1/2,\ \ 0\lt \theta \lt ...
0
votes
1answer
54 views

Torus in differential geometry.

I want to write separately parametrizations (surface patches) $\sigma$ for torus when (1) x-axis rotation in the first part of the picture and (2) y-axis rotation in the second part of the picture. ...
2
votes
1answer
76 views

Number of lines required to split a plane into N regions

I was wondering if there was any theorem out there that talks about the number of lines required to split a plane into any given number of regions. I don't really have much of a mathematical back ...
1
vote
1answer
308 views

Showing how to find the vertices of the circle.

Find that the circle has four vertices. $$\gamma (t)=\langle R\cos (t/R), R \sin (t/R)\rangle$$ for $t\in [0,2\pi]$ I know the theorem: Every simple closed convex curve has atleast four ...
3
votes
1answer
124 views

Showing the parametrically representation of hyperbolic paraboloid. And how to find the curves $u$ and $v$ be constant.

Show that the hyperbolic paraboloid can be represented parametrically as $$r(u,v)=\langle a(u+v), b(u-v), uv\rangle$$ Find the curves $u$ is constant and $v$ is constant. I guess I need to use the ...
4
votes
1answer
88 views

The curve has constant torsion.

Question: Show that when the curve $c_1=c_1(t)$ has constant torsion $\tau$, the curve $$c_2=c_2(t)=-\frac{1}{\tau}N+\int_{t_0}^{t}B(u)du$$ has constant curvature $-\tau$ or $+\tau$. What I ...
0
votes
1answer
85 views

$(\frac{1}{\kappa})^2+(\frac{\dot{\kappa}}{\kappa^2\tau})^2=r^2$

Show that for a curve lying on a sphere of radius r with nowhere vanishing torsion, the following equation is satisfied: $$(\frac{1}{\kappa})^2+(\frac{\dot{\kappa}}{\kappa^2\tau})^2=r^2$$ Please ...
3
votes
1answer
53 views

Show that $\dot{n_s}=-\kappa_s t$

I found the question in a differential geometry textbook while studying. This question seems so intesting to me. So please help me solving it. Show that, if $\gamma$ is a unit-speed plane curve, ...
3
votes
1answer
77 views

If $\gamma$ is spherical, then the equation $\frac{\tau}{\kappa}=\frac{d}{ds}(\frac{\dot{\kappa}}{\tau \kappa^2})$ holds.

Question: Let $\gamma (t)$ be a unit-speed curve with $\kappa(t)\gt0$ and $\tau(t)\neq0$ for all $t$. Show that, if $\gamma$ is spherical, i.e., if it lies on the surface of a sphere, then ...
8
votes
2answers
122 views

How to obtain $y$

The question was written with dark-blue pen. And I tried to solve this question. I obtained $x$ as it is below. But I cannot obtain $y$ Please show me how to do this. By the way, $\gamma (t)$ ...
-1
votes
1answer
44 views

how to find the signed normal

$$\gamma (t)= (R\cos (t/R), R\sin (t/R))$$ $$\dot {\gamma (t)}=(-\sin (t/R), \cos (t/R))$$ $$n_s= (-\cos (t/R), -\sin (t/R))$$ where $n_s$ is the signed normal. the instructor has found the $n_s$. ...
1
vote
1answer
75 views

How to show that the limaçon has only two vertices.

Question: Show that the limaçon has only two vertices. I researched what is limaçon. And I reached the following result; Note that I only know that The limaçon is the parametrized curve ...
2
votes
1answer
92 views

Writing a parametrization of the cissoid by using $\theta$

The cissoid of Diocles is the curve whose equation in terms of polar coordinates $(r,\theta)$ is $$r = \sin\theta \tan\theta, −\pi/2 < \theta < \theta/2$$ Write down a parametrization of the ...
0
votes
1answer
95 views

Question on Geometry

I have a B.Ed in Management, but recently developed interest in geometry & would love to study it on my own. Do i need or must have a B.A in Maths before doing this ?
0
votes
3answers
391 views

Given a line segment. Construct an equilateral triangle with one side the given line segment.

I found this problem in a website, but I don't know how to solve it. Given a line segment $AB$. Construct an equilateral triangle with one side being $AB$.
5
votes
1answer
403 views

Want to learn differential geometry and Riemannian geometry by myself

I want to follow a Ph.D. programme on geometric analysis. The main focus are Monge-Ampere equations and convex level set of $p$-harmonic equations. The theory of PDEs is very familar to me. However, I ...
5
votes
2answers
194 views

The area problem!

We have to find area of the quadrilateral formed by joining the point of intersection of the four quarter circles that are drawn from each vertex in a unit square. $\hspace{4cm}$ The challenge is ...
3
votes
5answers
171 views

What prerequisite material should I work through to understand four dimensional space?

I understand (at least to a comfortable degree) dimensions which are less than or equal to 3. For the past several years, I have been hearing a lot about four dimensional space. I'm intrigued and ...
6
votes
2answers
426 views

Self-Teaching: Is Geometry the Nexus of all Mathematics?

Necessary prologue: I'd really like to become more fluent in the language of mathematics. I don't have a schedule that permits me taking a class and any on-line tutors that I find seem relatively ...
6
votes
5answers
1k views

Coxeter's “Geometry Revisited” vs. “Introduction to Geometry”

Which of these two books should I read first? Is one of them more advanced than the other? I'm planning to use them to self-study geometry. Thanks.
6
votes
4answers
703 views

What are some good resources for brushing up on geometry and trigonometry?

I'm brushing up on my calculus, and geometry and trigonometry keep coming up. However, these are probably my weakest areas mathematically. Are there any good sites or free PDFs/ebooks that I can use ...