curvature final
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TRANSCRIPT
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