What is Geometrical: Definition and 143 Discussions

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.

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1. I How does EM wave geometrical attenuation affect atomic absorption?

Let's say we have a point source of an EM wave in a vacuum of total energy E, and an absorber atom at some distance from this source, whose first excited state is at the energy B, with B < or = E. The energy of the wave is constant as a whole, but at each point around the source the energy...
2. B Geometrical Optics: Explaining the Effects of Small Wavelengths

Read this in my textbook:- The reason Geometrical optics works in case of formation of shadows, reflection and rarefaction is that the wavelength of light is much smaller compared to the reflecting/refracting surfaces as well as shadow causing objects that we use in day-to-day life. I...
3. I MOND from MacDowell-Mansouri geometrical formulation

One poster is a strong promoter of Deur and Deur theory of how to get MOND out of GR via self-interaction and analogy to QCD. There is intense skepticism of Deur's approach, which has 0 citations other than the author. I saw this paper, High Energy Physics - Theory...
4. Geometrical optics: using Snell's law, find the depth of the pool

α=30°; l=0.5 m; n1=1; n2=1.33 α+β=90°, so β=90°-30°=60°. Using Snell's law: sinβ/sinγ = n2/n1 sinγ≈0.651 γ≈41°. β=γ+θ (vertical angles) θ=60°-41°=19° tan(θ+β)=l/h h=l/tan(θ+γ) h=0.5/(tan(19+41))≈0.289 m
5. What is the focal point of a lens in a geometrical optics problem?

I have recently started with geometric optics and I do not quite understand what this problem asks of me. According to the statement, the focal point of the lens would be -25.5cm, right? That is, it is only a problem of concepts where it is not necessary to take into account the radii of the...
6. B Geometrical meaning of magnitude of vector product

My notes says that the geometrical meaning of $$|\vec v \times \vec w |$$ is the perpendicular distance from point ##V## to line passing through ##O## and ##W## (all vectors are position vectors) $$|\vec v \times \vec w | = |\vec v| |\vec w| \sin \theta$$ From the picture, the perpendicular...
7. Calculation of orbital velocity -- Geometrical solution

Hello, I would like to calculate the orbital velocity using the geometrical way of reasoning. But I have a hard time to understand and apply some basics into my calculations. The reasoning is pretty simple. After some time: dt ,the particle travels the distance: Vtot1 * dt = R*sinθ (see the...
8. Geometry Geometrical books (differential geometry, tensors, variational mech.)

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...
9. Is there some geometrical interpretation of force from Newton's Laws?

dP = F dt dE = F dr or if we introduce ds = (dt, dr) (dP, dE) = F ds And both dP and dE are constant in closed system. Some questions: - How does its implies on definition of Force? - Is there some clever geometrical interpretation of Force? - Why P and E seems almost interchengable?
10. Classical Geometrical Mechanics by Talman

Does anyone have any experience with this book? https://www.amazon.com/gp/product/3527406832/?tag=pfamazon01-20
11. I Is it possible to calculate this geometrical relationship between circles?

A large cirlcle with radius 50 m contains a smaller circle with radius 7.4 m that is tangent to its surface internally. Is it possible to calculate what number of the small circle the larger circle can contain iside it in which all are tangent to its surface ... but without using trig. Functions
12. MHB Prove Triangle Inequality: AB/MZ + AC/ME + BC/MD ≥ 2t/r

Given a triangle ABC and a point M inside the triangle ,draw perpendiculars MZ,MD,ME at the sides AB,BC,AC respectively. Then prove:\frac{AB}{MZ}+\frac{AC}{ME}+\frac{BC}{MD}\geq\frac{2t}{r} Where t is half the perimeter of the triangle and r is the radius of the inscribed circle
13. A Are there conditions for the vanishing of geometrical phases in QM?

Are there theorems for sufficient and necessary conditions for the vanishing of Berry and/or Wilzeck-Zee phases for a given quantum mechanical system?
14. A Crossing degeneracies and geometrical phases

Assume all the usual things for the usual things for the geometric phases: A Hamiltonian that depend on external parameters, Adiabatic evolution, cyclic evolution in parameter space and all that If through the evolution in parameter space there is no energy level crossing, then a eigenvector of...
15. I Variation of geometrical quantities under infinitesimal deformation

This question is about 2-d surfaces embedded inR3It's easy to find information on how the metric tensor changes when $$x_{\mu}\rightarrow x_{\mu}+\varepsilon\xi(x)$$ So, what about the variation of the second fundamental form, the Gauss and the mean curvature? how they change? I found some...
16. B Struggles with the geometrical analogy

{Moderators note: thread split from https://www.physicsforums.com/threads/a-geometrical-view-of-time-dilation-and-the-twin-paradox-comments.842793 } I am a novice about relativity, but I found this convoluted and very difficult to follow. I think the lengths in the different coordinate systems...
17. Geometrical Optics - Light ray angles on a spherical mirror

I can't see how the textbook produces the following relationships between angles: $$\theta = \phi + \alpha \qquad (1)$$ $$2\theta = \alpha + \alpha ' \qquad (2)$$ My thinking is that the exterior angle theorem for triangles was used to create expression ##(1)##, but I am unsure as to how...
18. Stability of geometrical isomers

Homework Statement Among the following, which should be the most stable compound? 1)Cis-cyclohexane-1,2-diol 2)Trans-cyclohexane-1,2-diol 3)Cis-cyclohexane-1,3-diol 4)Trans-cyclohexane-1,3-diolHomework Equations -- The Attempt at a Solution My thought process is-cis isomers with adjacent OH...
19. I'm struggling with an Arithmetic Geometrical question.

Homework Statement The angles in triangle ABC form an increasing arithmetic sequence. The ratio of angles A:B:C can be written in the form 185:370:555 respectively. You are told that the total area of the triangle is 9 Length BC is Given the area of...
20. B Any geometrical meaning of multiplication of quaternions?

Let's just talk about unit quaternions. I know that $$\left(\cos{\frac{\theta}{2}}+v\sin{\frac{\theta}{2}}\right)\cdot p \cdot \left(\cos{\frac{\theta}{2}}-v\sin{\frac{\theta}{2}}\right)$$ where ##p## and ##v## are purely imaginary quaternions, gives another purely imaginary quaternion which...
21. MHB Complex number geometrical problem

Show geometrically that if |z|=1 then, $Im[z/(z+1)^2]=0$ I am unsure how to begin this problem. I have sketched out |z|=1 but can't work out how to sketch the Imaginary part of the question.
22. MHB TikZ Challenge 1 - Geometrical Diagram - Votes

Hey all, 2 weeks ago I created a challenge to create a geometrical diagram, like a triangle, that is somehow interesting or impressive. Now the moment of truth is here. Please everyone, give your vote! Voting will close in 2 weeks time. Let me recap the submissions.I like Serena...
23. MHB TikZ Challenge 1 - Geometrical Diagram

Who can make the most impressive, interesting, or pretty TikZ picture? This first challenge is to create a geometrical diagram, like a triangle, that is somehow interesting or impressive. We might make it a very complicated figure, or an 'impossible' figure, or use pretty TikZ embellishments...

47. Curve Matching Techniques for Rotated Curves in Geometric Analysis

I need to perform geometry matching of curves (see http://www.tiikoni.com/tis/view/?id=c54d9b8 ). As it can be seen, the big problem is that curves might be rotated, though they have similar shape. Do I need to make curve fitting and look at the parameters of analytical models? But, I guess...
48. Application of advanced spectrometer in geometrical optics?

We have an advanced spectrometer in our geometrical optics lab! I'm seeking for any experiment in geometrical optics to include it!
49. Geometrical Proof: Prove Intersection Point on Line CM

Homework Statement Consider an triangle ABC with M as the middle point of the side AB. On the straight line through AB you put the angle ∠ ACM at A and the angle ∠ MCB at B. Now you have two new lines. The new lines should be on the same side of AB as C. Proof that the intersection point of the...
50. Geometrical properties of circle

Homework Statement Dear Mentors/PF helpers, Please help me with part (ii), I couldn't find a way through this part. Homework Equations The Attempt at a Solution My solutions: (i) angle RPS = 65 deg (angles in the same segment are equal) (iii) angle PRS = 110 - 65 = 45 ( exterior...