Section 16.6 Parametric Surfaces and Their Areas

Xin Li

MAC2313 Summer 2020

Xin Li (FSU) Section 16.6 MAC2313 Summer 2020 1/20 parametric curve and surface

Recall: When we parameterized a curve, we took value of t from some interval [a, b]andplugtheminto →~r(t)=x(t)~i + y(t)~j + z(t)~k It and the resulting set of vectors will be the position vectors for the points on the curve. To parameterize a surface, we take points (u, v) on region D in the uv-plane and plug them into ¥0000~r(u, v)=x(u, v)~i + y(u, v)~j + z(u, v)~k ¥H#E#### The resulting set of vectors will be the position vectors for the points on the surface S that we are trying to parameterize. This is often called the parameteric representation of the parametric surface S

Xin Li (FSU) Section 16.6 MAC2313 Summer 2020 2/20 parametric surface

The parameteric equation for a surface is x = x(u, v), y = y(u, v), z =(u, v) - - -

" ' -2 ← T . ) luv) O

Xin Li (FSU) Section 16.6 MAC2313 Summer 2020 3/20 Example 1

Identify and sketch the surface with vector equation

~r(u, v) = 2 cos u~i + v~j +2sinu~k TT TT I ¥ {Fz÷g xz=4as2ut4sin2u= I Evan ..

Xin Li (FSU) Section 16.6 MAC2313 Summer 2020 4/20 Parametric surface

If a parametric surface S is given by a vector function ~r(u, v), then - there are two useful families of curves that lie on S, one family with u constant and the other with v constant. These families correspond to vertical and horizontal lines in the uv-plane.

If we keep u constant by putting u = u0,then~r(u0, v) becomes a vector function of the single parameter v and deﬁnes a curve C1 lying on S. .ee#EeffFeE*

Xin Li (FSU) Section 16.6 MAC2313 Summer 2020 5/20 Grid Curves

Similarly, if we keep v constant by putting v = v0,then~r(u, v0) becomes a vector function of the single parameter u and deﬁnes a curve C2 lying on S

We call these ~r(u0, v)and~Ir(u, v0eeEEo) Grid Curves In fact, when a computer graphs a parametric surface, it usually depicts the surface by plotting these grid curves

Xin Li (FSU) Section 16.6 MAC2313 Summer 2020 6/20 Example 2

Give parametric representations for the- sphere x2 + y 2 + z2 = 30 Solution: Equation of sphere is x2 + y 2 + z2 = ⇢2,so⇢ = p30

- Recall the parametric equation of the spherical

x = ⇢ sin cos ✓

( ) → (x. y Z ) y = ⇢ sin sin ✓ 8,0 , ✓z = ⇢ cos We have the corresponding vector equation

~r(✓,)=p30 sin cos ✓~i + p30 sin sin ✓~j + p30 cos ~k - - - where 0 ✓ 2⇡, 0 ⇡ mm  - 

Xin Li (FSU) Section 16.6 MAC2313 Summer 2020 7/20 Example 2

D~r(✓,)=p30 sin cos ✓~i + p30 sin sin ✓~j + p30 cos ~k The grid curves for a sphere are curves of constant latitude or constant longitude.

⑧ TOEWS DECK of # Ea , ,

¥¥ . .⇐

Xin Li (FSU) Section 16.6 MAC2313 Summer 2020 8/20 Tangent Plane

Recall the equation of a plane

a(x x )+b(y y )+c(z z )=0 0 0 0 # PT . = where the point pot o (x0, y0, z0) O tax. is on the plane and the normal vector of the plane is 0-600n~ =< a, b, c >

Xin Li (FSU) Section 16.6 MAC2313 Summer 2020 9/20 Tangent Plane

We now ﬁnd the tangent plane to the parametric surface S at a point P0 with position vector ~r(u0, v0). The surface S is given by a vector function

~r(u, v)=x(u, v)~i + y(u, v)~j + z(u, v)~k - Deﬁnition @x @y @z ~r = (u , v )~i + (u , v )~j + (u , v )~k u @u 0 0 @u 0 0 @u 0 0 @x @y @z ~r = (u , v )~i + (u , v )~j + (u , v )~k v @v 0 0 @v 0 0 @v 0 0 n=r# O

Xin Li (FSU) Section 16.6 MAC2313 Summer 2020 10 / 20 Example 3

Find the tangent plane to the surface with parameteric equations x = u2, y = v 2, z = u +2v Tat the point (1, 1, 3) - mm Solution: ' Itv ' ZHI Tcu , v ) = u j t cut

= zu it O = C ZU o I > Fu j t I , ,

= 2V = 2. C O 27 Fu OF t zvjt I , ,

= n' Fux I - incult

- zig ! (zu )C2)- Co =/ 1=4%4( (1) )) T

- I t ( @a) ( 24 G) co) )

= - 2 'T - V 4 Uj t 4kV

Xin Li (FSU) Section 16.6 MAC2313 Summer 2020 11 / 20 Example 3

Find the tangent plane to the surface with parameteric equations x = u2, y = v 2, z = u +2v at the point (1, 1, 3)

I = - - 4 2 Vj Uj t Kurt

at point ) u2=I → a- It qq.az so V' = - =/ → V -11 V - I y z y, ' " Ut2V= -Y 3 utzv =3 ur fr utzu

ICI = - 2T , 1) -14kt =L -2 -4T , -4,43

Tangent plane

-2 - - (x l ) 4cg - l ) -14 (2--3)=0

Xin Li (FSU) Section 16.6 MAC2313 Summer 2020 12 / 20 Surface Area

Now we deﬁne the surface area of a general parametric surface. Let’s

choose (ui⇤, vj⇤) to be the lower left color of small region Rij enter .

O sij f

Xin Li (FSU) Section 16.6 MAC2313 Summer 2020 13 / 20 Surface Area

Let ~ru⇤ = ~ru(ui⇤, vj⇤)and~rv⇤ = ~rv (ui⇤, vj⇤) be the tangent vectors at point Pij . The vector of the two edges of the region can be approximated by the vectors u~ru⇤ and v~rv⇤

÷t÷%T .

Xin Li (FSU) Section 16.6 MAC2313 Summer 2020 14 / 20 Surface Area

So we approximate Sij by the parallelogram determined by the vectors u~ru⇤ and v~rv⇤. ← ←

The area of this parallelogram is the magnitude of the cross product.

(u~r ⇤) (v~r ⇤) = ~r ⇤ ~r ⇤ uv | u ⇥ v | | u ⇥ v | So the Area of S can be approximated by m n ~r ⇤ ~r ⇤ uv I SEE | u ⇥ v | i=1 j=1⇒ X X .÷sii¥÷o , Xin Li (FSU) Section 16.6 MAC2313 Summer 2020 15 / 20 Surface Area Deﬁnition

Deﬁnition If a smooth parametric surface S is given by the equation

~r(u, v)=x(u, v)~i + y(u, v)~j + z(u, v)~k, (u, v) D 2 and S is covered just once as (u, v) ranges throughout the parameter domain D, then the surface area of S is

A(S)= ~r ~r dA | u ⇥ v | -ao•ZZD ⇒# @x ~ @y ~ @z ~ @x ~ @y ~ @z ~ where ~ru = @u i + @u j + @u k and ~rv = @v i + @v j + @v k ① D define region . ② Fax RT I . Fa . . truant compute .

Xin Li (FSU) Section 16.6 MAC2313 Summer 2020 16 / 20 Example 4

Find the surface area of a sphere of radius a. Solution:

O = as ¥0 . ) .no/.asoTtaano/smoj

I 'E÷÷÷:←i=H¥#1- a asf I D= ( { 0,0 ) off E.ae # ) , this . F-

= a OT cosy as t a Tp - asf snot a Singh

= - A To Sino since i a as t sing OTT OIG

Xin Li (FSU) Section 16.6 MAC2313 Summer 2020 17 / 20 Example 4

Find the surface area of a sphere of radius a. * xri=l÷ - ÷:X

= a' as sing so it a's.io/smoj-a2smo/oso/k g-

a' ' hipxro #( snip 's f- a) t (a .no/smoj4-@smoeos8T

" ' = 't -a Saito as O Sm ' ta sin o 't - 44 ta - s.io/cos2oT = " a 't #ssh a 401 + cos -sink 201 so = FIE Xin Li (FSU) Section 16.6 MAC2313 Summer 2020 18 / 20 Example 4

Find the surface area of a sphere of radius a.

try x Fo I = fatso = AZ sin of

azotfsm.co/3o for #EK .

- ALS) - try xtro IDA Sf,

" = fo [ a 's.no/do1dI - " = a' (f. do ) ( S? sing doll

' " = " a (off ) fast to ) ' ' = = ( 22 ( - a ) 2) a ( 22 ) ( asset cos o ) =4T⑦

Xin Li (FSU) Section 16.6 MAC2313 Summer 2020 19 / 20 Example 4

Find the surface area of a sphere of radius a.

Xin Li (FSU) Section 16.6 MAC2313 Summer 2020 20 / 20