curvature (2)
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 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