Tangent and Normal Vector - (.5). Principal Unit Normal Vector Let C be the curve traced out bythe vector-valued function r!!t # f!t, g!t, h!t $. The vector T!!t r! % r! %!t!t is the unit tangent vector to the curve C. Define N!!t. The vector N!!t is called the principal unit normal vector. Observe that! N!!t is a unit vector! N!!t is in the same direction as T! %!t! N!!t and T!!t are orthogonal It is known if v!!t c then v!!t # v! %!t 0. Since T!!t, T!!t # T! %!t 0. N!!t # T!!t # T!t # T!t 0! N!!t! dt!!t N!!t dt!!t dt!!t dt!!t dt!!t dt!!t dt!!t dt!!t dt!!t! dt!!t,! 0. N!!t will always point to the direction in which T!!t is turning as arc length increases and point to the concave side of the curve. Example Let the helix C be traced out by r!t # cos t, sint, t $. Find the unit tangent and principal unit normal vectors to the curve at anytime t. Sketch the helix C and the unit tangent and principal unit normal vectors when t # and t #. r! %!t # sint, cos t, $, r! %!t sin t & cos t & T!!t sin t & cos t & # sint, cos t, $ sin t & sin t & # sint, cos t, $ # cos t, sint, 0$ 8sintcos t cos t sint sin t & / # sint, cos t, $ sin t & # cos t, sint, 0$ sintcos t sin t & / # sint, cos t, $
x : cos t sin t & & sin t cos t cos t y : sint sin t & sintcos t sint sin t & cos t sint 5sint z : sintcos t sin t & / # cos t, 5sint, sin!t $ sin t & / N! sin t & /!t & 9sin t! & cos t & 9sin t! & cos t When t #, cos t & 5sin t & 9sin t cos t sin t & / & 9sin t! & cos t sin t & / # cos t, 5sint, sin!t $ # cos t, 5sint, sin!t $ When t #, T! # 5 #, 0, $, N! # 5 T!!# # 0,, $, N!!# # 0, 5, 0 $# 0,, 0 $ #, 0, 0 $#, 0, 0 $ - -0.5 y 0 t 0.5 - - r!!t cos t, sint, t, t #/, -.- t #. Binormal Vector Note that B!!t T!!t ' N!!t
! B!!t is a unit vector Since T!!t, N!!t, and the angle $ between T!!t and N!!t is #, B!!t T!!t N!!t sin$!!!! B!!t is orthogonal to both T!!t and N!!t by definition.! TNB frame - a frame of reference formed by vectors T!, N! and B!. Vectors T!, N! and B! form a basis of the -D space (as! i,! j and! k form a basis of the -D space).. Normal Plane and Osculating Plane The plane spanned by vectors N!!t and B!!t is called the normal plane. The plane spanned by T!!t and N!!t is called the osculating plane. Note that! The normal vector to the normal plane is T!!t and the normal vector to the osculating plane is B!!t. The osculating circle (or the circle of curvature) is the circle of radius %! for! # 0 lying completely in the osculating plane. % is called the radius of curvature and the center of curvature is the center of the osculating circle. For example:! in a -D plane: http://poncelet.math.nthu.edu.tw/disk/cabrijava/osc-ell.html! in a -D space: http://www.lsus.edu/sc/math/rmabry/lived/helix.htm Example Find the osculating circle (the equation) of the curve traced out by r!t # t, t $ at t. r!! #, $, r! %!t # t, t $, r! %! #, $, r! %%!t #, t $ T!!t r! % r! %!t # t, t $ # t, t $!t t & 9t t & 9t #, t $, t $ 0 & 9t T!! #, $, r! %! # 0, $ 8t #, t $ & 9t! & 9t /! & 9t!! & / 9t # 0, $ 9t #,t $ # 8t, $! & 9t / T! %! # 8, $, T! %! 8 &! /! /! / Find & such that!! N!! T! %! T! %! r! %!, %! T! %!! / # 8, $ #, $ center of curvature: w r!! & &N!! #, $&& #, $ & N!! %, & % N!! %
center of curvature: w #, $& #, $#, $ the equation of the circle of curvature: x r!!t # & & y or cos t, & sint $ 0 Example Let the helix C be traced out by r!t # cos t, sint, t $. circle at t #. When t #, r!!# #, 0, # $ Find the equation of the osculating r! %!t # sint, cos t, $, r! %!# # 0,, $, r! %!# & T!!# # 0,, $, T! %!# #, 0, 0 $! / #,0,0 $, T %!#!! T %!# r! %!# N!!# #, 0, 0 $, radius of curvature: %! center of curvature: w r!!# & &N!!# #, 0, # $&& #, 0, 0 $ Find & so that &N!!#, & N!, w #, 0, # $!#. Tangential and Normal Components of Acceleration Suppose that the position of an object at the time t can be described by the end point of r!!t. Then the velocity and acceleration vectors of the object are v!!t and a!!t, respectively. Recall that vectors T!!t and N!!t are orthogonal and form a basis for a -dimensional plane in space. Consider v!!t r! %!t r! %!t T!!t T!!t a!!t v! %!t d T!!t d s T!!t &
N!!t $ T! %!t N!!t dt! dt!!t dt!!t! a!!t d s T!!t & d s T!!t &! N!!t Tangential component of acceleration: a T d s, depending on the rate of change of the speed Normal components of acceleration: a N!, depending on the curvature and speed square Compute a N without computing! : Fact: if u! and v! are orthogonal then &u! & 'v! & u! & ' v! by the Pythegorean Theorem. Here T! and N! are orhogonal and are unit vectors ( T!!t and N!!t ). So, a!!t at & a N $ a N a!!t a T since a N! 0. Example Find tangential and normal components of accelearation for an object with position vector r!t # sint, cos t, t $. r! %!t # cos t, sint, $, r! %!t cos t & sin t & & 0 r! %!t 0, d s 0, a T 0! b b & a & 0, a N 0 0 a!!t N!!t The other way: a!!t # sint, cos t, 0$, a!!t, a N 0 Applications:! Force acting on a space curve: F!!t ma!!t m d s T!!t & m! N!!t 5