Arclength, Parametric lines-- {cal3}

How to paramaterize a cruve?

• L(T) = <$x_0+xt,y_0+yt+\dots$>

(See photos notes)

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Arclength between 2 pts!!

• Arclength btwn 2 time cords OR btwn 2 pts (2 ways 2 think abt it)
• Phyiscs: dst btwn obj travel btwn 2 time val, geo: dst btwn 2 pts on curve

1. Solve eq P=(0,0) → r(a) = P → $<asin(a),acos(a)>$,then $<0,0>$ or a=0
2. Then solve for Q = $(0,4\pi )$ → r(b) = Q → $<bsin(b),bcos(b)>$=$<0,4\pi >$ → Sys of EQ, $bsinb=0,bcos(b)=4\pi$
   1. Solve for a&b to get LOI
   2. Get b=4 for LOI, then plug into aLength formula

Take shape: Apply formula → get ans

• arclength!
  • ${\int }_b^a|s(t)|dx$ (abs val)aap
  • eg. circumfrence of unit circ?
  • v(t) = <cost,sint>, then s(t) = ←sint,cost>
• Eg.2 arclength,
  • What is arclength btwn P=(1,1), Q=(-1,1), then
  • solve r(a) = P, so a=1, and r(b) = Q, so b = -1
  • therefore arclength = ${\int }_{-1}^1\sqrt{1+4t^2}dt$

Q1) Create @param line 4 2D circle s/t clockwise + center on (3,2) + rad=5 + const speed = 3

• Take $<\sin t,\cos t>$ to center on (3,2) get: $<\sin t+3,\cos t+2>$
• Radius: $<5\sin t+3,5\cos t+2>=r(t)$
• To find speed: take r’(t) =$5\cos t,-5\sin t$ → r’(t) =5, Obtain: $\frac{3}{5}=t$, then plug into original sin(t) for radius EQ
  • Re@param 4 diff speed: 1) calc speed 2) undo original speed, multiply by new speed (hence how we get 3/5)

Formulas! KNOW

• Slope of a @param $\frac{dy}{dx}=\frac{y^{\prime }}{x^{\prime }}$, and second derative if wanted would be = y”/x’
  • ex: $r(t)=<t^2,t^3>$
  • Take derative of both x&y, then take of y again putting ontop into y” / x’