Local Theory of Surfaces

Introduction

In this lab we introduce the local theory of surfaces in three-dimensional space. This subject is the natural extension of the study of the local theory of plane curves and space curves. As we did for plane curves and space curves, we will introduce coordinate representation and vector notation for surfaces. Next, we will define a particular family of plane curves on a surface, the parametric curves, and use these to define the metric coefficients, which help us to characterize classes of surfaces. The metric coefficients provide us with a tool for computing not only lengths of curves but also areas of regions and angles between curves on a surface.

Surfaces and Their Representation

By a surface in three-space, we mean a mapping of a domain in the plane into ordinary three-dimensional Euclidean space. In most of our examples, the domain of a surface will be a rectangle, and can be considered with or without its boundary edges.

We will consider the points of three-dimensional space as vectors emanating from the origin. A surface is usually represented as a parametric vector function X of two parameters (u,v) . For a particular value (u₀,v₀) in the domain of the function, the vector corresponding to the point (u,v) in the domain is denoted by X(u,v) .

In terms of the standard basis E₁ ,E₂ ,E₃ of Euclidean three-space, we have

X(u,v) =x(u,v)E₁ +y(u,v)E₂ +z(u,v)E₃

The functions x(u,v) , y(u,v) and z(u,v) are called the coordinate functions of the surface.

Demonstration 1: Inputting Surfaces

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The default function is a function graph given by x(u,v)=u , y(u,v)=v and z(u,v)=u²-v² , with domain given by the rectangle where u and v are between -1 and 1. The demonstration shows the domain in one window and the surface in three-dimensional space in the second window. Selecting a point in the domain highlights a small rectangle, and simultaneously shows the image of that rectangle on the surface.

As in the case of curves, we can enter in the equations of the coordinate functions of a surface, and the values defining the rectangular domain of these functions, and the program will display the surface in three-dimensional space.

If we fix a value of v=v₀ , then as u changes, we have a curve X(u,v₀) on the surface called a u-parameter curve . Similarly, if we fix a value of u=u₀ , then as v changes, we have a curve X(u₀,v) on the surface called a v-parameter curve .

Demonstration 2: Parameter Curves Demo

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To display the parameter curves, there are two checkboxes which allow you to toggle the display of either one. Below each checkbox is a tapedeck that controls the position of the u- or v-parameter curve by scaling between the minimum and the maximum of the domain in u or the domain in v . In the bottom right corner of the control panel, the actual coordinates of the point are displayed together.

As we move a point around in the domain, the demonstration shows how the u- and v-parameter curves through that point move around on the surface.

Velocity and Arc Length of Parameter Curves

The definition of the partial derivative of a vector function is very similar to the definition of a partial derivative of a function of two variables. The u-partial derivative of the position vector is defined by

X_u(u₀,v₀)=lim[h->0][X(u₀+h,v₀) -X(u₀,v₀)]/h

if this limit exists. The v-partial derivative X_v(u₀,v₀) is defined analogously. If these two vectors exist and are linearly independent, they define a plane called the tangent plane of the surface at X(u₀,v₀). We avoid pathological situations by assuming that our functions have an appropriate number of derivatives.

If the two partial derivative vectors are linearly independent at each point of a domain, then the parametrization is said to be regular; if the two partial derivative vectors are linearly dependent at a point of a domain, then the parametrization is said to be singular at the point.

In certain cases a singularity occurs as a result of the parametrization, while in others, it reflects a geometric property of the surface itself. We have already seen this phenomenon for curves: the velocity vector of the plane curve X(t) =(t³,t³) is X^'( t)=(3t²,3t²) , so the curve has a singularity when t=0 . However we may reparametrize the curve as X(s) =(s,s) , and with this parametrization, the velocity vector is non-zero at each point.

Consider the parametrization of the plane given by

X(u,v) =(vcos(u),vsin( u),0)

defined over the domain with u between 0 and 2p , and v non-negative. Then the partial derivatives are

X_u(u ,v)=(-vsin(u) ,vcos(u),0)

and

X_v(u ,v)=(cos(u) ,sin(u),0)

Since

X_u(u ,0)=(0,0,0)

for any u , the two partial derivative vectors are automatically dependent at (u,0) , so the origin is a singularity for this parametrization. On the other hand, it is easy to give a parametrization of the plane which has no singularities, for example

X(u,v) =(u,v,0)

On the other hand,

X(u,v) =(vcos(u),vsin( u),v)

over the same domain describes a cone with a singularity that can not be removed by reparametrization.

Demonstration 3: Partial Derivatives Demo

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A point widget marked by an x in light blue enables you to choose a point in the domain. As we move the point around in the domain, we see the u- and v- partial derivative vectors moving along the surface depending on whethter or not u-curve or v-curve are on. If both of these checkboxes are turned on, we display the tangent plane by drawing the parallelogram determined by the partial derivative vectors at the point. To facilitate the viewing of the tangent plane the surface will not solidify, only the plane will solidify if you press the space bar.

In this demonstration, when we choose a point in the domain of a surface, in addition to the two parameter curves, the partial derivative vectors are also displayed.

With this demo, quickly look at some smooth surfaces and observe how the tangent plane behaves. Then, look a surfaces which you know to have singularities. For example, try the Cone, which is entered as [u,v]&maps;[v cos(u),v sin(u),v]. What happens at the poles of a sphere? Why? Analyze also the tangential surface of a circular helix. This latter has what is called a cuspidal edge . One parametrization is [u,v]&maps;[cos(u)-v*sin(u),sin(u)+v*cos(u),0.5*u+v].

Arc Length of Parameter Curves

The velocity vector of a u-parameter curve is the u-partial derivative vector at that point, so we can integrate the length of that vector to calculate the length of the u-parameter curve with v=v_o. We obtain

s(b)-s(a) =[Integral(a,b)][sqrt]X_u(u,v₀) ·X_u( u,v₀)du

The expression

X_u(u ,v)·X_u(u,v)

occurs so frequently in the differential geometry of surfaces that it has a symbol g₁₁(u,v) . The length of the u-parameter curve is then given by

s(b)-s(a) =[integral(a,b)][sqrt]g₁₁(u,v₀) du

Similarly we define the symbol

g₂₂(u,v)= X_v(u,v) ·X_v( u,v)

and this can be used to compute the length of v-parameter curves. For example, if u=u₀ , then the length of this v- parameter curve is

s(d)-s(c) =[integral(c,d)][sqrt]g₂₂(u₀,v) dv

Find some examples of surfaces for which at certain points the length functions of the parameter curves have an inflection point. When will this happen?

Velocity and Arc Length of General Curves on Surfaces

Given a curve (u(t),v( t)) in the domain of a surface, we have a curve

X(t) =X(u(t ),v(t))

lying on the surface. By the chain rule for vector functions, we have

X^'( t)=X_u(u(t),v( t))u^'(t)
+X_v( u(t),v(t))v^'(t)

(This can be proved directly, or coordinatewise.)

Demonstration 4: Illustration of the Chain Rule

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This demo allows you to input a surface in the X(u,v) type-in, and a curve in the Curve in UV domain type-in. The curve's domain is defined in the type-in called Domt. Finally, the tapedeck will allow you to move along the curve in the uv-plane and the corresponding curve on the surface. (The tapedeck only goes between 0 and 1 but is scaled to go effectively between the minimum and the maximum of the domain of the curve.) Then X_u(u( t),v(t)) u^'(t) and X_v(u( t),v(t)) v^'(t) are displayed on the curve in The Surfacewindow.

In this demonstration, we show a point moving along a parametrized curve (u(t),v( t)) in the domain, with velocity vector (u^'(t) ,v^'(t)) . We show how the velocity vector is the sum of vectors (u^'(t),0) and (0,v^'(t)) parallel to the coordinate vectors in the plane. We then show the decomposition of the velocity vector X^'( t) as a sum of two basis vectors in the tangent plane, X_u(u( t),v(t)) u^'(t) and X_v(u( t),v(t)) v^'(t) .

To compute the length of the curve X(t) , we must integrate the length of the vector X^'( t) over the domain of the curve. The dot product of this vector with itself can be expressed in terms of the dot products of the partial derivative vectors, which we define in terms of the metric coefficients g_ij(u,v) . We obtain

|X^'(t)|²=g₁₁(u(t) ,v(t))u^'(t)²
+2g₁₂(u(t) ,v(t))u^'(t)v^'(t)
+g₂₂(u(t) ,v(t))v^'(t)²

where g12(u,v) =Xu(u,v)·Xv(u,v) and g21(u,v) =Xv(u,v)·Xu(u,v) , and so g12(u,v) =g21(u,v) .

Here we calculate of the lengths of curves on a surface for the simple case of lines in the u,v domain. Metric of lines in the uv-Domain

(u₀,v₀) will be determined by the type-in [u0,v0]. c now determines the angle that the line makes with the u-axis in the parameter domain.

In this demonstration, we choose a point (u₀,v₀) in the domain and exhibit the lengths of the curves

(u(t),v( t))=(u₀+tcos(c),v₀ +tsin(c))

where t runs from a to b, chosen so that the curve lies in the domain. For this special case,

|X^'(t)|²=g₁₁(u(t) ,v(t))cos²(c)
+2g₁₂(t(t) ,v(t))cos(c)sin(c)
+g₂₂(u(t) ,v(t))sin²(c)

Linear Independence in Terms of Metric Coefficients

The two partial derivative vectors are linearly dependent if and only if their cross product is zero. Thus the condition for regularity is that X_u(u,v) ·X_v( u,v) is not zero.

Recall that for any two vectors A and B , we have

|A· B|²=|A|²|B|²sin²(q)

where q is the angle between the vectors. Since sin²(q)=1- cos²(q) , we may write

|A· B|²=|A|²|B|²-|A|²|B|²cos²(q)
=|A|²|B|²-|A· B|²

In the case of the partial derivative vectors of a vector function, this gives

|X_u( u,v) x X_v(u,v)|=g₁₁(u,v)g₂₂( u,v)-g₁₂(u,v)²

Thus, the expression to the right in particular is never negative, and it is zero at some point if and only if the parametrization is singular at that point.

Our notation and this last formula suggest a matrix representation of what are formally called the metric coefficients :

(g_ij)=(g₁₁g₁₂g₂₁g₂₂)

We often leave out the parameters and recognize that the metric coefficients are always functions of the parameters of the surface. The matrix of metric coefficients is also called "the first fundamental form".

Since the determinant of the metric coefficients appears in many different places it has its own name:

g(u,v)=det(g₁₁(u,v)g₁₂(u,v)g₂₁(u,v)g₂₂(u,v))

Areas of Regions

To find the area of a domain on a surface, we compute the double integral of the areas of the parallelograms with sides given by the partial derivative vectors. Thus the area of a region is the integral of the length of the cross product, i.e. the integral of the square root of g₁₁(u,v)g₂₂(u,v)-g₁₂( u,v)²=g(u,v) . This function represents how much the domain is being stretched at a given point.

Demonstration 6: Area Function Graph Demo

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In this demo, The Area Element Graph shows that area element graphed over the domain. This means that we draw the function graph for g(u,v) . The total area is shown in the Area of Surfaces control panel window.

You may notice that the value of the area is slightly overestimated for compact regions, because fnord counts bits of the edges of the domain twice where they come together on the surface. You can minimize this problem by increasing the number of divisions in the domain.

Lengths And Areas Given Intrinsically

In our study of surfaces we categorize various properties as either intrinsic or extrinsic. A property is said to be intrinsic if it can be expressed only in terms of the metric coefficients. If this is not the case, then it is said to be extrinsic. If an object is not intrinsic, one needs a parametrization of the surface to study it. Many interesting problems in the theory of surfaces arise in determining whether or not a given property is intrinsic.

Given the metric coefficients g_ij(u,v) , it is possible to find lengths of curves on the domain even if we do not know the definition of a surface in three-space that determines the coefficients. For example, suppose we are given g₁₁(u,v)= 1/v² , g₁₂(u,v)=0 , and g₂₂(u,v)= 1/v² , where u can be any real number and v is strictly positive. Then g₁₁(u,v)g₂₂(u,v)-g₁₂( u,v)²=1/v⁴ , and since this is never zero, it is possible for these to be the metric coefficients of a regular surface. This g that we have obtained can be used as a metric.

Hyberbolic Metric in the Upper Half-Plane

The total length is printed in the Length of Curves on Surfaces control panel window. The arclength function is drawn in the window that has that name.

This demo looks at curves in the upper half plane with the metric given in the previous paragraph, i.e g₁₁=g₂₂=1/v² and g₁₂=g₂₁=0.

Find the lengths of a number of curves joining the points (-1,1) and (1,1) in the domain. One is the straight line (u(t),v( t))=(t,1 ) as t goes from -1 to 1 . Another is an arc of the circle centered at the origin with radius 2 . Show that, using this metric, the straight line segment is a longer curve than the arc of circle. What will the "shortest" curve be joining these two points?

Since the integrand for the area of a region of a surface is expressed in terms of the metric coefficients of that surface, the same is true of our formula to compute the area of rectangles on the surface. The metric coefficients completely determine the effects a parametrization of the surface would have on the parameter domain.

Metric Coefficients to Calculate Area in the Hyperbolic Upper Half Plane

To select the rectangle in the domain we just change the domain boundaries and observe the changes. The total area of the domain chosen using our hyperbolic metric is displayed in the control panel under Area of Surface.

This demo shows the domain and area element graph for a region in the upper half plane, using the hyperbolic metric.

Culculate the area of rectangles in the hyperbolic upper half-plane using the previous expression for g.

Angles between Parameter Curves

To find the angle between the u- and v-parameter curves at a point u₀,v₀ , we use the formula

cos(q)= X_u(u,v) ·X_v( u,v)/[sqrt]g₁₁( u,v)[sqrt]g₂₂( u,v)
=g12( u,v)/[sqrt]g₁₁( u,v)/[sqrt]g₂₂( u,v)

It is important to note right here that if g₁₂(u,v) ≡0 then the parameter curves form an orthogonal net: the intersections of the u- and v-curves are all right angles.

The Angle Between Parameter Lines

To choose a point [u,v], you may click on the middle mouse button on the desired point in TheDomain window. The value of the point will appear in the Parameter Curves Demo control panel. The point is marked in the domain by a small red circle, and its image under the mapping X(u,v) is also marked by a red circle.

Then, the parameter lines through the chosen point are drawn in green on the surface in order to view the angle in question. The exact value of the angle between the parameter curves is printed in the control panel also. To get a global view of the angle variations on the surface, we have plotted the graph of q(u,v) where the point is also marked by a red circle.

The demonstration shows the domain, the surface in question and the graph of q(u,v) over the domain of the surface.