Exercises

Find the first fundamental form of the given surfaces. Explain the relationship of the fundamental form to the given surface.

1. r = á u,v,u ñ 2. r = á u,v,u+v ñ 3. r = á vsin( u) ,vcos( u),v ñ 4. r = á vsin( u),v,vcos( u) ñ 5. r = á sin( u) cos( v),cos( u) ,sin( u) sin( v) ñ 6. r = á sin( v) sin( u) ,cos( v) sin( u) ,cos(u) ñ 7. r = á sin( u) cosh( v),sinh( v) ,cos( u) cosh( v) ñ 8. r = á sin( u)cosh( v) ,sin( u) sinh( v) ,cos( u) ñ

Find the length of the following curves in polar coordinates:

9. r = 2,     q in [ 0,p] 10. r = eq,     q in [ 0,p] 11. r = sec( q) ,    q  in  [ 0,p/4] 12. r = csc( q) , q = t 13. r = cos( q) ,    q in [ 0,p] 14. r = sin( q) ,    q  in  [ 0,p] 15. r =  1 cos( q) -sin( q) ,    q  in  [ 0,p/6] 16. r = sec(q) tan2( q) ,    q  in  [0,p/4]

 

The fundamental form of the unit sphere in spherical coordinates is given by

ds2 = df2+sin2( f) dq2

 (i.e., R=1) Determine the image of each of the following curves on the unit sphere and then find the arclength of the curve.

17. q = t,f =  p 2 , t  in [ 0,2p] 18. q = 0,f = t, t  in [ 0,p] 19. q = t,f =  p 6 ,  t  in [ 0,2p] 20. q = t,f =  p 4 , t  in [ 0,2p]

Calculate and simplify r^¢¢·r_u and r^¢¢·r_v. Which of the following curves are geodesics on the given surface?

21. r( t) = r( t2,t2+2)  on 22. r( t) = r(t2,t2+2) on  r( u,v) = á u,v,u ñ r( u,v) = á u,v,u+v ñ 23. r( t) = r( t,4t+3)   on   24. r( t) = r( 1,t) r( u,v) = á vsin( u) ,vcos( u) ,v ñ r( u,v) = á vsin( u) ,v,vcos( u) ñ 25. r( t) = r( t,t2)   on   26. r( t) = r( t,t) r( u,v) = á sin( u) cos( v) ,cos( u) ,sin( u) sin(v) ñ r( u,v) = ásin( v) sin( u) ,cos( v) sin(u) ,cos( u) ñ 27. r( t) = r( t,0)  on 28. r( t) = r( p,t) r( u,v) = á u,sin( u) sin( v) ,sin( u) cos( v) ñ r( u,v) = á sin( u) cosh( v) ,sin( u) sinh( v) ,cos(u) ñ

Find the great circle that passes through each of the two points on a sphere centered at the origin. What is the shortest distance from P to Q on the sphere that contains them both?

29. P = ( 0,0,Ö2) ,     Q = (1,1,0) 30. P = ( 2,3,6) ,     Q = ( 0,7,0) 31. P = ( 2,2,1) ,     Q = (2,1,2) 32. P = ( 2,3,6) ,     Q = ( 6,2,3) 33. P = ( a,b,0) ,     Q = (0,c,d) 34. P = ( a,b,0) ,     Q = ( c,d,0)     a2+b2 = c2+d2 = 1 a2+b2 = c2+d2 = 1

35. New York City is located at 73^°56^¢38^¢¢ W and 40^°40^¢11^¢¢ N. Atlanta, Georgia is located at 84^°25^¢21^¢¢ W and 33^°45^¢46^¢¢ N. Find the parametrization of the great circle between New York and Atlanta. What is the shortest distance between these two cities, assuming the earth is a sphere with radius R = 3963 miles?.

36. Memphis, TN, is located at 90^°00^¢25^¢¢ W and 35^°6^¢20^¢¢ N. Seattle, Washington, is located at 122^°21^¢1^¢¢ W and 47^°37^¢18^¢¢ N. Find the parametrization of the great circle between Memphis and Seattle. What is the shortest distance between these two cities, assuming the earth is a sphere with radius R = 3963 miles?.

37. Kingsport, TN, is located at 82^°33^¢25^¢¢ W and 36^°31^¢46^¢¢ N.     Bristol, TN, is located at 82^°11^¢51^¢¢ W and 36^°34^¢04^¢¢ N.     Johnson City, TN, is located at 82^°22^¢07^¢¢ W and 36^°19^¢53^¢¢ N. If we assume the earth is a sphere with radius R = 3963 miles, then is Johnson City closer to Bristol or to Kingsport?

38. *Bristol-Johnson City-Kingsport is often referred to as the Tri-Cities. What is the approximate area of the triangle formed by the Tri-Cities (see exercise 37)?  

39. The latitude-longitude parameterization of a sphere of radius R is given by

r( q,j) = á Rcos( j) cos( q) ,Rcos( j) sin(q) ,Rsin( j) ñ

What is the fundamental form of the sphere with respect to this parameterization.

40. Mercator Projection maps are based on the parameterization of the sphere given by

r( q,m) = á R sech( m) cos( q) ,R sech( m) sin( q) ,Rtanh( f) ñ

What is the fundamental form of the Mercator parameterization?

41. The helicoid is the surface parametrized by

r( u,v) = á sinh( v) cos(u) ,sinh( v) sin( u) ,u ñ

What is its fundamental form?

42. A catenoid is the surface parametrized by

r( u,v) = á cosh( v) cos(u) ,cosh( v) sin( u) ,v ñ

Show that it has the same fundamental form as the helicoid (exercise 41). What does this mean?

Maple Graphics Export          Maple Graphics Export Ex. 41: A Helicoid          Ex. 42: A Catenoid

43.  Let f( u) ³ 0 for all u in [ a,b] , and consider the surface of revolution given by

r( u,v) = á u,f( u) sin(v) ,f( u) cos( v) ñ

where ( u,v) is in [ a,b] ×[ 0,2p] . Show that

g( t) = r( t,c) ,  t in [ a,b]

is a geodesic on the surface for all c in [ 0,2p] .

44.   Let f( u) ³ 0 for all u in [ a,b] , and consider the surface of revolution given by

r( u,v) = á u,f( u) sin(v) ,f( u) cos( v) ñ

where ( u,v) is in [ a,b] ×[ 0,2p] . Show that if f^¢( p) = 0, then

g( t) = r( p,t) ,  t in [ 0,2p]

is a geodesic on the surface.

45. Show that any horizontal plane intersects the helicoid

r( u,v) = á sinh( v) cos(u) ,sinh( v) sin( u) ,u ñ , u in [ 0,2p] , v in [0,¥)

along a straight line, thus implying that the geodesics of the helicoid include infinitely many straight lines.

46. Show that for any constant m, the plane

z = 2mx-m2

intersects the surface z = x²-y² in a pair of straight lines.

Exercises 47 - 50 deal with conformal parameterizations, which are parameterizations r( u,v) for which

ru·ru = rv·rv   and    ru·rv = 0

In particular, we relate conformality to the Mercator projection and Deviations of actual paths from those plotted on a map. The Application Worksheet for this section uses computer algebra systems to explore this topic in more detail.

47. Latitude-Longitude: On a rectangular latitude-longitude map, a line j = mq+b for m and b constant crosses the meridians (the vertical lines) at a fixed angle g, where cot( g) = m.

image

In the latitude-longitude parameterization of a sphere of radius R, which is given by

r( j,q) = R á cos( j) cos( q) ,cos( j) sin(q) ,sin( j) ñ

the curve r( q) = r( mq+b,q) is the image of the line j = mq+b and r_j( q) is in the direction vector of a meridian. The b angle between r^¢ and r_j satisfies

cos( b) =  r¢·rj | | r¢||  | | rj||

Show that b depends on q and how it implies that the actual path on the sphere deviates from the plotted linear path on the latitude-longitude map.

image

48. Mercator: Repeat exercise 47 with the parameterization

r( m,q) = R á sech( m) cos( q) ,sech( m) sin( q) ,tanh( m) ñ

and the vector r_m in place of r_j (i.e., m = mq+b). Does the angle b between r^¢ and r_m depend on q?

49. Write to Learn: Suppose that r( u,v) is a conformal parameterization of a surface S. For fixed g, the curve r( t) = r( sin(g) t+p,cos( g) t+q) is the image of the line through ( u,v) = ( p,q) that crosses the v-axis at an angle g.

conformpic

In a short essay, calculate the angle between r_v and r^¢( t) and use the result to explain what is meant by the statement Conformality implies that angles are preserved ..

50. Conformal Metrics: A conformal mapping r(u,v) has a conformal metric of

ds2 = g( u,v)  ( du2+dv2)

where g = r_u·r_u = r_v·r_v. Show that the latitude-longitude parameterization is not conformal but that the Mercator projection is conformal. How is this related to exercises 47 and 48?