Parameterizations of Surfaces

Plane a(x-x0)+b(y-y0)+c(z-z0)=0

Clear[a,b,c]
a(x-x0)+b(y-y0)+c(z-z0)==0
Solve[%,z]

a=1;
b=2;
c=2;
x0=1;
y0=-3;
z0=0;
r[u_,v_]={u,v,((-a u+a x0-b v+b y0+c z0)/c)}
ParametricPlot3D[r[u,v],{u,-10,10},{v,-10,10},ColorFunction->"CherryTones",AxesLabel->{"x","y","z"}]

Sphere (x^2)+(y^2)+(z^2)=25

a=5;
r[ϕ_,θ_]={a Sin[ϕ]Cos[θ],a Sin[ϕ] Sin[θ],a Cos[ϕ]}
ParametricPlot3D[r[ϕ,θ],{θ,0,2Pi},{ϕ,0,Pi},ColorFunction->"CherryTones",AxesLabel->{"x","y","z"}]

Cone (z^2)=(( (x^2))/(a^2))+((y^2)/(b^2))

a=1;
b=2;
r[u_,θ_]={u a Cos[θ],u b Sin[θ],u}
ParametricPlot3D[r[u,θ],{θ,0,2Pi},{u,-10,10},ColorFunction->"CherryTones",AxesLabel->{"x","y","z"}]

Paraboloid z=(( (x^2))/(a^2))+((y^2)/(b^2))

a=1;
b=2;
r[u_,θ_]={(Sqrt[u]) a Cos[θ],(Sqrt[u]) b Sin[θ],u}
ParametricPlot3D[r[u,θ],{θ,0,2Pi},{u,-10,10},ColorFunction->"CherryTones",AxesLabel->{"x","y","z"}]

Cylinder (c^2)= (((x-a))^2)+(((y-b))^2)

Clear[a,b,c,r]
c=1;
a=-1;
b=2;
r[u_,θ_]={c Cos[θ]+a,c Sin[θ]+b,u}
ParametricPlot3D[r[u,θ],{θ,0,2Pi},{u,-1,1},ColorFunction->"CherryTones",AxesLabel->{"x","y","z"}]

Examples

Example 1: Tangent Plane

Find the tangent plane to the surface r(u,v)=<(u^2),u-(v^2),(v^2)> at the point (1,0,1).

Solution

The vector normal to the tangent plane is given by the cross product of the partial derivatives of the vector r. With this, we can construct the equation of the tangent plane!

r[u_,v_]={(u^2),u-(v^2),(v^2)}

Solve[r[u,v]=={1,0,1},{u,v}]

Cross[D[r[u,v],u],D[r[u,v],v]]/.u->1/.v->1
Simplify[%.({x,y,z}-r[1,1])==0]

The second solution gives the same plane:

Cross[D[r[u,v],u],D[r[u,v],v]]/.u->1/.v->-1
Simplify[%.({x,y,z}-r[1,-1])==0]

Here is a sketch:

plane=ContourPlot3D[1+x==2 (y+z),{x,-5,5},{y,-5,5},{z,-5,5}]

surface=ParametricPlot3D[r[u,v],{u,-5,5},{v,-10,10},ColorFunction->"CherryTones",AxesLabel->{"x","y","z"}]

Show[plane,surface]

Example 2: Surface Area from Shadow region

Find the surface area of the cone (z^2)= (x^2)+(y^2) above the region (x^2)+(y^2)≤1 and below z=1, excluding the top of the cone

Solution

This is a good place to start, since we know the surface area is (Sqrt[2])π.

Here, we need to parameterize the ellipse, will will carry on up and parameterize the surface for us! The ellipse is the important quantity to begin with, since we were not given a parameterization of anything.

x=u cos v
y=u sin v
0≤v<2π
0≤u≤1

The above choice was made to ensure (x^2)+(y^2)≤1 is satisfied.

Now that we have the parameterization of the region, we can parameterize the surface:

r[u_,v_]={u Cos[v],u Sin[v],(Sqrt[(((u Cos[v]))^2)+(((u Sin[v]))^2)])}

Let's get a picture of what we are doing:

surface=ParametricPlot3D[r[u,v],{v,0,2Pi},{u,0,1},ColorFunction->"CherryTones",AxesLabel->{"x","y","z"},ImageSize->{500,Automatic}]

We can easily gert a plot of the shadow region if we desire, just by setting z=0:

region[u_,v_]={u Cos[v],u Sin[v],0}
shadow=ParametricPlot3D[region[u,v],{v,0,2Pi},{u,0,1},ColorFunction->"CherryTones",AxesLabel->{"x","y","z"},ImageSize->{500,Automatic}]

Show[surface,shadow]

Now, we can blast head and get the surface area of the surface with a simple calculation:

Cross[D[r[u,v],u],D[r[u,v],v]]
integrand=Simplify[(Sqrt[Cross[D[r[u,v],u],D[r[u,v],v]].Cross[D[r[u,v],u],D[r[u,v],v]]])]
Integrate[Integrate[integrand,{u,0,1}],{v,0,2Pi}]

Example 3: Surface Area from Shadow region

Find the surface area of the paraboloid z= 2(x^2)+3(y^2) above the region (x^2)+4(y^2)≤4.

Solution

Here, we need to parameterize the ellipse, will will carry on up and parameterize the surface for us! The ellipse is the important quantity to begin with, since we were not given a parameterization of anything.

x=u cos v
y=(u/2) sin v
0≤v<2π
0≤u≤2

The above choice was made to ensure (x^2)+4(y^2)≤ 4 is satisfied.

Now that we have the parameterization of the region, we can parameterize the surface:

r[u_,v_]={u Cos[v],u Sin[v]/2,2(((u Cos[v]))^2)+3(((u Sin[v]/2))^2)}

Let's get a picture of what we are doing:

surface=ParametricPlot3D[r[u,v],{v,0,2Pi},{u,0,2},ColorFunction->"CherryTones",AxesLabel->{"x","y","z"},ImageSize->{500,Automatic}]

We can easily gert a plot of the shadow region if we desire, just by setting z=0:

region[u_,v_]={u Cos[v],u Sin[v]/2,(0^2)}
shadow=ParametricPlot3D[region[u,v],{v,0,2Pi},{u,0,2},ColorFunction->"CherryTones",AxesLabel->{"x","y","z"},ImageSize->{500,Automatic}]

Show[surface,shadow]

Now, we can blast head and get the surface area of the surface with a simple calculation (although the integral is not simple, and so we do it numerically):

Cross[D[r[u,v],u],D[r[u,v],v]]
integrand=(Sqrt[Cross[D[r[u,v],u],D[r[u,v],v]].Cross[D[r[u,v],u],D[r[u,v],v]]])
NIntegrate[Integrate[integrand,{u,0,2}],{v,0,2Pi}]