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 surfaceNext, 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.
| (a)Normal to the surface | (b)Normal section to the surface |
| (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\(\Gamma\) be a plane at P
Then intersection of \(S \cap \Gamma\) 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 \(\Gamma \) contains \(\vec{N}\) , then \(C\) is called normal section to the surface,
otherwise \(C\) is called oblique section to the surface
| (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 \(\kappa_n\).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\(\kappa \vec{n}=\vec{r}''\)
or\(\kappa_n \vec{n}=\vec{r}''\) [normal curvature is denoted by \( \kappa_n \)]
or\(\kappa_n \vec{N}=\vec{r}''\) [for normal section \(\vec{N}=\vec{N}\)]
Operating dot product on both sides by \(\vec{N}\), we get
\(\kappa_n= \vec{N}.\vec{r}''\) (A)
Next, equation of the surface is
\(\vec{r}=\vec{r}(u,v)\)
Differentiation of w. r. to. s, we get
\(\vec{r}'=\vec{r}_1 \frac{du}{ds} +\vec{r}_2 \frac{dv}{ds} \)
or\(\vec{r}'=\vec{r}_1 u' +\vec{r}_2 v' \) (i)
Again, differentiation of w. r. to. s, we get
\(\vec{r}''=\frac{d}{ds} ( \vec{r}_1 u' +\vec{r}_2 v') \)
or \(\vec{r}''=( \vec{r}_{11} u' +\vec{r}_{12} v')u'+ ( \vec{r}_{12} u' +\vec{r}_{22} v')v' +\vec{r}_1 u'' +\vec{r}_2 v'' \)
or \(\vec{r}''=\vec{r}_{11} u'^2 +2 \vec{r}_{12} u'v'+ \vec{r}_{22} v'^2 +\vec{r}_1 u'' +\vec{r}_2 v'' \) (ii)
By substitution of (ii) in (A), the expression of normal curvature is
\(\kappa_n= \vec{N}.\vec{r}''\)
or\(\kappa_n= \vec{N}. \left ( \vec{r}_{11} u'^2 +2 \vec{r}_{12} u'v'+ \vec{r}_{22} v'^2 +\vec{r}_1 u'' +\vec{r}_2 v'' \right ) \)
or\(\kappa_n= ( L u'^2 +2 M u'v'+ N v'^2) \)
or\(\kappa_n= L \left(\frac{du}{ds} \right )^2 +2 M \frac{du}{ds}\frac{dv}{ds}+ N \left(\frac{dv}{ds} \right )^2 \)
or \(\kappa_n= \frac{L du^2 +2 M dudv+ N dv^2}{ds^2} \)
or \(\kappa_n= \frac{L du^2 +2 M dudv+ N dv^2}{Edu^2+2Fdudv+Gdv^2} \)
or \(\kappa_n= \frac{II}{I} \)
Thus is required expression of normal curvature.
Note
If first and second fundamental coefficients are proportional, the normal curvature has expression
\( \kappa_n=\frac{L}{E}=\frac{M}{F}=\frac{N}{G}\)
Find normal curvature of a curve \(u=t^2,v=t\) on \(\vec{r}=(u,v,u^2+v^2) \)at t=1
SolutionGiven surface is
\( \vec{r}=(u,v,u^2+v^2) \) (i)
By a bit of calculation, we get
\(E=5, F=4, G=5 \) at \(t=1\)
\(L=\frac{2}{5}, M=0 , N=\frac{2}{3}\) at \(t=1\)
\(\frac{du}{dt}=2 , \frac{dv}{dt}=1\) at \(t=1\)
Thus, normal curvature at t=1 is
\( \kappa_n=\frac{L{du}^2+2Mdudv+N{dv}^2}{E{du}^2+2Fdudv+G{dv}^2}\)
or \( \kappa_n=\frac{L\left(\frac{du}{dt}\right)^2+2M\frac{du}{dt}\frac{dv}{dt}+N\left(\frac{dv}{dt}\right)^2}{E\left(\frac{du}{dt}\right)^2+2F\frac{du}{dt}\frac{dv}{dt}+G\left(\frac{dv}{dt}\right)^2}\)
or \( \kappa_n=\frac{10}{123}\)
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
If \(\kappa\) and \(\kappa_n\) are curvatures of oblique and normal sections and \(\theta\) be angles between these sections then \(\kappa_n= \kappa \cos \theta \)| (a)Normal to the surface | (b)Oblique section to the surface |
Let \(S:\vec{r}=\vec{r}(u,v)\) be a surface and \(\vec{N}\) be the unit normal. Then, curvature of oblique section is
\(\vec{r}''=\kappa\vec{n} \) (i)
Taking dot product of both sides of (i) by \(\vec{N}\), we get
\( \vec{r}''.\vec{N}=\kappa\vec{n}.\vec{N} \) (A)
Next, curvature of normal section is
\(\vec{r}''=\kappa_n.\vec{N} \) (ii)
Taking dot product of both sides of (ii) by \(\vec{N}\), we get
\(\vec{r}''.\vec{N}=\kappa_n \) (B)
Also given that \(\theta\) is the angle between these sections, then
\( \cos \theta=\vec{n}.\vec{N}\) (C)
By substituting (B) and (C) in (A) we get
\( \kappa_n=\kappa \cos \theta \)
This completes the proof of the theorem.
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 \(\kappa_n\) be the principal curvature, then\( \kappa_n=\frac{Ldu^2+2Mdudv+Ndv^2}{Edu^2+2Fdudv+Gdv^2} \)
or\( (Ldu^2+2Mdudv+Ndv^2)-\kappa_n(Edu^2+2Fdudv+Gdv^2)=0 \)
Then, differentiating w r. to. \(du\) and \(dv\) separately, we get
\((Ldu+Mdv)-\kappa_n(Edu+Fdv)=0\) (i)
\((Mdu+Ndv)-\kappa_n(Fdu+Gdv)=0 \) (ii)
Eliminating \(k_n\) from (i) and (ii) we get
\( \frac{(Ldu+Mdv)}{(Mdu+Ndv)}=\frac{(Edu+Fdv)}{(Fdu+Gdv)} \)
or \((EM-FL)du^2+(EN-GL)dudv+(FN-GM)dv^2=0 \) (iii)
which is the required equation of principal sections/directions.
Note
The equation of principal section/directions can be written as
\(\begin{vmatrix}E&F&G\\L&M&N\\dv^2&-dudv&du^2 \end{vmatrix}=0 \)
Find principal sections on hyperboloid \(2z=7x^2+6xy-y^2\) at origin
Solution The position vector of the hyperboloid \(2z=7x^2+6xy-y^2\) is
\(\vec{r}=(x,y,\frac{1}{2}(7x^2+6xy-y^2)) \)
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
\((EM-FL)dx^2+(EN-GL)dxdy+(FN-GM)dy^2=0 \)
or\(3dx^2-8dxdy-3dy^2=0 \)
or\(dx-3dy=0,3dx+dy=0 \)
or\(x-3y=0,3x+y=0 \)
This completes the solution.
Show that principal directions are always orthogonal
ProofThe equation of principal directions is
\( (EM-FL)du^2+(EN-GL)dudv+(FN-GM)dv^2=0 \) (A)
The equation of double family of curves is
\(Pdu^2+Qdudv+Rdv^2=0 \) (B)
Comparing (A) and (B) we get
\(P=(EM-FL),Q=(EN-GL),R=(FN-GM)\)
Now, condition of orthogonally for double family of curves is
\(ER-QF+GP=0 \)
Hence, we have
\(ER-QF+GP=E(FN-GM)-F(EN-GL)+G(EM-FL) \)
or\(ER-QF+GP=0 \) 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\( \kappa_a\) and \(\kappa_b \)
The corresponding radii of principal curvatures are called principal radii and are denoted by
\( \rho_a\) and \(\rho_b\)
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 \(\kappa_n\) be the principal curvature, then\( \kappa_n=\frac{Ldu^2+2Mdudv+Ndv^2}{Edu^2+2Fdudv+Gdv^2}\)
or\((Ldu^2+2Mdudv+Ndv^2)-\kappa_n(Edu^2+2Fdudv+Gdv^2)=0 \)
Then, differentiating w r. to. duanddv separately, we get
\( (Ldu+Mdv)-\kappa_n(Edu+Fdv)=0\) and \( (Mdu+Ndv)-\kappa_n(Fdu+Gdv)=0 \)
or\((L-\kappa_nE)du+(M-\kappa_nF)dv=0\) and \( (M-\kappa_nF)du+(N-\kappa_nG)dv=0 \)
Eliminating du and dv, we get
\(\frac{(M-\kappa_nF)}{(L-\kappa_nE)}=\frac{(N-\kappa_nG)}{(M-\kappa_nF)} \)
or\((M-\kappa_nF)(M-\kappa_nF)=(L-\kappa_nE)(N-\kappa_nG) \)
or\({\kappa_n}^2(EG-F^2)-\kappa_n(EN-2FM+GL)+(LN-M^2)=0 \)
This is the equation of principal curvatures
Solving this quadratic equation for \(\kappa\) , we get two principal curvatures (maximum or minimum). Thus, principal curvatures \( \kappa_a \) and \( \kappa_b\) are obtained by
\(\kappa_a+\kappa_b=\frac{EN-2FM+GL}{EG-F^2}\)and
\(\kappa_a.\kappa_b=\frac{LN-M^2}{EG-F^2} \)
Note
- The sum of the principal curvatures i.e., \(\kappa_a+\kappa_b=\frac{EN-2FM+GL}{EG-F^2}\) is called first curvature, it is denoted by J.
- The arithmetic mean of principal curvatures i.e., \(\frac{1}{2}(\kappa_a+\kappa_b)=\frac{EN-2FM+GL}{2(EG-F^2)}\) is called mean curvature
or mean normal curvature, it is denoted by \(\mu\). - The product of principal curvatures i.e., \(\kappa _a \times \kappa _b = \frac{LN-M^2}{EG-F^2}\) 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 \( \kappa_a=\kappa_b=0\) , 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 \( 2z=7x^2+6xy-y^2\) at origin are 8 and -2.
SolutionThe position vector of the hyperboloid \(2z=7x^2+6xy-y^2\) is
\(\vec{r}=(x,y,\frac{1}{2}(7x^2+6xy-y^2)) \)
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
\({\kappa_n}^2(EG-F^2)-\kappa_n(EN-2FM+GL)+(LN-M^2)=0 \)
or\({\kappa_n}^2(1)-\kappa_n(6)+(-16)=0 \)
or\({\kappa_n}^2-5\kappa_n-16=0 \)
or\(\kappa_n=8 \) and \(\kappa_n=-2 \)
This completes the solution.
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