13.3 Arc Length
Review: curve in spce: r t f t i g t j h t k Tngent vector: r '( t ) f ' t i g ' t j h' t k Tngent line t t t : s r( t ) sr '( t ) Velocity: v( t) r '( t) speed = v( t) Accelertion ( t) v '( t) r ''( t) 1 Projectile motion: r( ) j v r t gt t initil velocity v, initil position r g 3 ft grvittionl constnt sec Initil speed v, initil height h, lunching ngle : 1 r( t) v cos t, h v sin t gt
Recll the length of curve in the plne: curve s grph y f x : length from t to t curve in prmetric form : x f t nd y g t : length 1 [ f '(x)] dx [ '(t)] [ '(t)] f g t f t g t ht A curve in 3-spce : r,, Length or Arc Length [ '(t)] [ '(t)] [ '(t)] f g h Length r '( t)
Exmple 1: A line from to : r( t) t( ), t 1 length 1 r t 1 = distnce from to Exmple : Compute the length (circumference) of circle: prmetric description: ( x x ) ( y y ) r x x r cos( t), y y r sin( t) t length = r t r sin ( t) r cos ( t) r r
Prolem : A drunken ee trvels long the pth r( t) cos( t),sin( t), t for 1 secon. It then trvels t constnt speed in stright line for 1 more secon. How fr did the ee trvel? velocity: v( t) r'( t) sin( t),cos( t),1 Speed = v( t) 9 3 distnce trveled long the helix: 1 r '( t) 3 At time t 1, it trvels long the tngent line tngent line: u( s) r( t ) sv( t ) length = u '(s) 1 v ( t ) 3 ( 1) 3 1 Totl distnce trveled: 6 ft
Arc length function descries distnce from eginning point t : s( t) t r u du Notice: '( t) r rt gives position on the curve s function of time its more nturl to look t the odometer or the mile mrkers to tell where you re fter you hve trveled certin distnce s we would like to hve t s function of s so tht we could reprmetrize in terms of s. Using the rc length function or just t t s giving us r t s r s s s t we cn find t s function of s
Exmple: t t Prmetrize the following curve y rc length: r 3sin(t ) i 3cos( ) k t t t t t r ' 6 cos( ) i 6 sin( ) k t t t t t r ' 36 cos ( ) 36 sin( ) 36t 6 t t s s( t) 6udu ut s Solve for t : t= 3 3 u 3t u s s 3 3 Plug into r( t) : r( t( s)) r( s/ 3) 3sin( ) i3cos( ) s s Arc length prmetriztion: r( s) 3sin( ) i 3cos( ) k 3 3 Notice: r '( s) 1 length from s to s : r s 1 t is n rc length prmeter if r '( t) 1! k
13.4 Curvture
The curvture of curve r( t) mesures the mount the tngent vector en. Q curvture t P> curvture t Q P Tngent vector r '( t) Unit tngent vector '( t) T r r '( t) ( r '( s)! ) Curvture - the mgnitude of the rte of chnge of the unit tngent vector T with respect to the rc length prmeter s. dt T s s dt d dr r if s is the rc length prmeter T r s
Exmples: ) The curvture of stright line: r () t r + t v () r' t v wht is the rc length prmetriztion of the line: v r (s) r + s r' v dt r s ( if not rc lenght: ( ), then ''( ) ) The curvture of circle of rdius r: r ( t) r cos( t), r sin( t) r '( t) r sin( t), r cos( t) ' t r t r + t v r t t v ) r r hence s u du rt or t r / s s s s s rc length prmetriztion: r (s) rcos( ), rsin( ) T r'(s) sin( ),cos( ) r r r r dt 1 s 1 s T' cos( ), sin( ) dt 1 1 r r r r r r smll rdius mens lrge curvture! t r v since then (s) 1 v
It is usully too difficult to prmetrize curve y rc length. Express curvture in terms of generl prmeter t : Recll s( t) dt d T dt Exmple: t r u du hence: '( t) r T r t t Compute the curvture of n ellipse: r ( t) cos( t),sin( t) r '( t) sin( t),cos( t) r T r t t x y 1 '( t) sin ( t) cos ( t) T( t) T'( t) sin( t),cos( t) t t sin ( ) cos ( ) sin ( ) cos ( ) t t cos( t), sin( t) T'( t) ( sin ( t) cos ( t)) 3/ T r t t ( lengthy computtion ) ( sin ( t) cos ( t)) 3/
Sometimes it is esier to use nother formul: Why is this true? r '( t) Recll: T nd r'( t) r '( t) T r t t r tr t rt 3 Hence: r '( t) r '( t) T T nd r''( t) rtr t T T T d s ' d T T' T d s T T' ' T T ut TT 1 implies T' T TT' TT' i.e., T is perpendiculr to T' This implies TT' T T' sin( ) T' r' Put together, we get: rtr t T T' r' = r' 3
Exmple: Ellipse revisited: r( t) cos( t),sin( t) or r( t) cos( t),sin( t), r '( t) sin( t),cos( t), r ''( t) cos( t), sin( t), i j k t t sin( t) cos( t) r r cos( t) sin( t) ( sin ( t) cos ( t)) k = k κ = r t r t r t 3 = sin (t) + cos (t) 3
Exmple: Curvture of the (ellipticl) helix: r ( t) cos( t),sin(t), c t r '( t) sin( t),cos( t),c r ''( t) cos( t), sin(t), i j k sin( ) cos( ) r t r t t t c r cos( t) sin( t) tr t c sin ( t) c cos ( t) r'( t) sin ( t) cos ( t) c κ = r t r t ( csin( t)) i ( c cos( t) ) j ( sin ( t) cos ( t)) k csin( t) i c cos( t) j k c sin ( t) c cos ( t) r t 3 sin ( t) cos ( t) c 3/ Circulr helix:, c 3/ c c c 3/ c constnt!
Apppliction: Curvture of plne curve: r tr t rt 3 r ( t) x( t), y(t), r' ( t) x '( t), y'(t), i j k rtr t x ' y ' ( x ' y '' x '' y ') k x'' y'' x ' y '' x '' y ' ( x') ( y') 3/
Summry: Length of curve r '( t) Arc length function s( t) t r u du '( t) r Arc length prmetriztion r( s) with r'( s) 1 r '( t) Unit tngent vector T r '( t) dt Curvture: r s r'(s) T r t t r tr t rt 3