Geometry Definition and 999 Threads

Homework Statement In Fano's Geometry, we have the following axioms a. There exists at least one line b. Every line has exactly three points on it c. Not all points are on the same line d. For two distinct points, there exists exactly one line on both of them e. Each two lines have at least one...

By the above question, I mean how would one effectively study synthetic geometry (geometry that makes no reference to explicit formulas or coordinate systems, like described in Euclid's Elements)? Do you just read through the propositions, try to reconstruct them later and perhaps more or are...

Hello! I would like to know if anybody here knows if there's any good book on academic-level dfferential geometry(of curves and surfaces preferably) that emphasizes on geometrical intuition(visualization)? For example, it would be great to have a technical textbook that explains the geometrical...

Good Day Early on, in Frankel's text "The Geometry of Physics" (in the introductory note on differential forms, in fact, on page 3) he writes: "We prefer the last expression with the components to the right of the basis vectors." Well, I do sort of like this notation and after reading a bit...

I wanted to study General Relativity, but when I started with it, I found that I must know tensor analysis and Differential geometry as prequisites, along with multivariable calculus. I already have books on tensors and multivariable calculus, but can anyone recommend me books on differential...

Homework Statement Let ## (M, \omega_M) ## be a symplectic manifold, ## C \subset M ## a submanifold, ## f: C \to \mathbb{R} ## a smooth function. Show that ## L = \{ p \in T^* M: \pi_M(p) \in C, \forall v \in TC <p, v> = <df, v> \} ## is a langrangian submanifold. In other words, you have to...

Hi there guys, Currently writing and comparing two separate Mathematica scripts which can be found here and also here. The first one I've slightly modified to suit my needs and the second one is meant to reproduce the same results. Both scripts are attempting to simulate the trajectory of a...

Does Geometry of space Changes?

Determine the area of the painted hexagon, knowing that the area of triangle ABC is 120cm^2 IMG Link: https://m.imgur.com/a/WtdsW I tried using Heron´s formula, but just ended up with a bunch of terms and one more variable. Sidenote: I guess part of it is figuring out that the side lenghts...

Lets assume we are mapping one face of earth. we place a plane touching the Earth at 0 lattitude and 0 longitude. Now we take the plane of projection. suppose that we expand the projection unevenly. The small projectional area of a certain lattitude and longitude is expanded by a factor which is...

Good morning (or evening), I have a geometrical tricky question, which I need your assistance with. Look at the following sketch: In the sketch you see a pyramid ABCD. Inside the space of ABCD, you see a plane MKPN, where M, K, P and N are points on the pyramid sides. Using the axioms of...

Yes, one more reason to be humble, I know. This is the simplest problem I couldn't solve so far. Assume we have a circle of center O, a ruler of arbitrary size and a pencil. We use the ruler and the pencil to choose 4 points on the circle - the extremities of two diametral/diagonal segments...

Homework Statement This picture: http://i.imgur.com/n015WjU.png It's drawn with exactly the amount of information from the worksheet. Specifically, the two secants meet at a point, with an angle of 28 degrees between them. Both secants partition off an arc of 120 degrees. The goal is to...

So I was reading this book, "Euclidean and non Euclidean geometries" by Greenberg I solved the first problems of the first chapter, and I would like to verify my solutions 1. Homework Statement Homework Equations [/B] Um, none that I can think of? The Attempt at a Solution (1) Correct...

Acute triangle $ABC$,3 points $D,E,F $ are on $\overline{BC},\overline{AC},\overline{AB}$ respectively , if $\overline{AD}\perp \overline{BC} ,\overline{DE}\perp \overline {AC} $ and $\overline{DF}\perp \overline {AB}$ prove : (1)$\triangle ABC \sim \triangle AEF$ (2) $\overline{AO}\perp...

(mentor note: posted in a non-homework forum hence no template) Hello! I have a problem I'm trying to solve. I'm transforming a circle with known radius. Knowing it's radius i can calculate the circumference. I transform it by squeezing one side, leveling it, creating a circle segment with a...

In a triangle ABC: A(-2,7) and C(7,-5). The length of the altitude of AC is 5, and the length of the altitude of BC is the square root of 45. I wish to find the vertex B, given that it is below the line AC. I need your help, I have no idea how to approach this. Thank you in advance. I have...

The Complete Idiot's Guide to Calculus INTRODUCTION I've never really been very good at math and when I found out I had to take a Calculus class I started to panic. Once I gathered myself I went to the local bookstore to see if I could get a book to read so i could get a heads start. We are...

Hello In India, SL Loney's Elements of Coordinate Geometry is very popular for entrance examinations. I wanted to refresh my coordinate geometry, so tried reading through the book. But I found that the language used is old. I found myself referring to current material on the topic to properly...

Hello. I am currently working with a beam with the following cross-section: It consist of three bended sections with the following parameters, alpha = 90 degrees, Thickness = 4 mm, Radius = 50.59 mm. The top section consist of a small triangle and a rectangle. the triangle have a width = 4 mm...

There are three "concentric" rectangles, one inside the other, like in this figure: The image is not perfect, but we know that the distance between the sides of the inner-most rectangle and the middle rectangle is always "X", while the distance between the middle rectangle and the outer...

Hello all, sorry for the basic question but I suck at math and it's been many years since I had a course. Anyway I've got two pieces of metal that bolt together and ideally are square and centered. However the pieces are not exactly square when bolted. The smaller piece is off by about .6mm...

This is a problem I thought of, and I was wondering how to mathematically solve it with an equation. I tried calling one leg x. So the other leg, because of Pitagora's theorem, is: √(52 - x2) The area is equal to the product of the legs divided by two, so: 6 = (x * √(52 - x2))/2 12 = x * √(52...

Assume an eternal, static black hole which has an event horizon, a spherical surface at which any object passes a point of no return and is condemned to move toward a mathematical singularity. One of the predictions of being inside a black hole is that every spatial direction points towards the...

Homework Statement If G be the centroid of ΔABC and O be any other point, prove that , ## 3(GA^2 + GB^2 + GC^2)=BC^2+CA^2+AB^2## ##and,## ##OA^2 + OB^2 + OC^2 = GA^2.GB^2+GC^2+3GO^2## Homework Equations i m practising from S L LONEY coordinate geometry first chapter ... only the equation...

If gravity was not geometry.. what conservation law(s) would be broken? For example.. if gravity was a force.. would other laws of physics be broken? But gravity as geometry may not be complete answer because it has to be made compatible with quantum. Its quite puzzling.

Homework Statement The Frenet frame of a curve in R 3 . For a regular plane curve (and more generally for a regular curve on a 2-dimensional surface - e.g. the 2-sphere above) we could construct a unique adapted frame F. This is not the case for curves in higher dimensional spaces. Besides the...

Homework Statement Let γ : I → R3 be an arclength parametrized curve whose image lies in the 2-sphere S2 , i.e. ||γ(t)||2 = 1 for all t ∈ I. Consider the “moving basis” {T, γ × T, γ} where T = γ'. (i) Writing the moving basis as a 3 × 3 matrix F := (T, γ × T, γ) (where we think of T and etc...

find : (1)$ tan \,\, 15^o$ (2)$cos\,\,72^o$ (using geometry)

Homework Statement Homework Equations I could really use a push on how to approach this problem. My primary problem is it asks for the heat flux into the page, which makes no sense to me as that is the z direction and this is in the x/y plane. If anyone could explain this problem and maybe...

An interesting article on the field of Sympletic Geometry built on a shaky foundation and current attempts to fix it: https://www.quantamagazine.org/20170209-the-fight-to-fix-symplectic-geometry/

Hello, I am currently a High School Senior who has completed Multivariable Calc (up to stokes theorem), basic Linear Algebra ( up to eigenvalues/vectors) and non-theory based ODE (up to Laplace transforms) at my local University. (All with A's) I am hell bent on taking either one of the courses...

Knowing that Gauss' law states that the closed integral of e * dA = q(enclosed)/e naut, how would you find exactly what A is in any given problem? I know it varies from situation to situation depending on the geometry of the charge. For instance, I know that for an infinite wire/line of...

1. Here is the Problem: A line passes through A(2,3) and B(5,7). Find: (a) the coordinates of the point P on AB extended through B to P so that P is twice as far from A as from B; (b) the coordinates if P is on AB extended through A so that P is twice as far from B as from A. Homework...

Homework Statement Let A and B be elements of the line EF such that A=/B prove that the line AB=EFHomework Equations Axiom that two points determine a unique line and that the intersection of two lines has two distinct points then these lines are the same. The Attempt at a Solution [/B] If A...

Homework Statement In the drawing you can see a circumference inscribed in the triangle ABC (See the picture in the following link). Calculate the value of X https://goo.gl/photos/CAacV2dJbUrywfXv92. The attempt at a solution It seems I found a solution for this exercise with the help of a...

trilobite submitted a new PF Insights post Simple Geometry, Deep Math Continue reading the Original PF Insights Post.

Before I go any further, I do understand the ways that mechanical engineering textbooks explain why stress is a tensor. But all of those explanations seem infused with geometry (which I do NOT mean in a negative way at all); and are demonsrtrations. I am searching for a more concise/abstract...

Homework Statement Imagine a life guard situated a distance d1 from the water. He sees a swimmer in distress a distance L to his left and distance d2 from the shore. Given that his speed on land and water are v1 and v2 respectively, with v1 > v2, what trajectory should he choose to get to the...

Hey Guys, through the past years you helped me a lot. But now i have to ask my first question. I am simulating a solenoid, a very easy one, with a core and a coil. And a translational band (2mm). The only problem is, that i cannot put the band correctly because my solenoid is in a housing, the...

So I always thought that geometry is somewhat different from the rest of math. I mean, most of math is about numbers and relations. While geometry is about space. Does analysis connect the two? For example, the hypotenuse of a triangle is just a truncated portion of the number line that has...

PeterDonis submitted a new PF Insights post The Schwarzschild Geometry: Part 4 Continue reading the Original PF Insights Post.

Hello All, I am an artist who is just beginning to learn how to think mathematically. Have studied the basics "The Golden Standard", Da Vinci, MC Esher. Given my interest, I was introduced to the work of the Dorothea Rockburne and given this work to critique. While I can do the all the art and...

PeterDonis submitted a new PF Insights post The Schwarzschild Geometry: Part 3 Continue reading the Original PF Insights Post.

PeterDonis submitted a new PF Insights post The Schwarzschild Geometry: Part 2 Continue reading the Original PF Insights Post.

Homework Statement So, I have this problem here that's pretty basic, but the solution manual sets different axes, and I'm having a bit of trouble understanding the geometry part, meaning how he applies the given forces to the new axes. A model airplane of mass 0.750 kg fl ies with a speed...

PeterDonis submitted a new PF Insights post The Schwarzschild Geometry: Part 1 Continue reading the Original PF Insights Post.

Homework Statement [/B] Given a rectangular parallelepiped ABCDEFGH, the diagonal [AG] crosses planes BDE and CFH in K and L. Show K and L are BDE's and CFH's centres of gravity. I think I have understood the problem, could you verify my demo please ? Thanks Homework Equations The Attempt at...

Consider two coordinate systems on a sphere. The metric tensors of the two coordinate systems are given. Now how can I check that both coordinate systems describe the same geometry (in this case spherical geometry)? (I used spherical geometry as an example. I would like to know the process in...

Homework Statement What is the next radius outwards of this Apollonian gasket? R = radius of outer circle = 5 r1 = radius of largest inner circle = 3 r2 = radius of second largest inner circle = 1 a = unknown radius Homework Equations C = 2πr A = πr2 d = 2r The Attempt at a Solution Make a...