Website owner: James Miller
CURVATURE OF A CURVE ON A SURFACE
Curvature of a curve on a surface. 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 there be given a curve C: u = u(s), v = v(s) on a
surface S. Let
be the position vector to point P on curve C. We wish to find an expression for the curvature k of curve C at point P in terms of the first and second fundamental coefficients E, F, G, L, M, N.
To apply space curve theory we can think of C as merely a curve in space, ignoring the fact that it
is lying on a surface. Let
be the unit tangent to C at P,
be the principal normal of C at P and
be the unit surface normal at P. Then we have
Taking the dot product of both sides of 1) with
we get
where f is the angle between the principal normal vector
and the surface normal
. Now
because
, it follows that
Combining 2) and 3) and remembering that
we get
or
Thus
Theorem. If two curves passing through a point P on a surface S have the same osculating plane at P, and their common direction at P is not an asymptotic direction, they have the same curvature at P.
This theorem says, in particular, that the curvature at point P of a given curve C on a surface S is equal to the curvature at P of the plane curve in which the osculating plane of C at P cuts S. All curves passing through P with the same osculating plane have the same curvature. Thus we can restrict ourselves to the consideration of curvatures of plane sections of S.
Curvature of a normal section. When f = 0 in 5) we have the case of a normal section. Thus it follows that the curvature of a normal section is given by
References.
1. Graustein. Differential Geometry.
Jesus Christ and His Teachings
Way of enlightenment, wisdom, and understanding
America, a corrupt, depraved, shameless country
On integrity and the lack of it
The test of a person's Christianity is what he is
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
On Self-sufficient Country Living, Homesteading
Topically Arranged Proverbs, Precepts, Quotations. Common Sayings. Poor Richard's Almanac.
Theory on the Formation of Character
People are like radio tuners --- they pick out and listen to one wavelength and ignore the rest
Cause of Character Traits --- According to Aristotle
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
Personal attributes of the true Christian
What determines a person's character?
Love of God and love of virtue are closely united
Intellectual disparities among people and the power in good habits
Tools of Satan. Tactics and Tricks used by the Devil.
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
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-control, self-restraint, self-discipline basic to so much in life