curvature final

CurvatureCurvatureJasmine HeVictoria de MetzRashi Ojha

What is curvature?What is curvature?Refers to how much a geometric

object deviates from being “flat” or “straight”

The measure of the amount of curving

The degree by which a non-linear or surface curves

Curvature in CalculusCurvature in CalculusThe rate of change of direction of

the tangent vectorHow fast a curve is changing

directionCurves that bend to the right, are

negativeCurves that bend to the left, are

positiveWe will focus on plane curves

Curvature equationCurvature equation

K-CurvatureS-Arc Length -Angle measured counterclockwiseT-

ProofProofR(θ )=rcosθ I + rsinθ jR(s)=rcos(s/r)I + rsin (s/r) jR’(s)=-sin(s/r)I + cos (s/r) JT(s) = r’(s)/||r’(s)||=-sin(s/r)i+cos(s/r)JK= ||T’(s)||

Gravity or Curvature?Gravity or Curvature? In Euclidian space, gravity in a force that attracts two

bodies together In reality, this effect is caused by the curvature of

space and time The object is traveling in a strait line Gravity does not redirect the object it redefines what

the straightest path is

Relationship between Relationship between Acceleration, Speed, and Acceleration, Speed, and CurvatureCurvature

a(t)=d2s T + K (ds/dt)^2 N dt2

K- Curvatureds/dt- Speed

How is curvature applied in How is curvature applied in real life?real life?Applied in mostly in physics and

engineeringIs used in frictional forceUsed in calculating space time –

orbitalsForce = mass x acceleration

Example textbook Example textbook problemproblemPage 876, Section: 12.5, #39r(t) = 4ti + 3 cos tj + 3 sin tk---r'(t) = 4i – 3 sin tj + 3 cos tkT(t) = [r'(t)] / [ II r'(t) II ] = (1/5)[ 4i –

3 sin t j + 3 cos t k ] T'(t) = (1/5)[ -3 cos tj – 3 sin tk ]

K = [ II T'(t) II ] / [ II r'(t) II ]K = (3/5) / 5K = 3/25

Sybase’s LogoSybase’s Logo

The Sybase logo is represented by the Archimedean spiral

This is represented by the parametric equations:

y = t cos tx = t sin t

Insert picture of archimedes.

Curvature of the Sybase Curvature of the Sybase LogoLogo Archimedes’ Spiral

Parametric Equation:x = t cos ty = t sin t

K = ( Iy''I )/ [1 + (y')²]³∕²

y' = [( t cos t) + sin t] / [ cos t – t sin t] y'' = (d/dt [ (sin t + t cos t) / ( cos t – t sin t)]) · (1 / dx/dt )y'' = [ (cos t + cos t – t sin t)(cos t – t sin t) – ( sin t + t cos t)(-sin t – sin t – t cos t) ] / (cos t – t sin t)³y'' = [ ( 2 cos t – t sin t)(cos t – t sin t) + ( sin t + t cos t)(2 sin t + t cos t)] / (cos t – t sin t)³y'' = [ (2 cos²t – 2t sin t cos t – t sin t cos t + t² sin²t) + (2 sin²t + 2t sin t cos t + t sin t cos t + t² cos²t) ] / (cos t – t sin t)³y'' = (2 + t²) / ( cos t – t sin t)³

Curvature of the Sybase Curvature of the Sybase Logo cont.Logo cont.

Finding the curvature:

K = I y'' I / [ 1 + (y')²]³∕²

K = I (2 + t²) / ( cos t – t sin t)³ I / ([ 1 + ([( t cos t) + sin t] / [ cos t – t sin t])²]³∕²K = (2 + t²) / [ 1 + (t cos t + sin t)²]³∕²

Graph the curvature:

SourcesSourceshttp://www.cs.iastate.edu/~cs577/

handouts/curvature.pdfhttp://tutorial.math.lamar.edu/classes/

calcII/Curvature.aspxhttp://

www.newworldencyclopedia.org/entry/Curvature

http://xahlee.org/SpecialPlaneCurves_dir/ArchimedeanSpiral_dir/archimedeanSpiral.html