Curvature Definition and 920 Threads

In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius. Smaller circles bend more sharply, and hence have higher curvature. The curvature at a point of a differentiable curve is the curvature of its osculating circle, that is the circle that best approximates the curve near this point. The curvature of a straight line is zero. In contrast to the tangent, which is a vector quantity, the curvature at a point is typically a scalar quantity, that is, it is expressed by a single real number.
For surfaces (and, more generally for higher-dimensional manifolds), that are embedded in a Euclidean space, the concept of curvature is more complex, as it depends on the choice of a direction on the surface or manifold. This leads to the concepts of maximal curvature, minimal curvature, and mean curvature.
For Riemannian manifolds (of dimension at least two) that are not necessarily embedded in a Euclidean space, one can define the curvature intrinsically, that is without referring to an external space. See Curvature of Riemannian manifolds for the definition, which is done in terms of lengths of curves traced on the manifold, and expressed, using linear algebra, by the Riemann curvature tensor.

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  1. S

    Zero curvature => straight line proof

    How would you prove that if the curvature of a 'curve' in R3 is zero that the line is straight? All I have learned about is the Serret Frenet equations which I thought only apply when the curvature is non-zero? How do you define normals/binormals in this case? I'm not sure if this is...
  2. P

    How Do You Compute Curvature Indices for a Polyhedron?

    Hello. I was wondering if anyone here could help me understand this method I found described in the paper: "A Practical Approach for 3D Model Indexing by combining Local and Global Invariants" by Jean-Philippe Vandeborre, Vincent Couillet and Mohamed Daoudi for computing curvature indices...
  3. W

    What is the difference between curvature and concavity in one or two dimensions?

    Hello- would someone mind clarifying the distinction between curvature of a function and the concavity? I would prefer if you could keep it to the one or two dimensional case, since my math background is just multivariable. Thank you very much.
  4. M

    Curvature in polar coordinates

    hi i need a affirm of curvature in polar coordinates. i need now please
  5. S

    Spacetime Curvature: Exploring General Relativity

    General relativity has it that the spacetime continuum is curved. The physics of continuum is dealt with [stress] tensors. My questions: (1) The presence of a mass creates the curvature in spacetime. By how? (2) If the curvature due to matter is positive, is the curvature due to antimatter...
  6. H

    .Exploring Spacetime Curvature to General Relativity

    I was just gong to learn general relativity(not with maths) but with some very basic tutorials given over internet. I also watched the animated series of general realtivity. Everywhere i see,matter bends spacetime( a fabric of spac and time woven ). And when there is matter than this...
  7. L

    What is the Gaussian Curvature of a Cone at its Vertex?

    Hi, I know that you can determine that the Gaussian curvature of a cone tends to infinity at the vertex, but seeing as the curvature anywhere else on the cone is zero, how is this possible?
  8. C

    What is 'Curvature' of Spacetime

    We have described the distortion in spacetime which Einstein derived in GR as a "curvature" of spacetime. This is barely more descriptive than "warping" spacetime. I understand that what this means is that spacetime varies from being Euclidean, having distortion caused around objects of mass...
  9. R

    Curvature of spacetime inside of stars and planets

    hi, how does general relativity work INSIDE stars and planets, since the mass is no longer concentrated within a point, so there are necessarily gravitationnal effects outwards and not only inwards?
  10. R

    Subatomic particle and space curvature

    Can we say that each subatomic particle affects space time such that collectively as big as a planet it explains why there is gravity? Thank you very much.
  11. P

    Curvature radius of alpha and beta particles

    Homework Statement An alpha particle and beta particle, each with kinetic energy 35keV , are sent through a 1.1T magnetic field. The particles move perpendicular to the field. Homework Equations no idea. The Attempt at a Solution theres nothing on curvature radius in my...
  12. Telemachus

    Polar coordinates and radius of curvature

    Homework Statement I've got this problem on polar coordinates which says: A particle moves along a plane trajectory on such a way that its polar coordinates are the next given functions of time: r=0.833t^3+5t \theta=0.3t^2 Determine the module of the speed and acceleration vectors for this...
  13. TrickyDicky

    Weyl curvature and tidal forces

    I'm a bit confused about this and would like for someone to help me get this straight. I read in wikipedia that a manifold with more than three dimensions, like spacetime, is conformally flat if its Weyl tensor vanishes. I think all FRW metrics are conformally flat, so I guess our universe is...
  14. G

    What is the minimum radius of curvature of the curve?

    Homework Statement A civil engineer is asked to design a curved section of roadway that meets the following conditions: With ice on the road, when the coefficient of static friction between the road and rubber is 0.1, a car at rest must not slide into the ditch and a car traveling less...
  15. G

    How would we express the curvature of the 2-dimensional surface of a

    How would we express the curvature of the 2-dimensional surface of a sphere without referring to a radius of curvature or any other extra-dimensional description?
  16. L

    Banking angle and curvature radius of an airplane

    Homework Statement If a plane is flying level at 950 km/h and the banking angle is not to exceed 40 degrees what is the minimum curvature radius for the turn?Homework Equations possibly F = ma = mv2 / r ? The Attempt at a Solution no idea where to start on this one, not sure where the angle...
  17. S

    How to calculate radius of curvature of a reflector

    Hi I need to calculate the radius of curvature of a reflector.I have a sound source (Ultrasonic transducer of 40 mm operating at 50 Khz) in air . I am trying to generate a standing wave using this sound source .As curved reflectors can help to amplify the sound pressure (I actually don't...
  18. Rasalhague

    Constancy of metric tensor components as a test of curvature

    In the previous section he derived the components, with respect to the coordinate bases associated with a polar coordinate system, of the Riemannian metric tensor field on S2, the unit 2-sphere: g = \begin{pmatrix}1 & 0 \\ 0 & \sin^2(\theta) \end{pmatrix} where \theta is the zenith angle...
  19. S

    What is the radius of curvature of the track at 6 seconds into the curve?

    Homework Statement A train enters a curved horizontal section of track at 100 km/hr and slows down with constant deceleration to 15 km/hr in 12 seconds. An accelerometer mounted inside the train measures a horizontal acceleration of 2 m/s^2 when the train is 6 seconds into the curve. Calculate...
  20. H

    Radius of Curvature: Homework Statement & Equation

    Homework Statement If you paint a dot on the rim of a rolling wheel, the coordinates of the dot may be written as (x,y)=(R\theta+Rsin(\theta), R+Rcos(\theta) where \theta is measured clockwise from the top. Assume that the wheel is rolling at constant speed, which implies \theta = \omegat...
  21. O

    Potentials, connections and curvature

    Hi everyone, I have a question related about the relation between potentials, connections and curvature in gauge theories. In Newtonian physics, the common starting point is Newton's law, which determines the motion in terms of the derivative of the potential, i.e. sth. like \ddot...
  22. M

    Space-Time curvature? the units?

    What would the units be on the curvature of spacetime? G(curvature)=8πGT/c^4
  23. Telemachus

    Acceleration and radius of curvature

    Homework Statement A particle moves along a curvilinear trajectory shown in Fig. When passing through the point O has a 3.6m/sec speed and slows down so that it passes through point A does so with a speed of 1.8m/sec. The distance from point A to point O measured along the path is 5.4meters and...
  24. A

    Calculating Curvature of 3D Hyperboloid - Parameters & Extrinsic Curvature

    How can I calculate the curvature of a 3D hyperboloid? I mean, what parameters do I need to calculate the intrinsic curvature? I guess to calculate the extrinsic curvature as seen from a 4D space I would just need a curvature radius, right? Thanks
  25. A

    Radius of curvature usng: M/I= σ/Y = E/R

    Question: How do I use this formula to find the Radius of curvature? Formula: M/I= σ/Y = E/R (M = bending moment, I = second moment of aria, σ = stress, y = distance from nutral axia, E = modulus of elasticity & R = radius of curvature) Attempt: In this question, I have all of the...
  26. M

    Extrinsic Curvature: Normal Vector & Sign Impact

    In the definition of the extrinsic curvature, there is the normal vector. It depends on the sign of the normal vector? Because a normal vector can be directed in two ways. For example the curvature of a circle on the plane has different curvature from inside and outside! But this is...
  27. jfy4

    Phase, Geodesics, and Space-Time Curvature

    Please read and critique this argument for me please, any help is appreciated. Imagine a geodesic, and a matter wave that traverses this geodesic. The action of this matter wave determines the motion of the matter wave along this geodesic over a given space-time interval, and is specified...
  28. jfy4

    Phase, Geodesics, and Space-Time Curvature

    Please read and critique this argument for me please, any help is appreciated. Imagine a geodesic, and a matter wave that traverses this geodesic. The action of this matter wave determines the motion of the matter wave along this geodesic over a given space-time interval, and is specified...
  29. F

    Curvature and Tangential angle

    In Differential Geometry by Heinrich Guggenheimer (if you have the book, the proof I am asking about is Theorem 2-19), he gives the angle between a chord through points s and s' and the tangent at s, as the integral of the curvature (with respect to arc length) from s' to s. I'm not sure how he...
  30. L

    Metric, connection, curvature oh my

    I just read a sentence in GRAVITATION by MTW (aka the "Princeton Phonebook") that made me realize a confusion wrt the metric, connection, and curvature. In short how are g_{\mu\nu}, \Gamma^{\alpha}_{\mu\nu}, and R^{\alpha}_{\beta\mu\nu} distinguished? They all include the description "how space...
  31. S

    Geodesic Curvature of a curve on a flat surface

    Sorry if this ends up being a naive question, but I have just a little conundrum. I'm dealing with curves in R2 and the Gauss-Bonnet theorem is a very useful result with what I'm currently doing, what with Gaussian curvature of a flat surface being zero, which is all fine...
  32. Jonathan Scott

    Local curvature of surface just outside Schwarzschild radius

    Suppose one had a solid sphere just slightly larger than its Schwarzschild radius. What would the curvature of the surface look like to a local observer? Would it curve downwards, or appear flat, or curve upwards? If my brain was working a bit better today, I'd calculate it myself from the...
  33. M

    How Do You Calculate the Curvature of a Point on a Wave Function?

    May i know how to find the curvature of a point on a wave function? e.g wave function: y=Asin(wx) and i want to find the curvature of a point at for example x=x0 and y= Asin (wx0) thank you very much
  34. T

    Does compressing a mass to a smaller sphere change the curvature of space?

    If you take a homogenously distributed spherical mass and compress it to a smaller radius while maintaining its overall total energy and momentum, will it change the curvature of space (gravity) outside of the original sphere? According to Newton, it stays the same. However, it seems like...
  35. D

    General relativity: constant curvature, characterizing equation

    Homework Statement Show, that a three-dimensional space with constant curvature K is charaterized by the following equation for the Riemann curvature tensor: R_{abcd} = K \cdot \left(g_{ac}g_{bd}-g_{ad}g_{bc}\right) Homework Equations The Attempt at a Solution Hi folks, I would like to...
  36. M

    'Direction' of space-time curvature ?

    Hi, I'm new here. I want post a specific question that's been rattling around in my head. Basically, if you consider the curvature of 3 dimensional space into a 4'th dimension due to gravitational field, has anyone considered the 'direction' of that curvature ? If you think about the...
  37. J

    Is a zero Ricci scalar the defining characteristic of a 'flat' spacetime?

    If the Ricci scalar R happens to be zero (everywhere according to our metric), is that the definition of a 'flat' space time? And how are flat space times related to Minkowski space precisely? ARE they the SR space exactly? Thanks. Just trying to understand why 'flat space time' is a...
  38. M

    Curvature of space and dark matter

    Assuming gravity is matter curving space as Einstein says, isn't our theory of dark matter just an assumption that because more gravity is required to explain galaxy formation that it must be caused by unseen matter? Why do we assume that the curvature of space required must be caused by matter...
  39. A

    Curvature of Space-Time: Understanding Bing Bang & f(t)

    hello i understand that in a flat space the metric is \eta_{uv}dx^udx^v...i know that this means that the light follows straight geodesic in this space time... but ¿what would means that metric is f(t)\eta_{uv}dx^udx^v where f(t)=infinite in t=0 and f(t)=0 in t=infinite...obvious i...
  40. e2m2a

    Does energy alone contribute to spacetime curvature?

    Do Einstein's field equations explicitly show that energy alone can curve the metric of spacetime? True, energy is included in the stress-energy tensor, but is it assumed that energy in of itself curves spacetime? Or, is it possible that only energy "embedded" in mass contributes to...
  41. J

    Divergence of curvature scalars * metric

    How can one work out what terms like: (g^{cd}R^{ab}R_{ab})_{;d} are in terms of the divergence of the Ricci curvature or Ricci scalar? One student noted that since: G^{ab} = R^{ab} - \frac12 g^{ab}R {G^{ab}}_{;b} = 0 that we could maybe use the fact that G^{ab}G_{ab} = R^{ab}R_{ab} - \frac12...
  42. A

    Where does all this spacetime curvature come from?

    Hi, From what I know, science is the study of the observable world, its theories are supported by evidence. Now GR is a theory, and it informs that mass curves the 'fabric' of space and time. The thing I don't understand is that there is no evidence of mass curving spacetime, then how is...
  43. R

    Radius of curvature of a function

    Homework Statement I have a graph of y=lg(x) which is supposed to mimic the curvature of a beam, or I can use y =√x to be more precise. But in essence between two points x2 and x1, I need to find the radius of curvature R so as to find the bending stress on it. Homework Equations...
  44. F

    Approximation of total curvature

    Hello, I am trying to find an interpolating curve between a few points that has minimal curvature. That means, as close to a straight line as possible. Reading a document about cubic splines, they say that \kappa \left ( x \right )=\frac{|f''\left ( x \right )|}{\left ( 1+\left [ f'\left...
  45. I

    Gaussian Curvature, Normal Curvature, and the Shape Operator

    Homework Statement Let u_1, u_2 be orthonormal tangent vectors at a point p of M. What geometric information can be deduced from each of the following conditions on S at p? a) S(u_1) \bullet u_2 = 0 b) S(u_1) + S(u_2) = 0 c) S(u_1) \times S(u_2) = 0 d) S(u_1) \bullet S(u_2) = 0...
  46. G

    Radius of Curvature Derivation Help

    Homework Statement This is not exactly a question, but I am trying to understand the derivation of radius of curvature from a boof I'm reading. I would be extremely grateful if someone is able to help me. Homework Equations Let u and n be the tangent and normal unit vectors respectively...
  47. L

    Radius of Curvature: Math Homework Help

    Homework Statement I had my college math courses in 1955-1957, so I'm rusty. Lately interested in Radius of Circle of Curvature. I don't have a math typing program, so I'll try to describe the equation that I found recently, but it's complexity [though so far, I can handle any common...
  48. H

    Proving Constant Curvature of r(s) is a Circle

    If we parameterize the arc length of a vector valued function, say, r(s) and r(s) has constant curvature (not equal to zero), then r(s) is a circle. Thus, |T'(s)| = K but to prove it we would need to show |T'(s)| = K => <-Kcos(s), -Ksin(s)> and integrate component-wise two times, right?
  49. F

    Find T,N,B Vectors & Curvature of Curve x=-4ty=-t2z=-2t3 at t=1

    Homework Statement Find the unit tangent, normal and binormal vectors T,N,B , and the curvature of the curve x=−4t y=−t2 z=−2t3 at t=1. Homework Equations The Attempt at a Solution I found T=(-4/sqrt(56),-2/sqrt(56),-6/sqrt(56)) which is correct. But I keep getting N wrong...
  50. C

    How Can We Visualize Ricci Curvature in Different Dimensions?

    "Visualizing" Ricci curvature Can someone help me visualize the Ricci curvature? Since it is easier to visualize a surface bending in 3-D, let's try to view this as a sheet with one spatial dimension and one time dimension and embedding into euclidean 3-D. Since the metric can always be...
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