Fundamentals on Space Curve

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In differential geometry of curves in space, three vectors are called fundamental vectors. These three vectors are
  1. tangent vector
  2. principal normal vector, and
  3. binormal vector
What is tangent Vector
Let \( C:\vec{r}=\vec{r}( s ) \) be a space curve and P be a point on it.
Also, let Q be a neighboring point on C.
Then tangent line at P is define as limiting position of a secant line PQ when \( Q \to P \)

For three dimensional curves in space, there is unique tangent line at each of its points.

What is Normal Vector

Let \(C:\vec{r}=\vec{r}(s) \) be a space curve and P be a point on it.
Then normal vector at P is a vector perpendicular to the tangent at P. For three dimensional curves in space, there are infinitely many normal vectors at a given point.


What is Principal Normal Vector
Let \( C: \vec{r}=\vec{r} (s) \) be a space curve and P be a point on it.
Then principal normal at P is a normal vector lying in the osculating plane at P. We denote unit vector along this principal normal by \( \vec{n} \).
In the figure below,
we see that principal normal lies in the osculating plane at P.
Three dimensional space मा infinitely many normal lines हरु हुन्छन। जसमध्ये osculating plane मा पर्ने normal line लाई principal normal भनिन्छ ।
principal normal को unit vector लाई \( \vec{n} \) ले जनाईन्छ ।
What is Binormal Vector
Let \( C: \vec{r}=\vec{r} (s) \) be a space curve and P be a given point on it.
Then binormal at P is a normal perpendicular to the osculating plane at P.
We denote unit vector along the binormal line by \( \vec{b} \).
In the figure below,
we see that binormal is also perpendicular to both tangent and principal normal at P.
Fundamental Vectors
Let \( C: \vec{r}=\vec{r} (s) \) be a space curve and P be a point on it.
Then unit vectors \( \vec{t} \) ,\( \vec{n} \) and \( \vec{b} \) which exists at each point on three dimensional space curve are called fundamental vectors.
  1. These vectors moves along the curve, so are called moving trihedrons.
  2. These vectors are mutually perpendicular, so are also called orthogonal triad.
The three fundamental vectors move in a space curve along positive directions and satisfies the relations.
  1. \( \vec{t} . \vec{t} =1, \vec{n} . \vec{n} =1, \vec{b} . \vec{b} =1 \)
  2. \( \vec{t} \times \vec{t} =0, \vec{n} \times \vec{n} =0, \vec{b} \times \vec{b} \)=0
  3. \( \vec{t} . \vec{n} =0, \vec{n} . \vec{b} =0, \vec{b} . \vec{t} =0 \)
  4. \( \vec{t} \times \vec{n} = \vec{b} , \vec{n} \times \vec{b} = \vec{t} , \vec{b} \times \vec{t} = \vec{n} \)
Three vectors vectors: tangent \( \vec{t} \) , principal normal \( \vec{n} \), and binormal \( \vec{b} \) are called fundamental vectors.
The equation of three lines associated to these unit vectors are given below.
  1. Tangent line: \( R=\vec{r}+\lambda \vec{t} \)
  2. Principal normal line:\( R=\vec{r}+\lambda \vec{n} \)
  3. Binormal line: \( R=\vec{r}+\lambda \vec{b} \)
Equation of Tangent Vector
Let \( C: \vec{r}=\vec{r} (s) \) be a space curve and P be a point on it with position vector,
\( \vec{OP} =\vec{r} \)
Since, \( \vec{n} \) is unit vector along tangent at P, we suppose R be the position of arbitrary point T on this tangent. Here position vector of T is
\( \vec{OT} =R \)
Then, equation of tangent at P is
\( \vec{OT} = \vec{OP}+ \vec{PT} \)
or \( R=\vec{r}+\lambda \vec{t} \)
where \( \lambda \) is scalar.
Equation of Principal Normal Vector
Let \( C: \vec{r}=\vec{r} (s) \) be a space curve and P be a point on it with position vector,
\( \vec{OP} =\vec{r} \)
Since, \( \vec{n} \) is unit vector along principal normal at P, we suppose R be the position of arbitrary point T on this principal normal. Here position vector of T is
\( \vec{OT} =R \)
Then, equation of principal normal at P is
\( \vec{OT} = \vec{OP}+ \vec{PT} \)
or \( R=\vec{r}+\lambda \vec{n} \)
where \( \lambda \) is scalar.
Equation of Binormal Vector
Let \( C: \vec{r}=\vec{r} (s) \) be a space curve and P be a point on it with position vector,
\( \vec{OP} =\vec{r} \)
Since, \( \vec{n} \) is unit vector along principal normal at P, we suppose R be the position of arbitrary point T on binormal vector. Here position vector of T is
\( \vec{OT} =R \)
Then, equation of binormal at P is
\( \vec{OT} = \vec{OP}+ \vec{PT} \)
or \( R=\vec{r}+\lambda \vec{b} \)
where \( \lambda \) is scalar.
Fundamental Planes
  1. Osculating Plane

    Let \( C: \vec{r}=\vec{r} (s) \) be a space curve and P be a point on it, then osculating plane at P is a plane passing through P and perpendicular to the binormal normal at P.
    It is the plane spanned by tangent and principal normal at P.

  2. Normal Plane

    Let \( C: \vec{r}=\vec{r} (s) \) be a space curve and P be a point on it, then normal plane at P is a plane generated by all normal lines at P.
    It is a plane perpendicular to the tangent line at P. This plane is spanned by principal normal and binormal at P.
    In the figure alongside, normal plane is shaded.

  3. Rectifying Plane

    Let \( C: \vec{r}=\vec{r} (s) \) be a space curve and P be a point on it, then rectifying plane at P is a plane passing through P and perpendicular to the principal normal at P.
    It is the plane spanned by tangent and binormal at P.
    In the figure alongside, rectifying plane is shaded.

Three planes: Osculating plane, Normal plane and Rectifying plane associated are called fundamental planes. The equation of these three planes are given below.

  1. Osculating plane: \( (R-\vec{r}). \vec{b} =0 \)
  2. Normal plane: \( (R-\vec{r}). \vec{t} =0 \)
  3. Rectifying plane: \( (R-\vec{r}). \vec{n} =0 \)
Equation of Normal Plane
Let \( C: \vec{r}=\vec{r} (s) \) be a space curve and P be a point on it with position vector.
\( \vec{OP} =\vec{r} \)
Since, \( \vec{t} \) ,\( \vec{n} \) and \( \vec{b} \) are the unit vectors along tangent, principal normal and binormal at P respectively, we suppose R be the position vector of arbitrary point T on the normal plane at P then
\( \vec{OT} =R \)
Here,
\( \vec{PT} \) lies in the normal plane
Since normal plane is perpendicular to tangent,
\( \vec{PT} \) is perpendicular to the tangent
or \( (\vec{OT}-\vec{OP} ) \) is perpendicular to the tangent
or \( R-\vec{r} \) is perpendicular to the tangent
or \( (R-\vec{r})\). \( \vec{t} \) =0
This is required equation of normal plane.
Equation of Rectifying Plane
Let \( C: \vec{r}=\vec{r} (s) \) be a space curve and P be a point on it with position vector.
\( \vec{OP} =\vec{r} \)
Since, \( \vec{t} \) ,\( \vec{n} \) and \( \vec{b} \) are the unit vectors along tangent, principal normal and binormal at P respectively, we suppose R be the position vector of arbitrary point T on the rectifying plane at P then
\( \vec{OT} =R \)
Here,
\( \vec{PT} \) lies in the rectifying plane
Since rectifying plane is perpendicular to principal normal,
\( \vec{PT} \) is perpendicular to the principal normal
or \( (\vec{OT}-\vec{OP} ) \) is perpendicular to the principal normal
or \( R-\vec{r} \) is perpendicular to the principal normal
or \( (R-\vec{r})\). \( \vec{n} \) =0
This is required equation of rectifying plane.
Equation of Osculating Plane
Let \( C: \vec{r}=\vec{r} (s) \) be a space curve and P be a point on it with position vector.
\( \vec{OP} =\vec{r} \)
Since, \( \vec{t} \) ,\( \vec{n} \) and \( \vec{b} \) are the unit vectors along tangent, principal normal and binormal at P respectively, we suppose R be the position vector of arbitrary point T on the osculating plane at P then
\( \vec{OT} =R \)
Here,
\( \vec{PT} \) lies in the osculating plane
Since rectifying plane is perpendicular to binormal,
\( \vec{PT} \) is perpendicular to the binormal
or \( (\vec{OT}-\vec{OP} ) \) is perpendicular to the binormal
or \( R-\vec{r} \) is perpendicular to the binormal
or \( (R-\vec{r})\). \( \vec{b} \) =0
This is required equation of osculating plane.

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