For a ‘smooth’ parametric curve: xftygt= (), == () and zht () for atb≤≤ OR for a ‘smooth’ vector function: rijk()tftgtht= ()++ () () for atb≤≤
the arc length of the curve for the t values atb≤ ≤ is defined as the integral b ('())('())('())f tgthtdt222++ ∫ a
b For vector functions, the integrand can be simplified this way: r'(tdt ) ∫a
Ex) Find the arc length along the curve rjk()tt= (cos())33+ (sin()) t for 0/2≤ t ≤π .
Unit Tangent and Unit Normal Vectors
r'(t ) Unit Tangent Vector Æ T()t = r'(t )
T'(t ) Unit Normal Vector Æ N()t = (can be a lengthy calculation) T'(t )
The unit normal vector is designed to be perpendicular to the unit tangent vector AND it points ‘into the turn’ along a vector curve’s path.
2 Ex) For the vector function rij()tt=++− (2 3) (5 t ), find T()t and N()t .
(this blank page was NOT a mistake ... we’ll need it) Ex) Graph the vector function rij()tt= (2++− 3) (5 t2 ) and its unit tangent and unit normal vectors at t = 1.
Curvature Curvature, denoted by the Greek letter ‘ κ ’ (kappa) measures the rate at which the ‘tilt’ of the unit tangent vector is changing with respect to the arc length. The derivation of the curvature formula is a rather lengthy one, so I will leave it out.
Curvature for a vector function r()t rr'(tt )× ''( ) Æ κ= r'(t ) 3
Curvature for a scalar function yfx= () fx''( ) Æ κ= (This numerator represents absolute value, not vector magnitude.) (1+ (fx '( ))23/2 )
YOU WILL BE GIVEN THESE FORMULAS ON THE TEST!! NO NEED TO MEMORIZE!! One visualization of curvature comes from using osculating circles. When an osculating circle is drawn inside a plane curve at a specific point P, the circle has: 1. to be tangent to the curve at the point P, 2. to have the same curvature as the curve at point P, 3. to lie inside the concavity of the curve at point P.
Here’s an example of a 2D curve (vector function) with osculating circles drawn at points A, B and C.
The sharpest ‘turn’ is at the point C. The higher the curvature, the smaller the osculating circle.
The curvature is least at the point B and it has the largest osculating circle.
Osculating Circle’s radius = 1/curvature
Ex) Calculate the curvature at t = 1 for the vector function ri()tt=+ (1 + t2 ) j. Get an exact value for the curvature first, then round to nearest tenth.
Ex) Determine the curvature function for f()xx= ln() (i.e. determine κ for any value of x ). Approximate the x value where the function f()xx= ln() has the maximum curvature. Round to the nearest tenth.