Theorems

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
LetS: \vec{r}=\vec{r}(u,v)be 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 \vec{N},\vec{N}+d \vec{N} intersects at consecutive points P(\vec{r}) and Q(\vec{r}+d\vec{r}) on C is
\vec{N},\vec{N}+d\vec{N} and d\vec{r} are coplanar
[\vec{N},\vec{N}+d\vec{N},d\vec{r}]=0
[\vec{N},d\vec{N},d\vec{r}]=0
[\vec{N},{\vec{N}}_1du+{\vec{N}}_2dv,{\vec{r}}_1du+{\vec{r}}_2dv]=0
[\vec{N},{\vec{N}}_1{\vec{r}}_1]du^2+\{[\vec{N},{\vec{N}}_1{\vec{r}}_2]+[\vec{N},{\vec{N}}_2{\vec{r}}_1]\}dudv+[\vec{N},{\vec{N}}_2{\vec{r}}_2]dv^2=0
Since,
[\vec{N},{\vec{N}}_1,{\vec{r}}_1]=\frac{EM-FL}{H}
[\vec{N},{\vec{N}}_1,{\vec{r}}_2]=\frac{FM-GL}{H}
[\vec{N},{\vec{N}}_2,{\vec{r}}_1]=\frac{EN-FM}{H}
[\vec{N},{\vec{N}}_2,{\vec{r}}_2]=\frac{FN-GM}{H}
Thus, we have
\frac{EM-FL}{H}du^2+\{\frac{EM-GL}{H}+\frac{EN-FM}{H}\}dudv+\frac{FN-GM}{H}dv^2=0
(EM-FL)du^2+(EN-GL)dudv+(FN-GM)dv^2=0
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 d \vec{N}+\kappa_n d \vec{r}=0, where \kappa_n is normal curvature.
Proof
Necessary condition
LetS: \vec{r}=\vec{r}(u,v)be a surface and C be a curve on it.
Necessary part
Assume that C is line of curvature, then
\kappa_n=\frac{Ldu^2+2Mdudv+Ndv^2}{Edu^2+2Fdudv+Gdv^2}
or \kappa_nd\vec{r}.\ d\vec{r}+d\vec{N.}\ d\vec{r}=0
or d\vec{N.}\ d\vec{r}+\kappa_nd\vec{r}.\ d\vec{r}=0
or d\vec{r}(d\vec{N}+\kappa_nd\vec{r})=0
or d\vec{N}+\kappa_nd\vec{r}=0
Sufficient condition
Assume that, curve C on a surface S holds
d\vec{N}+\kappa_nd\vec{r}=0 (i)
or ({\vec{N}}_1du+{\vec{N}}_2dv)+\kappa_n({\vec{r}}_1du+{\vec{r}}_2dv)=0
Operating dot product on both sides by \vec{r}_1 we get
(Ldu+Mdv)-\kappa_n(Edu+Fdv)=0 (A)
Again, operating dot product on both sides by \vec{r}_2 we get
(Mdu+Ndv)-\kappa_n(Fdu+Gdv)=0 (B)
Eliminating, \kappa_n between (A) and (B) we get
\frac{Ldu+Mdv}{Mdu+Ndv}=\frac{Edu+Fdv}{Fdu+Gdv}
or (EM-FL)du^2+(EN-GL)dudv+(FN-GM)dv^2=0
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 surfaceS: \vec{r}=\vec{r}(u,v) and \vec{N} 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
[\vec{N},d\vec{N},\vec{dr}]=0
Diff. w. r. to. s we get
[\vec{N},\vec{N}\prime,\vec{t}]=0
or [\vec{t},\vec{N},\vec{N}\prime]=0
This, surface normal along the curve form developable.
Sufficient condition
Assume that surface normal along the curve form developable then
[\vec{t},\vec{N},\vec{N}']=0
or [\vec{N},\vec{d{N}},d{\vec{r}}]=0
or [\vec{N},{\vec{N}}_1du+{\vec{N}}_2dv,{\vec{r}}_1du+{\vec{r}}_2dv]=0
or (EM-FL)du^2+(EN-GL)dudv+(FN-GM)dv^2=0
which is the differential equation of line of curvature.
Hence the theorem

Euler’s theorem

The normal curvature \kappa_n on a surface S: \vec{r}=\vec{r}(u,v) is given as where \psi is angle between direction of normal section (du,dv) and principal directiond v=0
Proof
Let S: \vec{r}=\vec{r}(u,v) be a surface then normal curvature is
\kappa_n=\frac{Ldu^2+2Mdudv+Ndv^2}{Edu^2+2Fdudv+Gdv^2}(i)
Since, line of curvature is taken along parametric curves, we have
M=0,F=0
Now, normal curvature is
\kappa_n=\frac{Ldu^2+Ndv^2}{Edu^2+Gdv^2}(A)
Along parametric curve v=constant, the principal curvature \kappa_a is
\kappa_a=\frac{Ldu^2}{Edu^2}=\frac{L}{E}
Along parametric curve u=constant, the principal curvature \kappa_b is
\kappa_b=\frac{Ndv^2}{Gdv^2}=\frac{N}{G}
Now, direction of normal section is(du,dv)
and, direction of parametric curve v=constant is
\left (\frac{1}{\sqrt E},0 \right )
Given,\psi is angle between(du,dv) and \left (\frac{1}{\sqrt E},0 \right), thus
cos\psi=E\frac{1}{\sqrt E}du+F\frac{1}{\sqrt E}dv and sin\psi=H\frac{1}{\sqrt E}dv
or cos\psi=\sqrt E du and sin\psi=\frac{\sqrt{EG-F^2}}{\sqrt E}dv
or cos\psi=\sqrt E du and sin\psi=\sqrt G dv
Thus
Edu^2+Gdv^2=1(ii)
Now, from (ii), we write
\kappa_n=Ldu^2+Ndv^2
or \kappa_n=L \frac{\cos^2 \psi}{E}+N \frac{\sin^2 \psi}{G}
or \kappa_n= \frac{L}{E}\cos^2 \psi+ \frac{N}{G} \sin^2 \psi
or \kappa_n= \kappa_a \cos^2 \psi+ \kappa_b \sin^2 \psi

Note

  1. 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
    \kappa_a=\frac{L}{E} and \kappa_b=\frac{N}{G}
  2. 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
    \kappa_a=\frac{L}{E} and \kappa_b=\frac{N}{G}

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 \vec{r}=\vec{r}(u,v) be the surface and p be a point.
Also if C1 and C2 are two normal sections in orthogonal directions with normal curvatures \kappa_{n_1} and \kappa_{n_2}
Then,
\kappa_{n_1}=\kappa_acos^2\psi+\kappa_bsin^2\psi(i)
And
\kappa_{n_2}=\kappa_acos^2(\frac{\pi}{2}+\psi)+\kappa_bsin^2(\frac{\pi}{2}+\psi)
or \kappa_{n_2}=\kappa_asin^2\psi+\kappa_b\ cos^2\psi(ii)
Adding (i) and (ii), we get
\kappa_{n_1}+\kappa_{n_2}=\kappa_a+\kappa_b
Thus the theorem

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