#### An important property

Prove that, if normal at two consecutive points on a curve of a surface do intersect then the curve is line of curvature. Conversely, normal at consecutive points on line of curvature of a surface do intersect.

Proof

First part

Letbe a surface and C be a curve on it.

Assume that, surface normal at consecutive points on C do intersect.

Then, necessary and sufficient condition for intersects
at consecutive points and on C is

and are coplanar

⇔

⇔

⇔

⇔

Since,

Thus, we have

⇔

which is differential equation of line of curvature.

#### Rodrigue’s formula

The necessary and sufficient condition for a curve on a surface be line of curvature is , where is normal curvature.

Proof

Necessary condition

Letbe a surface and C be a curve on it.

Necessary part

Assume that C is line of curvature, then

or

or

or

or

Sufficient condition

Assume that, curve C on a surface S holds

(i)

or

Operating dot product on both sides by we get

(A)

Again, operating dot product on both sides by we get

(B)

Eliminating, between (A) and (B) we get

or

which is equation of line of curvature.

Gaspard Monge (1746 -1818) a French mathematician is considered the father of differential geometry because of his work: concept of lines of curvature of a surface.

#### Monge’s theorem

The necessary and sufficient condition for a curve on a surface be line of curvature is that surface normal along the curve form developable.

Proof

Let C be a curve on surface and be surface normal.

Necessary condition

Assume that a curve C on a surface be line of curvature. Then consecutive surface normal along the curve do intersect.

Thus we have

Diff. w. r. to. s we get

or

This, surface normal along the curve form developable.

Sufficient condition

Assume that surface normal along the curve form developable then

or

or

or

which is the differential equation of
line of curvature.

Hence the theorem

#### Euler’s theorem

The normal curvature on a surface is given as where is angle between direction of normal section (du,dv) and principal directiond v=0

Proof

Let be a surface then normal curvature is

(i)

Since, line of curvature is taken along parametric curves, we have

M=0,F=0

Now, normal curvature is

(A)

Along parametric curve v=constant, the principal curvature is

Along parametric curve u=constant, the principal curvature is

Now, direction of normal section is(du,dv)

and, direction of parametric curve v=constant is

Given, is angle between(du,dv) and , thus

and

or
and

or
and

Thus

(ii)

Now, from (ii), we write

or

or

or

Note

- If the directions of u- and v-parameter curves at a point P on a surface are principal directions, then the principal curvatures at P are given by

and - If the u- and v-parameter curves at a point P on a surface are line of curvatures, then the principal curvatures at P are given by

and

#### Dupin’s theorem

The sum of the normal curvatures in two orthogonal directions is equal to the sum of the principal curvatures at that point.

Proof

Let be the surface and p be a point.

Also if C1 and C2 are two normal sections in orthogonal directions with normal curvatures and

Then,

(i)

And

or (ii)

Adding (i) and (ii), we get

Thus the theorem