Local non-intrinsic property of surface ~ Bed Prasad Dhakal

In differential geometry, the study of surfaces involves understanding the intrinsic and extrinsic properties that define a surface's shape, curvature, and topology. A surface in differential geometry is typically a two-dimensional manifold, meaning that locally around each point, it resembles a flat plane, but globally it may have more complex structure and curvature.

Local Non-Intrinsic Properties of Surface

Intrinsic property of a surface is an invariant, inherent or unchanging property of surface. Some example like Gaussian curvature, which remain unchanged in a surface

Next, non-intrinsic properties (or local non-intrinsic properties) of a surface are those characteristics that are not determined by the internal geometry of the surface (like distances and angles, area within the surface) but by how the surface interacts with the external space. Some example of non-intrinsic properties of a surface are (a) Normal Vector: the orientation of the normal vector is determined by the surface's embedding in space. (b) Curvature of sections: It is a measure of how the surface bends in the surrounding space. If a surface is bent in one direction more than another, the curvature of sections captures this asymmetry.

0,0 az = 1.00 el = 0.30 s_o s_p $\vec{N}$	0,0 az = 1.00 el = 0.30 s_q s_r $\vec{N}$
(a)Normal to the surface	(b)Normal section to the surface

0,0 az = 1.00 el = 0.30	0,0 az = 1.00 el = 0.30
(a)Normal to the surface	(b)Oblique section to the surface

Normal section of Surface

Let

S : \vec{r} = \vec{r} (u, v)

be a surface and P be a point on it, on which

Γ

be a plane at P
Then intersection of

S \cap Γ

is a curve

C

on the surface. This curve is called section of the surface.
Also, we knaow that

\vec{N}

is normal to the surface at P
If the plane

Γ

contains

\vec{N}

, then

C

is called normal section to the surface,
otherwise

C

is called oblique section to the surface

0,0 az = 1.00 el = 0.30 s_q s_r $\vec{N} = \vec{n}$	0,0 az = 1.00 el = 0.30 s_q s_r $\vec{N}$ $\vec{n}$
(a)Normal to the surface	(b)Oblique section to the surface

NOTE

There are infinitely many normal sections on the surface.
In a normal section, $\vec{n} = \vec{N}$

Normal Curvature and related Theorems

Let

S : \vec{r} = \vec{r} (u, v)

be a surface and C be a normal section on it. Then curvature of C is called normal curvature. The normal curvature of the surface is denoted by

κ_{n}

.
It is measure of how the surface bends in a particular direction.

Expression of Normal Curvature

Let

S : \vec{r} = \vec{r} (u, v)

be a surface and C be a normal section on it. Then curvature of C is

κ \vec{n} = {\vec{r}}^{″}

κ_{n} \vec{n} = {\vec{r}}^{″}

[normal curvature is denoted by

κ_{n}

]
or

κ_{n} \vec{N} = {\vec{r}}^{″}

[for normal section

\vec{N} = \vec{N}

]
Operating dot product on both sides by

\vec{N}

, we get

κ_{n} = \vec{N} . {\vec{r}}^{″}

(A)
Next, equation of the surface is

\vec{r} = \vec{r} (u, v)

Differentiation of w. r. to. s, we get

{\vec{r}}^{'} = {\vec{r}}_{1} \frac{d u}{d s} + {\vec{r}}_{2} \frac{d v}{d s}

{\vec{r}}^{'} = {\vec{r}}_{1} u^{'} + {\vec{r}}_{2} v^{'}

(i)
Again, differentiation of w. r. to. s, we get

{\vec{r}}^{″} = \frac{d}{d s} ({\vec{r}}_{1} u^{'} + {\vec{r}}_{2} v^{'})

{\vec{r}}^{″} = ({\vec{r}}_{11} u^{'} + {\vec{r}}_{12} v^{'}) u^{'} + ({\vec{r}}_{12} u^{'} + {\vec{r}}_{22} v^{'}) v^{'} + {\vec{r}}_{1} u^{″} + {\vec{r}}_{2} v^{″}

{\vec{r}}^{″} = {\vec{r}}_{11} u^{' 2} + 2 {\vec{r}}_{12} u^{'} v^{'} + {\vec{r}}_{22} v^{' 2} + {\vec{r}}_{1} u^{″} + {\vec{r}}_{2} v^{″}

(ii)
By substitution of (ii) in (A), the expression of normal curvature is

κ_{n} = \vec{N} . {\vec{r}}^{″}

κ_{n} = \vec{N} . ({\vec{r}}_{11} u^{' 2} + 2 {\vec{r}}_{12} u^{'} v^{'} + {\vec{r}}_{22} v^{' 2} + {\vec{r}}_{1} u^{″} + {\vec{r}}_{2} v^{″})

κ_{n} = (L u^{' 2} + 2 M u^{'} v^{'} + N v^{' 2})

κ_{n} = L {(\frac{d u}{d s})}^{2} + 2 M \frac{d u}{d s} \frac{d v}{d s} + N {(\frac{d v}{d s})}^{2}

κ_{n} = \frac{L d u^{2} + 2 M d u d v + N d v^{2}}{d s^{2}}

κ_{n} = \frac{L d u^{2} + 2 M d u d v + N d v^{2}}{E d u^{2} + 2 F d u d v + G d v^{2}}

κ_{n} = \frac{I I}{I}

Thus is required expression of normal curvature.
Note
If first and second fundamental coefficients are proportional, the normal curvature has expression

κ_{n} = \frac{L}{E} = \frac{M}{F} = \frac{N}{G}

Find normal curvature of a curve $u = t^{2}, v = t$ on $\vec{r} = (u, v, u^{2} + v^{2})$ at t=1

Solution
Given surface is

\vec{r} = (u, v, u^{2} + v^{2})

(i)
By a bit of calculation, we get

E = 5, F = 4, G = 5

t = 1

L = \frac{2}{5}, M = 0, N = \frac{2}{3}

t = 1

\frac{d u}{d t} = 2, \frac{d v}{d t} = 1

t = 1

Thus, normal curvature at t=1 is

κ_{n} = \frac{L {d u}^{2} + 2 M d u d v + N {d v}^{2}}{E {d u}^{2} + 2 F d u d v + G {d v}^{2}}

κ_{n} = \frac{L {(\frac{d u}{d t})}^{2} + 2 M \frac{d u}{d t} \frac{d v}{d t} + N {(\frac{d v}{d t})}^{2}}{E {(\frac{d u}{d t})}^{2} + 2 F \frac{d u}{d t} \frac{d v}{d t} + G {(\frac{d v}{d t})}^{2}}

κ_{n} = \frac{10}{123}

Meusnier’s Theorem

Meusnier’s theorem shows a relation between curvature of normal and oblique sections, it was first announced by Jean Basptiste Meusnier in 1776. The essence of the theorem is that, all curves lying on a surface S having same tangent have the same normal curvature.

Theorem

κ

and

κ_{n}

are curvatures of oblique and normal sections and

θ

be angles between these sections then

κ_{n} = κ \cos θ

0,0 az = 1.00 el = 0.30 s_q s_r $\vec{N} = \vec{n}$ $\vec{t}$	0,0 az = 1.00 el = 0.30 s_q s_r $\vec{N}$ $\vec{n}$ $\vec{t}$
(a)Normal to the surface	(b)Oblique section to the surface

Proof
Let

S : \vec{r} = \vec{r} (u, v)

be a surface and

\vec{N}

be the unit normal. Then, curvature of oblique section is

{\vec{r}}^{″} = κ \vec{n}

(i)
Taking dot product of both sides of (i) by

\vec{N}

, we get

{\vec{r}}^{″} . \vec{N} = κ \vec{n} . \vec{N}

(A)
Next, curvature of normal section is

{\vec{r}}^{″} = κ_{n} . \vec{N}

(ii)
Taking dot product of both sides of (ii) by

\vec{N}

, we get

{\vec{r}}^{″} . \vec{N} = κ_{n}

(B)
Also given that

θ

is the angle between these sections, then

\cos θ = \vec{n} . \vec{N}

κ_{n} = κ \cos θ

This completes the proof of the theorem.

0,0

az = 1.00

el = 0.30

Principal Sections

Let

S : \vec{r} = \vec{r} (u, v)

be a surface and P be a point on it. Then there are infinitely many normal sections at P. Now, two sections, which give maximum and minimum curvature (principal curvatures) at P, are called principal sections at P. These principal sections are always orthogonal.

Principal Directions

Let

S : \vec{r} = \vec{r} (u, v)

be a surface and P be a point on it. Then, tangents to principal sections are called principal directions.Since, there are two principal sections (in general), it is convenient to state that, there are two principal directions (in general) at every point on a surface and it will be always orthogonal.

Differential equation of Principal Section/direction

Let

S : \vec{r} = \vec{r} (u, v)

be a surface and

κ_{n}

be the principal curvature, then

κ_{n} = \frac{L d u^{2} + 2 M d u d v + N d v^{2}}{E d u^{2} + 2 F d u d v + G d v^{2}}

(L d u^{2} + 2 M d u d v + N d v^{2}) - κ_{n} (E d u^{2} + 2 F d u d v + G d v^{2}) = 0

Then, differentiating w r. to.

d u

and

d v

separately, we get

(L d u + M d v) - κ_{n} (E d u + F d v) = 0

(i)

(M d u + N d v) - κ_{n} (F d u + G d v) = 0

(ii)
Eliminating

k_{n}

from (i) and (ii) we get

\frac{(L d u + M d v)}{(M d u + N d v)} = \frac{(E d u + F d v)}{(F d u + G d v)}

(E M - F L) d u^{2} + (E N - G L) d u d v + (F N - G M) d v^{2} = 0

(iii)
which is the required equation of principal sections/directions.
Note
The equation of principal section/directions can be written as

| \begin{matrix} E & F & G \\ L & M & N \\ d v^{2} & - d u d v & d u^{2} \end{matrix} | = 0

Find principal sections on hyperboloid $2 z = 7 x^{2} + 6 x y - y^{2}$ at origin

Solution The position vector of the hyperboloid

2 z = 7 x^{2} + 6 x y - y^{2}

\vec{r} = (x, y, \frac{1}{2} (7 x^{2} + 6 x y - y^{2}))

By computing the fundamental coefficients of the surface, we get

E = 1, F = 0, G = 1, H = 1, L = 7, M = 3, N = - 1

Now the equation of principal sections is

(E M - F L) d x^{2} + (E N - G L) d x d y + (F N - G M) d y^{2} = 0

3 d x^{2} - 8 d x d y - 3 d y^{2} = 0

d x - 3 d y = 0, 3 d x + d y = 0

x - 3 y = 0, 3 x + y = 0

This completes the solution.

Show that principal directions are always orthogonal

Proof
The equation of principal directions is

(E M - F L) d u^{2} + (E N - G L) d u d v + (F N - G M) d v^{2} = 0

(A)
The equation of double family of curves is

P d u^{2} + Q d u d v + R d v^{2} = 0

(B)
Comparing (A) and (B) we get

P = (E M - F L), Q = (E N - G L), R = (F N - G M)

Now, condition of orthogonally for double family of curves is

E R - Q F + G P = 0

Hence, we have

E R - Q F + G P = E (F N - G M) - F (E N - G L) + G (E M - F L)

E R - Q F + G P = 0

TRUE
It shows that, principal directions are always orthogonal

Principal Curvature

Let

S : \vec{r} = \vec{r} (u, v)

be a surface and P be a point on it. Then there are infinitely many normal curvatures at P. Now, two curvatures, which are maximum and minimum at P, are called principal curvatures at P. These two curvatures are denoted by

κ_{a}

and

κ_{b}

The corresponding radii of principal curvatures are called principal radii and are denoted by

ρ_{a}

and

ρ_{b}

Since, Principal curvatures are the maximum and minimum (signed) curvatures of various normal slices, these maximum and minimum curvatures always occur at right angles to one another.

Differential equation of principal curvature

Let

S : \vec{r} = \vec{r} (u, v)

be a surface and

κ_{n}

be the principal curvature, then

κ_{n} = \frac{L d u^{2} + 2 M d u d v + N d v^{2}}{E d u^{2} + 2 F d u d v + G d v^{2}}

(L d u^{2} + 2 M d u d v + N d v^{2}) - κ_{n} (E d u^{2} + 2 F d u d v + G d v^{2}) = 0

Then, differentiating w r. to. duanddv separately, we get

(L d u + M d v) - κ_{n} (E d u + F d v) = 0

and

(M d u + N d v) - κ_{n} (F d u + G d v) = 0

(L - κ_{n} E) d u + (M - κ_{n} F) d v = 0

and

(M - κ_{n} F) d u + (N - κ_{n} G) d v = 0

Eliminating du and dv, we get

\frac{(M - κ_{n} F)}{(L - κ_{n} E)} = \frac{(N - κ_{n} G)}{(M - κ_{n} F)}

(M - κ_{n} F) (M - κ_{n} F) = (L - κ_{n} E) (N - κ_{n} G)

{κ_{n}}^{2} (E G - F^{2}) - κ_{n} (E N - 2 F M + G L) + (L N - M^{2}) = 0

This is the equation of principal curvatures
Solving this quadratic equation for

κ

, we get two principal curvatures (maximum or minimum). Thus, principal curvatures

κ_{a}

and

κ_{b}

are obtained by

κ_{a} + κ_{b} = \frac{E N - 2 F M + G L}{E G - F^{2}}

and

κ_{a} . κ_{b} = \frac{L N - M^{2}}{E G - F^{2}}

Note

The sum of the principal curvatures i.e., $κ_{a} + κ_{b} = \frac{E N - 2 F M + G L}{E G - F^{2}}$ is called first curvature, it is denoted by J.
The arithmetic mean of principal curvatures i.e., $\frac{1}{2} (κ_{a} + κ_{b}) = \frac{E N - 2 F M + G L}{2 (E G - F^{2})}$ is called mean curvature
or mean normal curvature, it is denoted by $μ$ .
The product of principal curvatures i.e., $κ_{a} \times κ_{b} = \frac{L N - M^{2}}{E G - F^{2}}$ is called Gaussian curvature, it is denoted by K.
It is also called specific curvature, second curvature or total curvature.
Given a point P
(a). if $κ_{a} = κ_{b} = 0$ , we say P is planner point
(b). if $K = 0$ ,we say P is parabolic point
(c). if $K > 0$ , we say P is elliptic point
(d). if $K < 0$ , we say P is hyperbolic point

Show that principal curvature of hyperboloid $2 z = 7 x^{2} + 6 x y - y^{2}$ at origin are 8 and -2.

Solution
The position vector of the hyperboloid

2 z = 7 x^{2} + 6 x y - y^{2}

\vec{r} = (x, y, \frac{1}{2} (7 x^{2} + 6 x y - y^{2}))

By computing the fundamental coefficients of the surface, we get

E = 1, F = 0, G = 1, H = 1, L = 7, M = 3, N = - 1

Now the equation of principal curvature is

{κ_{n}}^{2} (E G - F^{2}) - κ_{n} (E N - 2 F M + G L) + (L N - M^{2}) = 0

{κ_{n}}^{2} (1) - κ_{n} (6) + (- 16) = 0

{κ_{n}}^{2} - 5 κ_{n} - 16 = 0

κ_{n} = 8

and

κ_{n} = - 2

This completes the solution.

Local non-intrinsic property of surface

Local Non-Intrinsic Properties of Surface

Normal section of Surface

Normal Curvature and related Theorems

Expression of Normal Curvature

Find normal curvature of a curve $u = t^{2}, v = t$ on $\vec{r} = (u, v, u^{2} + v^{2})$ at t=1

Meusnier’s Theorem

Theorem

Principal Sections

Principal Directions

Differential equation of Principal Section/direction

Find principal sections on hyperboloid $2 z = 7 x^{2} + 6 x y - y^{2}$ at origin

Show that principal directions are always orthogonal

Principal Curvature

Differential equation of principal curvature

Show that principal curvature of hyperboloid $2 z = 7 x^{2} + 6 x y - y^{2}$ at origin are 8 and -2.

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Local non-intrinsic property of surface

Local Non-Intrinsic Properties of Surface

Normal section of Surface

Normal Curvature and related Theorems

Expression of Normal Curvature

Find normal curvature of a curve u=t2,v=t on r→=(u,v,u2+v2)at t=1

Meusnier’s Theorem

Theorem

Principal Sections

Principal Directions

Differential equation of Principal Section/direction

Find principal sections on hyperboloid 2z=7x2+6xy−y2 at origin

Show that principal directions are always orthogonal

Principal Curvature

Differential equation of principal curvature

Show that principal curvature of hyperboloid 2z=7x2+6xy−y2 at origin are 8 and -2.

No comments:

Post a Comment

Follow Me

SEARCH

LATEST

SECTIONS

Pageviews

Popular

Archive

Courses

Categories

Comments

Find normal curvature of a curve $u = t^{2}, v = t$ on $\vec{r} = (u, v, u^{2} + v^{2})$ at t=1

Find principal sections on hyperboloid $2 z = 7 x^{2} + 6 x y - y^{2}$ at origin

Show that principal curvature of hyperboloid $2 z = 7 x^{2} + 6 x y - y^{2}$ at origin are 8 and -2.