curvature

Examples of curvature in the following topics:

• Arc Length and Curvature

  • On any curve, there is a center of curvature, C.
  • The curvature can also be approximated using limits.
  • Curvature is the amount an object deviates from being flat.
  • The curvature of C at P is then defined to be the curvature of that circle or line.
  • The radius of curvature is defined as the reciprocal of the curvature.

• The Relativistic Universe

  • In particular, the curvature of space-time is directly related to the energy and momentum of whatever matter and radiation are present.
  • People can use the metric to calculate curvature and then use the Einstein field equations to relate the curvature to the energy and momentum of the space-time.
  • Going in the reverse order, energy and momentum affect the curvature and the space-time.
  • The precise definition of curvature requires knowledge of advanced mathematics, but an intuitive way to understand it is that the definition of a straight line changes in curved spacetime.

• Abnormal Curves of the Vertebral Column

  • Abnormal curvatures of the spine include kyphosis, lordosis, retrolisthesis, and scoliosis.
  • However, abnormal curvatures such as kyphosis, lordosis, retrolisthesis, and scoliosis may occur in some people.
  • Kyphosis is an exaggerated kyphotic (posterior) curvature in the thoracic region.
  • Scoliosis, lateral curvature, is the most common abnormal curvature, occurring in 0.5% of the population.
  • Distinguish among the types of abnormal curvature of the vertebral column

• The Lensmaker's Equation

  • The lensmaker's formula is used to relate the radii of curvature, the thickness, the refractive index, and the focal length of a thick lens.
  • Unlike idealized thin lenses, real lenses have a finite thickness between their two surfaces of curvature.
  • Lenses are classified by the curvature of the two optical surfaces.
  • R1is the radius of curvature of the lens surface closest to the light source,
  • The signs of the lens' radii of curvature indicate whether the corresponding surfaces are convex or concave.

• The Spine

  • Kyphosis is an exaggerated concave (kyphotic) curvature of the thoracic vertebral column; it is commonly known as "humpback."
  • Lordosis is an exaggerated convex (lordotic) curvature of the lumbar region; it is commonly known as "swayback."
  • The thoracic and sacral curvatures are termed primary curves because these are present in the fetus, yet remain the same in the adult.

• Surface Tension

  • When all the forces are balanced, the curvature of the surface is a good measure of the surface tension, which is described by the Young-Laplace equation:
  • where $\Delta P$ is the pressure differential across the interface, $\gamma$ is the measured surface tension, and $R_1, R_2$ are the principal radii of curvature, which indicate the degree of curvature.
  • This equation describes the shape and curvature of water bubbles and puddles, the "footprints" of water-walking insects, and the phenomenon of a needle floating on the surface of water.
  • In imagining the shape of a liquid droplet or the curvature of the surface of a liquid, one must keep in mind that the molecules at the surface are at a different level of potential energy than are those of the interior.

• Image Formation by Spherical Mirrors: Reflection and Sign Conventions

  • The center of curvature is the point at the center of the sphere and describes how big the sphere is.
  • This point is half way between the mirror and the center of curvature on the principal axis.
  • We can see from the figure that this focal length is also equal to half of the radius of the curvature. shows the ray diagram of a concave mirror.

• Gross Anatomy of the Stomach

  • The greater omentum hangs from the greater curvature.
  • They consist of a superficial and a deep set, and pass to the lymph glands found along the two curvatures of the organ.

• Kinematics of UCM

  • We define the rotation angle $\Delta\theta$ to be the ratio of the arc length to the radius of curvature:
  • The direction of a centripetal force is toward the center of curvature, the same as the direction of centripetal acceleration.

• Angular Position, Theta

  • We define the rotation angle$\Delta \theta$ to be the ratio of the arc length to the radius of curvature:
  • The arc length Δs is the distance traveled along a circular path. r is the radius of curvature of the circular path.