SolitaryRoad.com

Website owner:  James Miller


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

DUPIN INDICATRIX, ASYMPTOTIC DIRECTIONS AND LINES



ole.gif

Dupin indicatrix of a surface at a point. Consider an xy-coordinate system where the x-axis represents the direction of minimum curvature k1 and the y-axis represents the direction of maximum curvature k2 at some point P of a surface S. The Dupin indicatrix of the surface at point P depends on the value of the total (Gaussian) curvature K = k1k2 at point P as follows:


K > 0. The Dupin indicatrix is the ellipse

                                                                        

            |k1| x2 + |k2| y2 = 1 


ole1.gif

 See Fig. 1.


K < 0. The Dupin indicatrix is a set of conjugate hyperbolas


            k1 x2 + k2 y2 = 1                                             

            k1 x2 + k2 y2 = -1                                           

                                                                                    

See Fig. 2.


K = 0. The Dupin indicatrix is a set of parallel lines corresponding to one of the forms


            x2 = |1/k1| or y2 = |1/k2|

ole2.gif

See Fig. 3.




We will now examine these three cases in detail. 


Case 1. K > 0. k1 and k2 have the same sign, the surface at point P is bowl shaped, and point P is an elliptic point. Let us suppose that k1 and k2 are both positive. Then the Dupin indicatrix at point P is the ellipse


1)        k1x2 + k2y2 = 1


shown in Fig. 1. Let us compute the distance r from the origin to point A. The equation of line OA is

 

2)        y = x tan α .


We find the coordinates (x, y) of point A by solving equations 1) and 2) simultaneously. If we substitute 2) into 1) we obtain


             ole3.gif


             ole4.gif



The distance r is then given by


             ole5.gif


and since


 

            k1(α) = k1 cos2 α + k2 sin2 α ,



where kn(α) is the normal curvature at point P in the direction α , we have


             ole6.gif



where ρn(α) is the radius of normal curvature in the direction α. The x and y intercepts Oa and Ob are


             ole7.gif


             ole8.gif


where ρ1 and ρ2 are the radii of normal curvature corresponding to k1 and k2.



Case 2. K < 0. k1 and k2 have opposite signs, the surface at point P is saddle shaped, and point P is a hyperbolic point. The Dupin indicatrix is a set of conjugate hyperbolas


            k1 x2 + k2 y2 = 1 

            k1 x2 + k2 y2 = -1


shown in Fig. 2. As in Case 1, and by the same type analysis, the values of r, Oa and Ob are


             ole9.gif


             ole10.gif


             ole11.gif



The two asymptotes of the hyperbolas are lines representing directions at which kn(α) = 0. As the angle α increases and passes over an asymptote a passage occurs from one hyperbola to the other along with a change in sign of kn(α).



Case 3. K = 0. Either k1 = 0 or k2 = 0, the surface at point P is a parabolic cylinder, and point P is a parabolic point. If k1 = 0 the Dupin indicatrix is a set of parallel lines corresponding to the equation


            y2 = |1/k2| .


If k2 = 0 the Dupin indicatrix is a set of parallel lines corresponding to the equation


            x2 = |1/k1| .


As in the other cases, and by the same type analysis, the values of r, and Oa are


             ole12.gif

 


             ole13.gif


 

Overview. The above three cases represent the different locii to which the equation


            k1 x2 + k2 y2 = 1


can give rise depending on the various possible combinations of signs and values of the constants k1 and k2. The Dupin indicatrix of a surface S at a point P is a device for gaining insight into the character of the surface near P. The curve of intersection of S and a nearby plane parallel to the tangent plane at P is approximately similar to the Dupin indicatrix. In the case of a hyperbolic point the surface S lies on both sides of the tangent plane so we need to intersect the surface with two planes close to the tangent plane, parallel to it, one on each side. If S lies entirely on one side of the tangent plane intersection by a single plane is adequate.




Asymptotic directions. A direction at a point P on a surface for which


1)        L du2 + 2M du dv + N dv2 = 0


is called an asymptotic direction. Because the denominator of


             ole14.gif


is positive definite, the asymptotic directions are also the directions in which kn = 0. At an elliptic point there are no asymptotic directions, at a hyperbolic point there are two distinct asymptotic directions, at a parabolic point there is one asymptotic direction and at a planar point every direction is asymptotic.



Asymptotic lines. A curve on a surface which is tangent to an asymptotic direction at every point is called an asymptotic line. Thus a curve on a surface element is an asymptotic line if and only if at each point of the curve the direction of the tangent to the curve satisfies 1). At a hyperbolic point equation 1) has two real distinct factors of the form Adu + Bdv = 0 which can be regarded as first order differential equations of the asymptotic lines.


Theorem 1. The u and v coordinate curves on a surface element are asymptotic lines if and only if at each point L = N = 0.




References.

1. Graustein. Differential Geometry.

2. Lipschutz. Differential Geometry

3. James/James. Mathematics Dictionary.

4. 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 ]