SolitaryRoad.com

Website owner:  James Miller


[ Home ] [ Up ] [ Info ] [ Mail ]

SURFACES, SURFACE REPRESENTATION, SIMPLE SURFACE ELEMENTS, CURVILINEAR COORDINATES, SURFACE NORMALS, SURFACE CURVES, FIRST AND SECOND FUNDAMENTAL QUADRATIC FORMS, OSCULATING PARABOLOID, SURFACE AREA



Three common methods for the analytical representation of a surface. Three methods are commonly used to represent surfaces. They are


1] Surface representation 1. An equation of form


            z = f(x, y)


where f is a single-valued continuous function defined on a region R of the xy-plane.



2] Surface representation 2. An equation of the form


            f(x, y, z) = 0



3] Surface representation 3. Parametric equations of the form


            x = f(u, v)

1)        y = g(u, v)

            z = h(u, v)


where f, g, h are continuous functions defined on a connected region R of the uv-plane.


Equations 1) of Surface Representation 3] do not necessarily map in a one-to-one fashion. That is, more than one point (u, v) may map into the same point (x, y, z). Because of this fact the “surface” produced by the system may sometimes be something quite different from the usual conception of a surface. To avoid this, a stipulation that it is one-to-one needs to be appended. We do this in the concept of a simple surface element.


Simple surface element. Let R be a simply connected region (i.e. a bounded region such as a rectangular or circular region without “holes”) of the uv-plane. Let equations


            x = x(u, v)

            y = y(u, v)

            z = z(u, v)


define a bicontinuous one-to-one mapping of R onto a point set S in xyz-space. Let ole.gif


             ole1.gif


be the position vector to point P on the surface and let the condition

 

             ole2.gif


hold for all points in R. Then S is called a simple surface element.


For brevity we will refer to a simple surface element using the notation S: ole3.gif .


A simple surface element may be viewed as any configuration which may be obtained from a rectangular plane region by continuous deformation (bending, twisting, stretching, shrinking) without tearing and without bringing any points together which were originally distinct. A simple surface element can be described as a patch of surface in space enclosed by a boundary — where the boundary points are images of the boundary points of region R under the mapping. Thus this excludes surfaces of infinite extent such as an infinite cylinder and closed surfaces such as a sphere or a torus. It includes such surfaces as a plane circular region or a hemispherical surface.


Any surface, however, be it a torus, sphere, cylinder or whatever, may be viewed as consisting of a collection of simple surface elements. That is, any surface can be divided up into regions viewed as simple surface elements.



Def. Class of a surface. Let S be a simple surface element defined by the one-to-one mapping


            x = x(u, v)

1)        y = y(u, v)

            z = z(u, v)


of a region R of the uv-plane into xyz-space. The surface element is said to be of class C m if the defining functions 1) have continuous derivatives of the m-th order.




Curvilinear coordinates of a point on a surface. The parametric equations


            x = x(u, v)

2)        y = y(u, v)

            z = z(u, v)

 

ole4.gif

assign a point (x, y, z) to each number pair (u, v). The number pair (u, v) can be considered a set of coordinates for a point P(x, y, z). The numbers (u, v) are called the curvilinear coordinates of point P. If we regard v as fixed in system 2) i.e. v = c, a constant, then system 2) becomes a system in a single variable u describing a space curve where u is the varying parameter. For each different value of v there is a separate space curve. These curves are called u-curves. Similarly one can let u = k, a constant, and obtain a v-curve where v is the varying parameter. One thus obtains families of u-curves and v-curves. The u-curves and v-curves are called coordinate curves. These coordinate curves form a curvilinear net on the surface similar to the coordinate net on a plane. The locus of all these coordinate curves form the simple surface element S. See Fig. 1. In the figure


             ole5.gif


 

Property of being smooth. A surface is called smooth at a point P if it has a tangent plane at each point in the neighborhood of P, and if the direction of the normal to this plane varies continuously from point to point.



Theorem 1. Let S: ole6.gif be a simple surface element. Then at any point P on S

             ole7.gif

             ole8.gif


where ole9.gif are the following Jacobians:


             ole10.gif


             ole11.gif


             ole12.gif



Proof.





Corollary.


             ole13.gif




Directions of surface normals. If a surface is smooth at a point it will have a normal at that point.. The directions of the normals are as follows:


Surface representation 1. z = f(x, y). The direction of the normal at a point P is given by the set of direction numbers


                         ole14.gif


evaluated at point P. The surface is smooth at P if f has continuous first partial derivatives there.



Surface representation 2. f(x, y, z) = 0. The direction of the normal at a point P is given by the set of direction numbers


                         ole15.gif


evaluated at point P. The surface is smooth at P if f has continuous first partial derivatives there.



Surface representation 3. Parametric equations of the form


          x = x(u, v)

          y = y(u, v)

          z = z(u, v)



The direction of the normal at a point P is given by the vector


             ole16.gif


where ole17.gif is the position vector of point P i.e.


             ole18.gif



The direction of the normal at a point P is also given by the set of direction numbers

 

             ole19.gif


where ole20.gif are the Jacobians


             ole21.gif  


             ole22.gif


             ole23.gif



evaluated at point P. The surface is smooth at P if the three Jacobians do not vanish simultaneously there or, equivalently, if

 

             ole24.gif  



Surface curves. We pose a question. How does one represent a curve on a surface? Let S be a simple surface element defined by the one-to-one mapping


            x = x(u, v)

3)        y = y(u, v)

            z = z(u, v)


of a region R of the uv-plane into xyz-space. Let ole25.gif


             ole26.gif


be the position vector to point P on the surface. How does one represent a curve on surface S? We do it as follows. We define a curve in the uv-plane by the parametric equations


4)        u = u(t)

            v = v(t) .


The mapping 3) then maps this curve onto the surface S. Substituting 4) into 3) gives us the equations of the curve as inscribed on surface S i.e.



             ole27.gif


or, in vector form,


             ole28.gif



Surface curve tangents. Let S: ole29.gif be a simple surface element and let P be a point on S. Let ole30.gif at point P, where we are using subscripts to denote partial differentiation i.e.


             ole31.gif


Let C be any curve on surface S passing through point P. Then the tangent vector ole32.gif to curve C at point P is given by


             ole33.gif


or, equivalently,


             ole34.gif


which shows that ole35.gif is a linear combination of the vectors ole36.gif and ole37.gif — which means that it lies in the same plane as ole38.gif and ole39.gif . Now the vectors ole40.gif and ole41.gif are tangent to the u- and v-coordinate curves at point P and define the surface tangent plane at point P. Thus we see that all surface curves passing through point P have, at point P, tangents which lie in the surface tangent plane there.


 

First and second fundamental quadratic forms of a surface. A space curve is completely determined by two local invariant quantities, curvature and torsion, as a function of arc length. In the same way a surface is completely determined by certain local invariant quantities called the first and second fundamental quadratic forms.



First fundamental quadratic form. Let S be a simple surface element defined by the one-to-one mapping


            x = x(u, v)

            y = y(u, v)

            z = z(u, v)


of a region R of the uv-plane into xyz-space. Let ole42.gif


             ole43.gif


be the position vector to point P on the surface. The First Fundamental Quadratic Form is


ole44.gif



where:


             ole45.gif


             ole46.gif


             ole47.gif



This form expresses the relationship between the differential of arc length ds for any curve on the surface S and the variables u, v, du and dv in the uv-plane. The quantities E, F and G are called the Fundamental Coefficients of the First Order of the surface. The form is positive-definite, meaning it is positive for all values of the variables.


Derivation



Theorem 2. At any point P on the surface S: ole48.gif


             ole49.gif



Proof.


Corollary. At any point P on the surface S: ole50.gif the unit surface normal is given by


             ole51.gif



Second fundamental quadratic form. Let S be a simple surface element defined by the one-to-one mapping


            x = x(u, v)

            y = y(u, v)

            z = z(u, v)


of a region R of the uv-plane into xyz-space. Let ole52.gif


             ole53.gif


be the position vector to point P on the surface and let ole54.gif be the unit normal to the surface at point P. The Second Fundamental Quadratic Form is

 


ole55.gif



where


             ole56.gif  

 

             ole57.gif                                                                          

             ole58.gif



The quantities L, M and N are called the Fundamental Coefficients of the Second Order.


ole59.gif

                                                                                    

Physical interpretation of f. Let P be the point corresponding to (u, v) and Q be the point corresponding to(u + du, v + dv) on surface S. See Fig. 2. Let d be the perpendicular distance from Q :(u + du, v + dv) to the plane tangent to S at P. Then d is equal to ½f. Equation 6) gives f as a function of the variables (du, dv) in a du-dv coordinate system. See Fig. 3.

                                                                                    

Derivation


ole60.gif
ole61.gif

The values of d and f can be either positive or negative, depending on the direction in which the unit surface normal ole62.gif is pointing. If ole63.gif is pointing in the direction of surface concavity then d and f will be positive. Otherwise they will be negative. The direction in which ole64.gif points on a surface is something that is arbitrarily chosen, a matter of choice. Consulting Fig. 2, d is equal to ole65.gif PQ, which will be positive or negative depending on the direction of ole66.gif . In the figure ole67.gif is pointing in the direction of surface concavity and d is positive. The fact that the value of f depends on ole68.gif can be seen from the fact that f is a function of L, M and N which are dot products containing the vector ole69.gif .



Theorem. The quantities L, M, N are also given by


             ole70.gif  

             ole71.gif

             ole72.gif  


Proof. 



Theorem 4. The Second Fundamental Quadratic Form is equal to ole73.gif i.e.


             ole74.gif


Proof.



Osculating paraboloid at point P. The function


ole75.gif  

ole76.gif

is called the osculating paraboloid at point P of the surface. The nature of this paraboloid determines the nature of the deviation of the surface from the tangent plane in the neighborhood of point P. If δ is plotted as a function of du and dv in a du-dv coordinate system (see figures 3 and 4) one will obtain one of four parabolic surfaces depending on the value the discriminate LN - M2 of the quadratic form Ldu2 + 2Mdudv + Ndv2. See Fig 5.  

 

Case 1. LN - M2 > 0. Elliptic paraboloid. See Fig. 5(a). In this case the function represents a elliptic paraboloid and point P is referred to as an elliptic point. The surface lies completely on one side of the tangent plane. 

 

Case 2. LN - M2 < 0. Hyperbolic paraboloid. See Fig. 5(b). In this case the function represents a hyperbolic paraboloid and point P is called a hyperbolic point. Here there are two lines in the tangent plane passing through point P which divide the tangent plane into four sections in which δ is alternately positive and negative. See figure. On the two lines, δ = 0.


Case 3. LN - M2 = 0 and the coefficients L, M and N are not all zero (i.e. L2 + M2 + N2 ole77.gif ). Parabolic cylinder. See Fig. 5(c). In this case the function represents a parabolic cylinder and point P is called a parabolic point. Here there is a single line in the tangent plane passing through P along which δ = 0. Otherwise δ maintains the same sign.


 

Case 4. L = M = N = 0. Plane. In this case the degree of contact of the surface and the tangent plane is of higher order than in the other cases with the surface approximating a plane. Here point P is called a planar point.

 



Calculation of arc length, angles and surface area. The first fundamental coefficients E, F and G play a fundamental role in the calculation of arc length, angles and surface area on a simple surface element.


Arc length. Let S be a simple surface element defined by the one-to-one mapping


            x = x(u, v)

8)        y = y(u, v)

            z = z(u, v)


of a region R of the uv-plane into xyz-space. Let ole78.gif


             ole79.gif


be the position vector to point P on the surface. Let


            u = u(t)

            v = v(t)  


a ole80.gif t ole81.gif b define a curve in the uv-plane and let C be the image of this curve on surface S under mapping 8). A point on the curve is thus given by


             ole82.gif




Length on curve C is then given by the integral

             ole83.gif

             ole84.gif


or, on expanding,



ole85.gif



Angles between intersecting curves. Let two curves C1 and C2 issue from some point P on surface S with C1 given by the equations


            u = u1(t)

            v = v1(t)


and C2 given by the equations


            u = u2(t)

            v = v2(t) .


Then the tangents ole86.gif and ole87.gif to the two curves at point P are given by

  

   ole88.gif


             ole89.gif

 


Let ole90.gif be the angle between these two tangents. From the definition of a dot product it follows that

 


ole91.gif



Substituting equations 10) into 11) we obtain


ole92.gif




Angle between u-curves and v-curves. Let β be the angle between the u-curve and the v-curve at some point P. From the definition of a dot product it follows that


ole93.gif



From 13) we have the following theorem:


Theorem. The u-curves and v-curves are perpendicular at a point if and only if F = 0.


                                                                        


ole94.gif

Surface area. To obtain a formula for area on a simple surface element we consider a curvilinear rectangle bounded by the coordinate curves u = u0, v = v0, u = u0 + Δu, v = v0 + Δv and use as an approximation to it the parallelogram lying in the tangent plane and bounded by the vectors ole95.gif tangent to the coordinate curves. See Fig. 6. The area of this parallelogram is                                                                                

ole96.gif  


where f is the angle between ole97.gif and ole98.gif . Now


ole99.gif


Substituting 15) into 14) and get

  

ole100.gif


Now recall that


ole101.gif

 


Substituting 17) into16) gives us



ole102.gif


Now

 

ole103.gif


Substituting 19) into 18) we have


ole104.gif


Summing up the areas of the parallelograms and taking the limit as Δu, Δv ole105.gif 0 we obtain the formula for area


ole106.gif


where the integration takes place over that domain D of the variables u and v that corresponds to the area of interest on surface S.




Alternate formulas for surface area. The following are two other equivalent formulas for surface area: 



ole107.gif



ole108.gif





References.

1. Taylor. Advanced Calculus. Chapter 12.

2. Lipschutz. Differential Geometry. Chapter 9.

3. Mathematics, Its Contents, Methods and Meaning. Vol. II, Chapter VII.

4. Olmstead. Advanced Calculus. Chapter 18, 19.

5. Graustein. Differential Geometry.

6. Struik. Lectures on Classical Differential Geometry.



More from SolitaryRoad.com:

The Way of Truth and Life

God's message to the world

Jesus Christ and His Teachings

Words of Wisdom

Way of enlightenment, wisdom, and understanding

Way of true Christianity

America, a corrupt, depraved, shameless country

On integrity and the lack of it

The test of a person's Christianity is what he is

Who will go to heaven?

The superior person

On faith and works

Ninety five percent of the problems that most people have come from personal foolishness

Liberalism, socialism and the modern welfare state

The desire to harm, a motivation for conduct

The teaching is:

On modern intellectualism

On Homosexuality

On Self-sufficient Country Living, Homesteading

Principles for Living Life

Topically Arranged Proverbs, Precepts, Quotations. Common Sayings. Poor Richard's Almanac.

America has lost her way

The really big sins

Theory on the Formation of Character

Moral Perversion

You are what you eat

People are like radio tuners --- they pick out and listen to one wavelength and ignore the rest

Cause of Character Traits --- According to Aristotle

These things go together

Television

We are what we eat --- living under the discipline of a diet

Avoiding problems and trouble in life

Role of habit in formation of character

The True Christian

What is true Christianity?

Personal attributes of the true Christian

What determines a person's character?

Love of God and love of virtue are closely united

Walking a solitary road

Intellectual disparities among people and the power in good habits

Tools of Satan. Tactics and Tricks used by the Devil.

On responding to wrongs

Real Christian Faith

The Natural Way -- The Unnatural Way

Wisdom, Reason and Virtue are closely related

Knowledge is one thing, wisdom is another

My views on Christianity in America

The most important thing in life is understanding

Sizing up people

We are all examples --- for good or for bad

Television --- spiritual poison

The Prime Mover that decides "What We Are"

Where do our outlooks, attitudes and values come from?

Sin is serious business. The punishment for it is real. Hell is real.

Self-imposed discipline and regimentation

Achieving happiness in life --- a matter of the right strategies

Self-discipline

Self-control, self-restraint, self-discipline basic to so much in life

We are our habits

What creates moral character?


[ Home ] [ Up ] [ Info ] [ Mail ]