In differential geometry, an intrinsic equation describes a curve or a surface based solely on properties that are inherent to the shape itself, independent of the surrounding space. These properties are often expressed in terms of quantities like arc length, curvature, and torsion. For curves, an intrinsic equation can be given using the arc length \( s\) as as the parameter, and it typically involves the curvature 𝜅(𝑠) and the torsion 𝜏(𝑠). The Frenet-Serret formulas are a classical example of intrinsic equations for curves in three-dimensional space, defined as:
\( \vec{t}'=\kappa \vec{n}\)
\( \vec{b}'=-\tau \vec{n}\)
\( \vec{n}'=\tau \vec{b}-\kappa \vec{t}\)
For surfaces, intrinsic equations often involve Gaussian curvature 𝐾 and mean curvature \(\mu\). An intrinsic characterization of a surface is provided by the Gauss-Codazzi equations, which relate these curvatures to the intrinsic metric of the surface.
Fundamental TheoremIf two single-valued continuous functions \( \kappa(s) \) (curvature) and \( \tau(s) \) (torsion) are given for \( s>0 \), then there exists exactly one space curve, determined except for orientation and position in space (i.e., up to a Euclidean motion), where s is the arc length, \( \kappa \) is a is the curvature, and \( \tau \) is the torsion. This fundamental theorem is divided in two parts
- Existence theorem (this theorem ensures existence of space curve)
If \( \kappa =f( s ),\tau =g( s ) \) are continuous functions of a real variable \( s ( s\geq 0 ) \) then there exists a space curve for which \( \kappa \) is curvature, \( \tau\) is torsion and \( s \) is the arc length measured from some suitable base point. - Uniqueness theorem (this theorem ensures uniqueness of space curve)
A curve is unequally determined, except as to the position in space, when its curvature and torsion are given functions of its arc length s (or two curves with same intrinsic equations are necessarily congruent)
Existence theorem
If \( \kappa =f( s ),\tau =g( s ) \) are continuous functions of a real variable \( s ( s\geq 0 ) \) then there exists a space curve for which \( \kappa \) is curvature, \( \tau\) is torsion and \( s \) is the arc length measured from some suitable base point.
Proof
The linear differential equations
\( \frac{dx}{ds}=\kappa y\)
\( \frac{dy}{ds}=\tau z-\kappa x\)
\( \frac{dz}{ds}=-\tau y\)
which is taken as
\( \vec{t}'=\kappa \vec{n}\)
\( \vec{n}'=\tau \vec{b}-\kappa \vec{t}\)
\(\vec{b}'=-\tau \vec{n}\)
admits a unique set of solutions for
\( ( x_1,y_1,z_1 ), ( x_2,y_2,z_2 )\) and \( ( {x_3},y_3,z_3 )\)
taking values
\( ( 1,0,0 ),( 0,1,0 ) \) and \( ( 0,0,1 )\) respectively at \( s=0\). Say.
Now
\( \frac{d}{ds}( x_1^2+y_1^2+z_1^2 )=2x_1\frac{dx_1}{ds}+2y_1\frac{dy_1}{ds}+2z_1\frac{dz_1}{ds}\)
or \( \frac{d}{ds}( x_1^2+y_1^2+z_1^2 )=2( \kappa x_1y_1+\tau y_1z_1-\kappa x_1y_1-\tau y_1z_1 )\)
or \( \frac{d}{ds}( x_1^2+y_1^2+z_1^2 ) =0\)
Hence,
\( x_1^2+y_1^2+z_1^2=\) constant
At \( s=0\), the value of \( x_1^2+y_1^2+z_1^2\) is
\( x_1^2+y_1^2+z_1^2={1^2}+{0^2}+{0^2}\)
or \( x_1^2+y_1^2+z_1^2=1\)
Hence,
value of \( x_1^2+y_1^2+z_1^2\) is 1 for all values of s (1)
Similarly, we can show that
value of \( x_2^2+y_2^2+z_2^2\) is 1 for all values of s (2)
value of \( x_3^2+y_3^2+z_3^2\) is 1 for all values of s (3)
Again
\( \frac{d}{ds}( x_1x_2+y_1y_2+z_1z_2 )=(x_1\frac{dx_2}{ds}+y_1\frac{dy_2}{ds}+z_1\frac{dz_2}{ds}+x_2\frac{dx_1}{ds}+y_2\frac{dy_1}{ds}+z_2\frac{dz_1}{ds} )\)
or \( \frac{d}{ds}( x_1x_2+y_1y_2+z_1z_2 )=x_1\kappa x_2+y_1( \tau z_2-\kappa x_2 )-z_1\tau y_2+x_2\kappa x_1 +y_2( \tau z_1-\kappa x_1 )-z_2\tau y_1=0 \)
Hence
\( x_1x_2+y_1y_2+z_1z_2=\) constant
At \( s=0\), the value of \( x_1x_2+y_1y_2+z_1z_2\) is
\( x_1x_2+y_1y_2+z_1z_2= 1.0+0.1+0.0\)
or \( x_1x_2+y_1y_2+z_1z_2= 0\)
Hence,
value of \( x_1x_2+y_1y_2+z_1z_2\) is 0 for all values of s (4)
Similarly, we can show that
value of \( x_1x_3+y_1y_3+z_1z_3=0\) is 0 for all values of s (5)
value of \(x_2x_3+y_2y_3+z_2z_3=0\) is 0 for all values of s (6)
Now,
Six equations (1-6) an be represented in matrix form as
\( \begin{pmatrix} x_1 & y_1 & z_1 \\ x_2 & y_2 & z_2 \\ {x_3} & y_3 & z_3 \end{pmatrix} \begin{pmatrix} x_1 & x_2 & {x_3} \\ y_1 & y_2 & y_3 \\ z_1 & z_2 & z_3 \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \)
Transposing the left matrices, the equation is
\( \begin{pmatrix} x_1 & x_2 & {x_3} \\ y_1 & y_2 & y_3 \\ z_1 & z_2 & z_3 \end{pmatrix} \begin{pmatrix} x_1 & y_1 & z_1 \\ x_2 & y_2 & z_2 \\ {x_3} & y_3 & z_3 \end{pmatrix} =\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \)
The above matrix equation gives
\( x_1^2+x_2^2+x_3^2=1 \)
\( y_1^2+y_2^2+y_3^2=1 \)
\( z_1^2+z_2^2+z_3^2=1 \)
and
\( x_1y_1+x_2y_2+x_3 y_3=0 \)
\( x_1z_1+x_2z_2+{x_3}z_3=0 \)
\( y_1z_1+y_2z_2+y_3z_3=0 \)
The equation given by (7) and (8) gives,
\( \vec{t}=x=( x_1,x_2,{x_3} )\)
\( \vec{n}=y=( y_1,y_2,y_3 )\)
\( \vec{b}=z=( z_1,z_2,z_3 )\)
which are defined for each values of \( s\), and satisfying
\( \vec{t}.\vec{t}=\vec{n}.\vec{n}=\vec{b}.\vec{b}=1\) and
\( \vec{t}.\vec{n}=\vec{n}.\vec{b}=\vec{b}.\vec{t}=0\)
Now, there exists a space curve \( S: \vec{r}=\vec{r}(s) \) for which
\( \vec{r}=\vec{r}( s )= \displaystyle \int_0^s \vec{t}( s )ds \) (A)
For which
\( \vec{r}'=\vec{t}\)
\( \vec{t}’=\frac{d\vec{t}}{ds}=\frac{dx}{ds}=\kappa y=\kappa \vec{n}\)
\( \vec{b}’=\frac{d\vec{b}}{ds}=\frac{dz}{ds}=-\tau y=-\tau \vec{n}\) and
\( \vec{n}’=\frac{d\vec{n}}{ds}=\frac{dy}{ds}=\tau z-\kappa x=\tau \vec{b}-\kappa \vec{t}\)
This completes the proof.
Uniqueness theoremA curve is unequally determined, except as to the position in space, when its curvature and torsion are given functions of its arc length s (or two curves with same intrinsic equations are necessarily congruent) Proof
Let \( C\) and \( {C_1}\) are two space curves whose curvature and torsions at corresponding points are equal. If \( {C_1}\) be moved without deformation so that corresponding points \( P\) and \( P_1\) on \( C\) and \( C_1\) coincide at \( s=0\). Also \( C_1 \) is suitably oriented so that two triads \( \vec{t},\vec{n},\vec{b}\) and \( \vec{t}_1,\vec{n}_1,\vec{b}_1 \) coincide at \( s=0\).
Thus,
\( \vec{t}=\vec{t}_1,\vec{n}=\vec{n}_1,\vec{b}=\vec{b}_1\) at \( s=0\).
Now,
\( \frac{d}{ds} ( \vec{t}\vec{t}_1+\vec{n}\vec{n}_1+\vec{b}\vec{b}_1 )=\vec{t}' \vec{t}_1+ \vec{n}' \vec{n}_1+\vec{b}' \vec{b}_1+\vec{t} \vec{t}_1 '+\vec{n} \vec{n}_1'+\vec{b} \vec{b}'_1 \)
or \( \frac{d}{ds}( \vec{t}\vec{t}_1+\vec{n}\vec{n}_1+\vec{b}\vec{b}_1 )=0\)
Thus,
\( \vec{t}\vec{t}_1+\vec{n}\vec{n}_1+\vec{b}\vec{b}_1=\) constant
At \(s=0\), the value of \( \vec{t}\vec{t}_1+\vec{n}\vec{n}_1+\vec{b}\vec{b}_1=3\)
At \(s=0\), \( \vec{t}=\vec{t}_1 \) भएकोले \( \vec{t}.\vec{t}_1=1 \) यसै गरि \( \vec{n}.\vec{n}_1=1,\vec{b}.\vec{b}_1=1 \)
Hence,
value of \( \vec{t}\vec{t}_1+\vec{n}\vec{n}_1+\vec{b}\vec{b}_1=3\) for all \( s \).
Now,
If \( \alpha ,\beta ,\gamma \) are angles between triads \( \vec{t},\vec{n},\vec{b}\) and \( \vec{t}_1,\vec{n}_1,\vec{b}_1\) Then
\( cos\alpha +cos\beta +cos\gamma =3\) for all \(s \)
This shows that
\( \vec{t}=\vec{t}_1,\vec{n}=\vec{n}_1,\vec{b}=\vec{b}_1\)
or \( \vec{t}-\vec{t}_1=0 \)
or \( \frac{d}{ds}( \vec{r}-\vec{r}_1)=0\)
It shows that \( ( \vec{r}-\vec{r}_1) \) is constant,
At \( s=0\), value of \( ( \vec{r}-\vec{r}_1) \) is 0.
Thus,
value of \( ( \vec{r}-\vec{r}_1=0) \) for all \( s \)
or value of \( \vec{r}=\vec{r}_1 \) for all \( s \)
So, curves \( C\) and \( C_1\) are identical except their positions in space.
This completes the proof.
ExplanationIf two single-valued continuous functions \( \kappa(s) \) (curvature) and \( \tau(s) \) (torsion) are given for \( s>0 \), then there exists exactly one space curve, determined except for orientation and position in space (i.e., up to a Euclidean motion), where s is the arc length, \( \kappa \) is a is the curvature, and \( \tau \) is the torsion.
-
Existence theorem (this theorem ensures existence of space curve)
If \( \kappa =f( s ),\tau =g( s ) \) are continuous functions of a real variable \( s ( s\geq 0 ) \) then there exists a space curve for which \( \kappa \) is curvature, \( \tau\) is torsion and \( s \) is the arc length measured from some suitable base point. This existence theorem deals that for a continuous functions \( \kappa =f( s ),\tau =g( s ) \), there exists- three unit vectors (3 entities)
\( \vec{t}=x=( x_1,x_2,{x_3} )\) where \( \vec{t}. \vec{t} =1 \)
\( \vec{n}=y=( y_1,y_2,y_3 )\) where \( \vec{n}. \vec{n} =1 \)
\( \vec{b}=z=( z_1,z_2,z_3 )\) where \( \vec{b}. \vec{b} =1 \)
- these three vectors satisfy orthogonal traids (3 entities)
\( \vec{t}.\vec{n}=0 \)
\(\vec{n}.\vec{b}=0 \)
\( \vec{b}.\vec{t}=0\)
- these three vectors also satisfy expression of curvature and torsion (3 entities)
\( \vec{t}’=\kappa \vec{n}\)
\( \vec{b}’=-\tau \vec{n}\) and
\( \vec{n}’=\tau \vec{b}-\kappa \vec{t}\)
where
\( \vec{r}'=\vec{t}\) (1 entitiy)
provided the definition of space curve
\( \vec{r}( s )= \displaystyle \int_0^s \vec{t}( s )ds \)
Since this theorem deals about all 10 fundatal entities of space curve, it is called fundamental theorem.
- three unit vectors (3 entities)
- Uniqueness theorem (this theorem ensures uniqueness of space curve)
A curve is unequally determined, except as to the position in space, when its curvature and torsion are given functions of its arc length s (or two curves with same intrinsic equations are necessarily congruent)
In addition, uniqueness theorem ensures that existence of two space curves are identical apart from position in space if they have identical curvature and torsion functions at corresponding points, therefore, it is also part of fundamental theorem.
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