# Vector Geometry

## Arc Length

We look again at the idea of a curve &Gamma. in space, which is defined by the equation

#~r _ = _ #~v ( &theta. )

where #~r represents the position vector of the points of the curve, #~v ( ) is a vector function of the scalar (real) parameter &theta., where &theta._1 =< &theta. =< &theta._2.

If ~A is a point on &Gamma. with position vector #~r_~A, then &exist. &theta._~A &in. [ &theta._1 , &theta._2 ] such that _ #~v ( &theta._~A ) = #~r_~A.

If ~A, ~B are two points on &Gamma. corresponding to parameter values &theta._~A, &theta._~B respectively, _ where _ &theta._~A < &theta._~B, _ then put

&delta._~n&theta. _ #:= _ fract{&theta._~B - &theta._~A , ~n}

- i.e. we split the interval [ &theta._~A , &theta._~B ] into ~n equal parts. The distance along the curve from ~A to ~B is approximately

sum{mod{#~r^{(~n)}_~i - #~r^{(~n)}_{~i - 1}},~i = 1,~n} , _ _ _ _ _ _ where _ #~r^{(~n)}_~i = #~v ( &theta._~A + ~i &delta._~n&theta. )

We define the #~{arc length} from ~A to ~B along &Gamma. as

lim{,~n -> &infty.}sum{mod{#~r^{(~n)}_~i - #~r^{(~n)}_{~i - 1}},~i = 1,~n}