Parametric Curves and Parametric Surfaces	
Parametric Curve	Parametric Surface
A parametric curve in R3 is given by	A parametric surface in ]R3 is given by
r(r) = *(r)i + i/(Oj + z(Ok	r{utv) = x{u,v)i + y{utv)) + z{u,v)\a
where a < i <b	where (u, v) lie in a region D of the uv plane.
There is one parameter, because a curve is a one-dimensional object	There are two parameters, because a surface is a two-dimensional object
There are three component functions, because the curve Lives in three-dimensional space.	There are three component functions, because the surface lives in three-dimensional space.
You Are Living on a Parametric Surface
Let u be your longitude (in radians, for this course) Let v be your latitude (in radians) Let R be the radius of the Earth Your position is
r{u,v) = R cos(v) cos(u)i + Rcos(v) $m(u}] + R sin(y)k
tt/2
0
tt/2
271
More Parameterized Surfaces: Planes
Problem: Find a parametric representation for the plane through (1,0,1) that contains the vectors (2,0,1) and (0,2,0)
Solution: Let tq — (1,0,1). Any point in the plane is given by
r(M) = (1,0,1) + s(2,0,l)+*(0,2,0)
y      Now you try it:
Find a parametric representation for the plane through the point (0, —1,5) that contains the vectors (2,1,4) and (—3,2,5).
More Parameterized Surfaces: Planes
Problem: Find a parametric representation for the plane through (1,0,1) that contains the vectors (2,0,1) and (0,2,0)
Solution: Let tq — (1,0,1). Any point in the plane is given by
r(M) = (1,0,1) + s(2,0,l)+*(0,2,0)
y      Now you try it:
Find a parametric representation for the plane through the point (0, —1,5) that contains the vectors (2,1,4) and (—3,2,5).
The Tangent Vectors ru and rv	
Suppose r{u,v) = x(u, v)i + y{u, v)j + z(w,z?)k, is a parameterized surface.	{ufv) e d
At a point Pq = r(uo, ^o)/ the vectors	
ru(u0,v0) = ^(tt0,z70)i + ^(ti0^o)j +	|(*o,*o)k
	
are both tangent to the surface.	
The Tangent Plane	
r«(«o^o) = ^(«0/w0)i+ ^(Mo,Po)j + ^	
M"0/Po) = 5-          i+ 3-(uo/»o)j + 5-dz;                 du d£>	
The tangent plane to a parameterized surface at Pq = r(wo/yo) through Po and perpendicular to ru(wQ,z7o) x tv(uq,vq).	is the plane passing
Find the equation of the tangent plane to the surface	
r{u, v) — u2i + 2u sin v\-\- u cos	
at u — lf v — 0.	
Tu{tlfV) TV(U,V)
r(l,0) r„(l,0) r,(l,0)
u2f2u sin v, u cos v) 1uf 2 sin v, cos v) 0, 2u cos v,—u sin u)
1,0,1) 2,0,1) 0,2,0)
The normal to the plane is
rH x r-L, = (-1,0,2)
so the equation of the plane is
(-l)(:t-l)+2(z-l) = 0 The tangent plane to the surface at (1,0,1) is parameterized by
(l+2s,2U+s)
r(u,v) = sin(y) cos(/Oi + sin(z>) sin(«)j + cos(r?)k
0<u<2ti, 0<v<n
r„ = - sm(v) sin(u)i + sin(u) tos(w)j =   cos(^) cos(u)i + co$(v) sin(«)j - sin(i;)k
Find the tangent plane to the sphere at (u, v) = {n/A, 7r/4)
r(n/4,7z/4) ru(n/4,7i/4)
tv{n/4tn/4)
-2i+2j 1.1.
n = r, xrn = --
1        1 "
s/2 V2l
1111
0=^(^2^^^-2)
f A