Surfaces in 3D

A ΔtΔs-increment of a parametric surface

Following are s-varies curves for t = t_0fixed and a nearby fixed  t_0 + Δt, then t-varies curves for  s = s_0 fixed and a nearby  s_0 + Δs:

[Graphics:../HTMLFiles/Lect3_134.gif]

The portion of the surface between these curves is the set

{X[σ, τ] : s≤σ≤s + Δs , t≤τ≤t + Δt}

We want to compare this small patch of the surface with its "best local linear" approximation as the size of the patch gets smaller.

The linear case: geometric meaning of cross product

A linear parallelogram with sides  U  and  V  is shown below with the cross product  UV.

[Graphics:../HTMLFiles/Lect3_139.gif]

The cross product  UV  is perpendicular to the parallelogram with edges  U  and  V  and has area | UV |.  We can think of the vector UV as measuring both the area and orientation of the parallelogram by giving the direction perpendicular to it with length equal to the area.

The differential approximation of a smooth function is

X[s + δs, t + δt] - X[s, t] = ∂X[s, t]/∂s δs + ∂X[s, t]/∂t δt + ε (δs^2 + δt^2)^(1/2)

with  ε≈0, when  δs≈0  and  δt≈0.  This says that if we magnify a sufficiently small piece of a parametric surface by (δs^2 + δt^2)^(1/2) , then the linear combination  ∂X[s, t]/∂s δs + ∂X[s, t]/∂t δt  appears the same as the displacement  X[s + δs, t + δt] - X[s, t] on the surface.

Click here to Zoom in  

[Graphics:../HTMLFiles/Lect3_153.gif]

As long as the partial derivatives are linearly independent, this approximation says that on a scale of  (δs^2 + δt^2)^(1/2), each point in the image of  the s-t-rectangle {(σ, τ) : s≤σ≤s + δs, t≤τ≤t + δt},  under the nonlinear map,

    δsδt-SurfaceIncrement = { Ξ : Ξ = X[s + σ, t + τ], 0≤σ≤δs, 0≤τ≤δt}

differs from  the image under the linear map Ζ = M σ + N τ + C  by a term  ε (δs^2 + δt^2)^(1/2)  (or  ε  after magnification by  1/(δs^2 + δt^2)^(1/2)) where  M = ∂X[s, t]/∂s, N = ∂X[s, t]/∂t, C = X[s, t].

    δsδt-LinearIncrement = { Ξ : Ξ = M σ + N τ + C, 0≤σ≤δs, 0≤τ≤δt}

  We can re-scale the linear map and express the image in local coordinates by

δsδt-LinearIncrement = { Ζ + C : Ζ = μ · σ + ν · τ, 0≤σ≤1, 0≤τ≤1},

μ = (δs ∂X[s, t]/∂s), ν = (δt ∂X[s, t]/∂t)

δsδt-LinearIncrement = { Ζ + C : Ζ = (δs ∂X[s, t]/∂s) σ + (δt ∂X[s, t]/∂t) τ, 0≤σ≤1, 0≤τ≤1}

[Graphics:../HTMLFiles/Lect3_169.gif]

The scalar and vector area differentials da[s,t] & dA[s,t]

We summarize this approximation by saying that the vector area differential is

    dA[s, t] = ∂X[s, t]/∂s∂X[s, t]/∂t ds dt

and the scalar area differential is

    da[s, t] = | ∂X[s, t]/∂s∂X[s, t]/∂t | ds dt

A surface with a crease and points

The portion of the implicit x-z-curve, (1 - x)^3 = z^2 where  x≥0 can be parametrized by

    

x = 1 - s^2
z = s^3
, -1≤s≤1

We can rotate this curve about the z-axis by letting  r = 1 - s^2  and  z = s^3  in the cylindrical coordinate equations and using  t  as the other parameter.

FormBox[Cell[TextData[Cell[BoxData[FormBox[InterpretationBox[RowBox[{System`Convert`CommonDump ... s

This parametrization gives

∂X[s, t] -------------- =  ( ) ∂s ... -2 s Sin[t] 2 3 s, ∂X[s, t] -------------- =  ( 2 ) ∂t ... 2 (1 - s ) Cos[t] 0, & ∂X[s, t] ∂X[s, t] --------------  -------------- =  (> ... 2 2 s (s - 1)

The surface has "sharp points" at  s = ± 1  and a "crease" at  s = 0.  

Notice that ∂X[s, t] -------------- =  ( 0 ) ∂s 0 0 at s = 0FormBox[Cell[TextData[Cell[BoxData[FormBox[Cell[TextData[Cell[BoxData[FormBox[Cell[], TraditionalForm]]]]], TraditionalForm]]]]], TraditionalForm].  The collapsed  magnified view of the surface can be seen by double clicking on the next graph.

Click here to Zoom in  

[Graphics:../HTMLFiles/Lect3_190.gif]

Notice that ∂X[s, t] -------------- =  ( 0 ) ∂t 0 0 at s = ± 1.  The collapsed  magnified view of the surface can be seen by double clicking on the next graph.

Click here to Zoom in  

[Graphics:../HTMLFiles/Lect3_194.gif]

Created by Mathematica  (July 2, 2004)