Geometry (from the Ancient Greek: γεωμετρία; geo- "earth", -metron "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a geometer.
Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts.During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Gauss' Theorema Egregium ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in an Euclidean space. This implies that surfaces can be studied intrinsically, that is as stand alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry.
Later in the 19th century, it appeared that geometries without the parallel postulate (non-Euclidean geometries) can be developed without introducing any contradiction. The geometry that underlies general relativity is a famous application of non-Euclidean geometry.
Since then, the scope of geometry has been greatly expanded, and the field has been split in many subfields that depend on the underlying methods—differential geometry, algebraic geometry, computational geometry, algebraic topology, discrete geometry (also known as combinatorial geometry), etc.—or on the properties of Euclidean spaces that are disregarded—projective geometry that consider only alignment of points but not distance and parallelism, affine geometry that omits the concept of angle and distance, finite geometry that omits continuity, etc.
Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, a problem that was stated in terms of elementary arithmetic, and remained unsolved for several centuries.
Dear all,
I am writing a vehicle dynamics simulation for my thesis topic. However, I came into a conundrum when testing the cornering behavior of my vehicle. The problem is inherently complex due to its many subsystems, but I'll try to give as much detail without bogging the thread down...
Dear all,
the following problem is not a home-work problem. I have come up with this question for myself. Nevertheless, I am stuck and need your help.
The question is: Can I calculate the distance between points A and B from this information? And if yes, how?
I think it should be possible...
Consider the following example:
Point A has coordinates 45 lat, 0 long. Point B has coordinates 45 lat, 2 long. Both points are 5000 ft above sea level. The distance between them is X.
Point C has coordinates 45 lat, 100 long. Point D has coordinates 45 lat, 102 long. Both points are at sea...
I want to use this to design a parabolic (optical) mirror;
The problem is that in my application I need both D and f to be a parameter, but I need to specify f only as a perpendicular distance from D. In other words, I need to specify some f_2=f-d, and calculate d. I can't seem to come up with...
Trying to calculate a circumference of a sphere from a radius of 3.09 inches. Is 19.4 a correct answer? Just ran numbers in the first circumference calculator I found http://calcurator.org/circumference-calculator/. Can I use the same formula for a sphere? What can I say ...Geometry is not my...
Hello everyone,
I wanted some help deciding which elective to choose. I am a junior and for my next semester I have the option to pick either Differential Geometry-I or Quantum Information. I am confused which one to choose. We will be doing QMII as a compulsory course next semester and I have...
Hi, I'm wondering why shorted circuit geometry like figure 2 did not sense photocurrent?
Even if the the circuit composed like 2, I guess that by the Kirchhoff's Law, voltage should apply to the ampere meter and photocurrent should be sensed. But in real experiment, I found that shorted circuit...
Problem: Given the line L: x = (-3, 1) + t(1,-2) find all x on L that lie 2 units from (-3, 1).
I know the answer is (3 ± 2 / √5, -1 ± 4/√5) but I don't know where to start. I found that if t=2, x= (-5, 5) and the normal vector is (2, 1) but I am not sure if this information is useful or how...
I'm reading about excitation of surface plasmons, and there's a claim in the derivation I don't know how to prove. The geometry is two infinite slabs of material with negligible permeability (##\mu_1 = \mu_2 = 1##) and different permittivity ##(\epsilon_1 \neq \epsilon_2 \neq 1)##. The claim...
Hello, so I saw this problem on a website while looking up trigonometric identities and trying to solve it.
what I know:
The internal angles add up to pi
Let the tangent point between A and B be X
Let the tangent point between B and C be Y
Let the tangent point between C and A be Z
##...
I have calculated the height of the segment using the Pythagorean Theorem and that's currently where I am right now. I don't seem to find any equations that can help me. Though I might be not trying hard enough or using the wrong words because I'm not really fluent in mathematical terms as you...
This is jut an example to illustrate my doubt. I don't know how to obtain the tracjectory given only the acceleration in this format. I realized that if i can show that there is an constat vector 'a' that satisfy a•r=constant, than the motion would be on the surface of a cone. So i tried to make...
Pictured below are two hinged panels that can rotate upward to form an upside-down V. In position 1, the panels are lying flat. In position 2, the panels have folded together and the joined edge is raised up.
Normally, in order to actuate this hinging motion, one would need to manually lift the...
Saw this on "Who Wants to Be a Millionaire" which, of course, I'd never trust on its own, so I verified:
It is common practice to cut flower stems at an angle, but I never thought to confirm why. I assumed it had something to do with cutting across the grain like one does with meat...
Hello folks,
I want to simulate a 2D heat transfer process in the subsurface on a region which is infinite on the r-direction. So, as you know, the very basic way to model this is to draw a geometry that is very long in the r direction. I have done this, and the results that I obtain is correct...
Dear Everyone,
So I would like some recommendation for high school geometry books that are affordable and preferably e-books.
Why do I need some books on high school geometry? I would like to improve my geometric reasoning. When I took high school geometry a decade and half ago, I was...
I have been working on a problem for a while and my progress has slowed enough I figured I'd try reaching out for some more experience. I am trying to map a point on an ellipsoid to its corresponding point on a sphere of arbitrary size centered at the origin. I would like to be able to shift any...
I was just browsing through the textbooks forum a few days ago when I came across a post on differential geometry books.
Among the others these two books by the same author seem to be the most widely recommended:
Elementary Differential Geometry (Barret O' Neill)
Semi-Riemannian Geometry with...
Computing timelike geodesics in the Schwarzschild geometry is pretty straightforward using conserved quantities. You can treat the problem as a variational problem with an effective Lagrangian of
##\mathcal{L} = \frac{1}{2} (Q \frac{dt}{d\tau}^2 - \frac{1}{Q} \frac{dr}{d\tau}^2 - r^2...
I'm imagining something like this:
The image was taken from the following paper, and is described as a rhombicuboctahedral quasicrystal. The paper itself gets very technical (at least for me), describing projecting a 4D crystal into 3D space. It seems to me based off of a rhombicuboctahedron...
I am looking for math books that focus on geometrical interpretations. Sadly most of the modern books lack these interpretations and only consists out of theorems and proofs. It seems to me that most modern mathematicians are pure left-brain sequential thinkers that do not have a lot of...
I am in the middle of a problem for the Kerr geometry, I need to do the integral ##\int_{\mathcal{N}} \star J## over a null hypersurface ##\mathcal{N}## which is a subset of ##\mathcal{H}^+##, where ##J_a = -T_{ab} k^b## and the orientation on ##\mathcal{N}## is ##dv \wedge d\theta \wedge...
Can anyone recommend a good on-line class for differential geometry? I'd like to start studying GR but want a good background in differential geometry before doing so. Many thanks.
Hello everyone. I was browsing through Amazon and found the aforementioned book by Theodore Frankel. As it is available at a relatively cheap price and covers a TON of material I was considering buying it for future use . Although the author says the prerequisites are only multivariable...
In Miles Reid's book on commutative algebra, he says that, given a ring of functions on a space X, the space X can be recovered from the maximal or prime ideals of that ring. How does this work?
Here is my attempt to draw a diagram for this problem:
I'm confused about the "the perpendicular bisector of ##BC## cuts ##BA##, ##CA## produced at ##P, \ Q##" part of the problem.
How does perpendicular bisector of ##BC## cut the side ##CA##?
In terms of diff geo it seems like an obvious fact, that a manifold can be equipped with quite a variety of different Riemann metrics. But when it comes to physics (relativity theory in particular) it seems there is a very specific metric singled out. Now i do not entirely understand the...
Hi,
My name is Cam and I've just literally joined so wanted to say hi first 🙂
I'm self studying Maths and Physics and wanted to know a good textbook that deals with arithmetic/pre-algebra/basic geometry? I know Physics mainly used Applied Maths, but I'm wanting to educate myself as thoroughly...
Hello,
I am studying geometry with an app on my phone. There was a difficult problem, which had two different explanations for solving. I correctly understood one explanation. I reviewed later without memory of the problem at all. There was an obvious attempt from what was learned previously...
Dear Everybody,
I am in the process of relearning high school geometry through Khan Academy. I am current an graduated undergraduate student in mathematics. I am doing this because geometry is one of my weakest subject in mathematics. Second reason is that I want to reason out a problem...
Dear Everybody,
I am in the process of relearning high school geometry through Khan Academy. I am current an graduated undergraduate student in mathematics. I am doing this because geometry is one of my weakest subject in mathematics. Second reason is that I want to reason out a problem...
Hello. Questions: do you know any applications of spherical geometry in physics? Are there any relations between spherical geometry and hyperbolic geometry? Why does Riemannian geometry use sphere theorems so much? Thank you.
In the book: The Elements Euclid defined 5 postulates:
1) A straight line segment can be drawn joining any two points.
2) Any straight line segment can be extended indefinitely in a straight line
3) Given any straight line segment, a circle can be drawn having the segment as radius and one...
Hi guys,
Hopefully, no geometry-enthusiasts are going to read these next few lines, but if that's the case, be lenient :)
I have always hated high-school geometry, those basic boring theorems about triangles, polygons, circles, and so on, and I have always "skipped" such classes, studying...
I am new to Comsol. I want to draw my model which is a Tesla valve. The geometry is little complicated and I don't know how to draw a semi-circle tangent to a line. Is it possible? I draw it in Solidworks and imported it into Comsol but it gives error and I think it is better to draw inside...
Recently came across this concept. It looks like a combination of dg and statistics. It sounds interesting, but I do not feel competent enough to make an informed decision.
For example see
https://arxiv.org/abs/1808.08271
Instead of talking about the simple of case of reflection interference due to a single film, this book starts off with two films with an angled air wedge between them. They talk about the "thickness", ##t##, of the wedge, but this thickness varies along the length of the films (Figure 35.`12)...
Hello dear PhysicsForums attendees!
I tried to solve for somebody the aforementioned problem. But I am not sure if my attempt is correct. So I am writing down what I suggested.
Looking at eq 2.46 in Carrolls book; The metric is Lorentzian in General Relativity so that ##g^{\mu \nu} =...
i'm trying to find what sort of 2-d geometry this system is in, I've been given the line element
𝑑𝑠2=−sin𝜃cos𝜃sin𝜙cos𝜙[𝑑𝜃2+𝑑𝜙2]+(sin2𝜃sin2𝜙+cos2𝜃cos2𝜙)𝑑𝜃𝑑𝜙
where
0≤𝜙<2𝜋
and
0≤𝜃<𝜋/2
Im just not sure where to start. I've tried converting the coordinates to cartesian to see if it yields a...
Is there a general method to determine what geometry some line element is describing? I realize that you can tell whether a space is flat or not (by diagonalising the matrix, rescaling etc), but given some arbitrary line element, how does one determine the shape of the space?
Thanks
Hi PF,
What would you say is the level of importance of Geometry, as opposed to say, Algebra I or II, in a future Mathematician's mathematical foundation? Would not covering it thoroughly leave holes that would show up later in higher mathematics? Any thoughts?Thank you,
Chandller
I started off by indicating ##p=-\frac{1}{m}## since it's perpendicular. The sum ##m+p## is now ##\frac{m^2-1}{m}##.
Honestly, I can't go beyond that. The interceptions with the y-axis are of course unuseful, I tried algebraically intersecting the two lines but I came up with nothing... and I...
I'm currently studying Algebra and have collected Euclid's Elements, Lang's Geometry, Gelfand's Trigonometry, and Rhoad and et al's Geometry for Enjoyment and challenge. A quick perusal of the books seem to involve coordinates and algebra knowledge. From what I've heard Euclid is closer to Pure...