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 equation Curvature equation (rectangular)(rectangular)The function must be a twice-

differentiable equationCan be used with rectangular

coordinates, polar coordinates, parametric equations, or vectors

Represented by “K”Equation:

◦ K = ( Iy''I )/ [1 + (y')²]³∕² ◦ Where y'' is the second derivative of the

function and y' is the first derivative of the function

Curvature equation (polar)Curvature equation (polar)

K-CurvatureS-Arc Length -Angle measured counterclockwiseT-

ProofProofII r'(x) II = √1 + [f '(x)]²

K = II r'(x) x r''(x) II / [II r'(x) II]³

K = I f''(x) I / { 1 + [f '(x)]²}³∕²

K = I y'' I / [ 1 + (y') ²]³∕²

Polar equation

Rectangular equation

Relationship between Relationship between Acceleration, Speed, and Acceleration, Speed, and CurvatureCurvaturea(t) = d²s / dt² T + K(ds/dt)² N

a(t) represents the acceleration vector

K represents the curvatureds/dt represents the speed

If r(t) is the position vector for a smooth cruve C, then the acceleration vector is given by the above equation

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

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

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 straight line Gravity does not redirect the object it redefines what

the straightest path is

Sybase’s LogoSybase’s Logo

We will:- Find the equation of this curve- Calculate the function that represents the

curvature of this logo

So Where Did This Come So Where Did This Come From?From?The Sybase logo is represented

by the Archimedean spiralThis is represented by the

parametric equations:y = t cos tx = t sin t

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∕²

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

Graph the curvature:y

x

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

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

calcII/Curvature.aspxhttp://www.newworldencyclopedia.org/

entry/Curvaturehttp://xahlee.org/

SpecialPlaneCurves_dir/ArchimedeanSpiral_dir/archimedeanSpiral.html

Larson, Ron, Hostetler, Robert, Edwards, Bruce, Calculus with Analytic Geometry. Boston, New York: Houghton Mifflin Company, 2006