Arclength and the Fundamental Form

Since the fundamental form is intrinsic to a surface, any property that can be directly derived from the fundamental form is also intrinsic to the surface. For example, suppose that r( u,v) parameterizes a surface and that r( t) = r( u( t) ,v( t) ) , t in [ a,b] , is a curve on the surface. Then the velocity of r( t) is

v = ru du dt  + rv dv dt

and the square of the speed is

v·v = æ è ru du dt +rv dv dt ö ø · æ è ru du dt +rv dv dt ö ø = ru·ru æ è du dt ö ø 2   +2rv·ru du dt dv dt +rv·rv æ è dv dt ö ø 2   = g11 æ è du dt ö ø 2   +2g12 æ è du dt ö ø æ è dv dt ö ø +g22 æ è dv dt ö ø 2

Thus, the length of the curve r( t) is given by  

l = ó õ b a  g11 æ è du dt ö ø 2   +2g12 æ è du dt ö ø æ è dv dt ö ø +g22 æ è dv dt ö ø 2       dt

(3)

which show that the length of a curve on a surface is intrinsic to that surface.

For example, the xy-plane can be parameterized in polar coordinates by

r( r,q) = á rcos( q),rsin( q) ñ

Since r_r = á cos( q) ,sin(q) ñ and r_q = á -rsin( q) ,cos( q) ñ , the metric coefficients of the plane in polar coordinates are g₁₁ = r_r·r_r = 1, g₁₂ = r_r·r_q = 0, and

g22 = rq·rq = r2sin2(q) +r2cos2( q) = r2

Consequently, (3) for a curve r(t) = r( r( t) ,q( t) ), t in [ a,b] , is given by

l = ó õ b a  æ è  dr dt ö ø 2    + r2 æ è  dq dt ö ø 2     dt

In case of r = f( q) , q in [ a,b] , we let q = t and r = f( t) .       

EXAMPLE 3    What is the length of the curve r = e^-q, q in [ 0,6p] in polar coordinates?

Solution: If we let r = e^-t and let  q = t, then r'  = -e^-t and q ' = 1, so that

l = ó õ 6p 0     ( -e-t)2+( e-t)2 (1)2   dt = ó õ 6p 0     e-2t+e-2t   dt

Because e^-2t+e^-2t = 2e^-2t, this simplifies to

l = ó õ 6p 0     2e-2t   dt = ó õ 6p 0    e-qdq = Ö2( -e-6p+1)

which is approximately l = 1.414.

The fundamental form of a sphere of radius R in spherical coordinates is

ds2 = R2df2+R2sin2( f) dq2

Thus, if r( t) = Re_r( f( t) ,q( t) ) , t in [ a,b], is a curve on the sphere, then its length is

l = ó õ b a  R2 æ è df dt ö ø 2    + R2sin2(f) æ è dq dt ö ø 2       dt

(4)

In case of f = g( q) , q in [ a,b] , we let q = t and f = g( t) .        

EXAMPLE 4    What is the length of the curve r( t) = 2Ö2e_r( p/4,t) , t in [ 0,p/2] , which is an arc at a constant 45^° latitude between the points P( 2,0,2) and Q(0,2,2) on the sphere of radius 2Ö2 centered at the origin?

image

 Solution: Since f = p/4 and q = t, the arclength formula (4) becomes

l = ó õ p/2 0  ( 2Ö2) 2 æ è  d dt  p 4 ö ø 2   +( 2Ö2) 2sin2 æ è  p 4 ö ø æ è  d dt t ö ø 2     dt = ó õ p/2 0  8( 0) 2+8 æ è  1 Ö2 ö ø 2   ( 1) 2   dt = ó õ p/2 0  Ö4dt = p