One of the basic problem in geometry is to determine geometric quantities exactly to distinguish from one another. For example, line segments are uniquely determined by their lengths, circles by their radii, triangles by side-angle-side axiom etc. We will now see that a space curve is also uniquely determined by two scalar quantities: called curvature and torsion
जसरी- दुईवटा line segment बराबर छ वा छैन थाहा पाउन line segment को lengths चाहिन्छ ( line segments are uniquely determined by their lengths),
- दुईवटा circles बराबर छ वा छैन थाहा पाउन circles को radius चाहिन्छ (circles are uniquely determined by their radii),
- दुईवटा triangle बराबर छ वा छैन थाहा पाउन side-angle-side को magnitude चाहिन्छ ( triangles are uniquely determined by their side-angle-side axiom)
दुईवटा space curve बराबर छ वा छैन थाहा पाउन curvature र torsion चाहिन्छ । Space curve are uniquely determined by two scalar quantities, called curvature and torsion
Curvature
Let
The magnitude of curvature is denoted by
where
More precisely, the curvature is
- Rate of change in the direction of tangent line
- Degree of curving along tangent line
- Scalar measure of bending nature of the curve along tangent direction
- Change in principal normal along tangent direction
Note
- Straight line has zero curvature
- A circle has constant curvature
- The curvature of small circle is large and vice versa
Expression of Curvature
Let
arcAP=s, arcAQ=s+𝛿s
Then
arcPQ=𝛿s
Also, we denote
where
Now, from isosceles triangle ∆QMN, we have
or
or
Taking limit as
or
Hence, curvature at P is
Since
This is the required expression of curvature
NOTE
The vector
The curvature is tendency of a curve to change their direction from point to point along tangent line. Thus along a curve which has rapidly changing tangent direction with respect to arc length, such as a circle with small radius, the curvature is relatively large, or equivalently, the radius of curvature is small.
Torsion
As the curvature measures deviation of the curve from lying along a straight line, the torsion measures deviation of the curve from lying in a plane. The precise definition is given below.
Let
The magnitude of torsion is denoted by
where
More precisely, the torsion is
- Rate of change in the direction of binormal line
- Degree of curving along binormal line
- Signed scalar measure of bending nature of the curve along binormal direction
- Change in principal normal along binormal direction
Note
- Plane curve has zero torsion
- Circle has zero torsion.
- A circular helix has constant torsion
- Torsion at any point on the curve is signed scalar
Expression of Torsion
Let
Then
Also, we denote
Now, from isosceles triangle QMN, we have
or
or
Taking limit as
or
Hence, torsion at P is
Since
This is the required expression of torsion. NOTE
The vector
The reciprocal of torsion is denoted by
The torsionis tendency of a curve to change their direction from point to point along principal normal line. Thus it measures the bending along a curve which has changing principal normal direction with respect to arc length
Expression of Screw-Curvature:
By the definition of fundamental unit vectors, we have
Differentiation w. r. to. s, we get
which is the required expression of screw curvature
Serret-frenet formula.
Given a space curveThe Serret Frenet formula is a matrix equation representing the fundamental unit vectors along with their derivatives, which give the derivatives of the trihedron vectors:
In summary, it is written as
It is to note that the coefficient matrix appearing on the right of (i) is skew-symmetric, also it is notifying that the curvature
Summary Table
Fundamental Vector | Expression | Magnitude |
Curvature Change in tangent line |
| |
Torsion Change in binormal line | | |
Screw-curvature Change in principal normal line | | |
Theorems
- A necessary and sufficient condition for a curve to be a straight line is that curvature
at all points on the curve.
Proof
Let be a space curve.
Then, necessary and sufficient condition for a curve to be a straight line is that, the position of arbitrary point on C is expressed as
...(i)
where and are constant vectors
⟺
⟺
⟺
⟺ at all points of the curve
This completes the proof
-
A necessary and sufficient condition for a curve to be a plane curve is that
at all points on the plane curve.
Proof
Let be a space curve.
Then, necessary and sufficient condition for a curve to be a straight line is that, binormal tvector is constant
⟺
⟺
Also
or
or
or
It shows that projection of arbitrary point of the curve on binormal is constant and this implies that the curve is a plane curve.
This completes the proof
-
Prove that
and .
Solution
Given that
...(i)
and
...(ii)
and
or ...(iii)
To find the curvature, we operate cross product between (i) and (ii), then
or
Taking magnitude, we get
...(A)
which is the required curvature
To find the torsion, we operate scalar triple product among (i),(ii) and (iii), then
or
or
or from (A)
which is the required torsion
-
Show that
and
Solution
Given that
or
or with ...(i)
and
...(ii)
and
or ...(iii)
To find the curvature, we operate cross product between (i) and (ii) then
or
or ...(A)
or since from (ii)
which is the required curvature
To find the torsion, we operate scalar triple product among (i), (ii), and (iii), then
or
or
or since from (A)
-
A necessary and sufficient condition for a curve to be a plane curve is that
at all points.
Proof
Let be a space curve.
Then, necessary and sufficient condition for a curve to be a plane curve is that
at all points of the curve
or at all points of the curve
or at all points of the curve
This completes the proof.
-
A necessary and sufficient condition for a curve to be a plane curve is that
at all points.
Proof
Let be a space curve.
Then, necessary and sufficient condition for a curve to be a plane curve is that
at all points of the curve
or at all points on the curve
or at all points on the curve
This completes the proof.
-
Show that principal normal at consecutive points on a curve do not intersect unless
Solution
Let be a space curve, also we consider two consecutive points P and Q on C with positions vectors
Let be unit vectors along principal normal at P and Q respectively, then these principal normals will intersect if and only if
are coplanar
or
or
or
or
or
or
Hence, principal normal at consecutive points do not intersect unless
-
If tangent and binormal on a space curve makes angles
and respectively with a fixed direction, then show that
Solution
Let a space curve on which and denote the unit vectors along tangent and binormal respectively.
Assume that, fixed direction (say ) make angles and with and respectively, then
Again
Dividing the equations , we get
-
If there is one-one correspondence between the points of two curves and tangents at corresponding points on these curves are parallel, show that the principal normal at these points are parallel and, therefore the binormal, also prove that
Solution
Let C and C1 be two curves such there is one-one correspondence between their points and the tangents at these corresponding points are parallel.
[we use suffix-unity to distinguish quantities belonging to curve c1]
Differentiating (i) w.r.to. s, we get
Taking magnitude we get
...(A)
Substituting the value of from (i) we get that
Thus principal normals at corresponding points of C and C1 are parallel.
1st relation established
Now taking cross product, we get
...(i)
Hence binormals at corresponding points of the curves C and are also parallel.
2nd relation established
Again differentiating (i) w. r. to. s, we get
Taking magnitude we get
...(B)
Hence equating (A) and (B) we get
3rd relation established
Hence, if tangents at corresponding points on the curves are parallel, then principal normals are parallel and, therefore, the binormals along with the given relation.
Exercise
-
Find the curvature and torsion of a following curve
-
Solution
Given space curve is
By successive differentiation w. r. to. t, we get
Since, curvature and torsion of space curve is given by the formula
Thus, taking magnitude of (i), we get
or
Taking cross product between (ii) and (iii), we get
or
Taking magnitude, we get
Operating scalar triple product among (i) ,(ii) (iii), we get
or
or
Hence, curvature and torsion of the curve are
or
or
and
or
or
This completes the solution.
Solution
Given space curve is
By successive differentiation w. r. to. t, we get
Since, curvature and torsion of space curve is given by the formula
Thus, taking magnitude of (i), we get
or
Taking cross product between (ii) and (iii), we get
or
or
or
Taking magnitude, we get
or
Operating scalar triple product among (i), (ii), (iii), we get
or
or
or
Hence, curvature and torsion of the curve are
or
or
and
or
or
This completes the solution.
Solution
Given space curve is
By successive differentiation w. r. to. t, we get
Since, curvature and torsion of space curve is given by the formula
Thus, taking magnitude of (i), we get
or
or
or
Taking cross product between (ii) and (iii), we get
or
or
Taking magnitude, we get
or
or
or
or
Operating scalar triple product among (i), (ii), (iii), we get
or
or
or
or
Hence, curvature and torsion of the curve are
or
or
or
or
and
or
or
or
or
This completes the solution.
Solution
Given space curve is
By successive differentiation w. r. to. t, we get
Since, curvature and torsion of space curve is given by the formula
Thus, taking magnitude of (i), we get
or
or
Taking cross product between (ii) and (iii), we get
or
or
Taking magnitude, we get
or
or
or
Operating scalar triple product among (i), (ii), (iii), we get
or
or
or
or
Hence, curvature and torsion of the curve are
or
or
and
or
or
or
This completes the solution.
Solution
Given space curve is
By successive differentiation w. r. to. t, we get
Since, curvature and torsion of space curve is given by the formula
Thus, taking magnitude of (i), we get
or
Taking cross product between (ii) and (iii), we get
or
or
Taking magnitude, we get
Operating scalar triple product among (i) ,(ii) (iii), we get
or
or
or
Hence, curvature and torsion of the curve are
or
or
and
or
or
This completes the solution.
Solution
Given space curve is
By successive differentiation w. r. to. , we get
Since, curvature and torsion of space curve is given by the formula
Thus, taking magnitude of (i), we get
or
Taking cross product between (ii) and (iii), we get
or
or
Taking magnitude, we get
Operating scalar triple product among (i) ,(ii) (iii), we get
or
or
or
Hence, curvature and torsion of the curve are
or
or
and
or
or
This completes the solution.
Solution
Given space curve is
By successive differentiation w. r. to. t, we get
Since, curvature and torsion of space curve is given by the formula
Thus, taking magnitude of (i), we get
or
Taking cross product between (ii) and (iii), we get
or
or
Taking magnitude, we get
Operating scalar triple product among (i) ,(ii) (iii), we get
or
or
or
Hence, curvature and torsion of the curve are
or
or
and
or
or
This completes the solution.-
Solution
Given space curve is
By successive differentiation w. r. to. t, we get
Since, curvature and torsion of space curve is given by the formula
Thus, taking magnitude of (i), we get
or
or
Taking cross product between (ii) and (iii), we get
or
Taking magnitude, we get
Operating scalar triple product among (i) ,(ii) (iii), we get
or
Hence, curvature and torsion of the curve are
or
or
and
or
or
This completes the solution.
-
- For a space curve
show that-
Solution
We know that:
By successive differentiation w.r.t. , we get:
and
and
Next,
This simplifies to:
This completes the proof. -
-
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