Bertrand curves and its properties
A pair of space curves
Principal Noemals are Same
Principal Noemals are Same
Properties of Bertrand curves
Let
- Distance between corresponding points of Bertrand curves is constant
- Tangent at corresponding points of Bertrand curves inclined at constant angle
- Curvature and torsion of either curves are connected by linear relation
- Torsion of Bertrand curves have same sign and their product is constant
Let
-
If
is distance between corresponding points of Bertrand curves, then - if
is tangent to and is tangent to , - if
and are curvature and torsion of then
; and are connected by linear relation
if and are curvature and torsion of then
; and are connected by linear relation - if
and are torsion of and then and have same sign, means and
Proof
- Distance between corresponding points of Bertrand curves is constant
, 7+2*Math.sin(t)];}; function (t) { return [ 7+5*Math.cos(t), 7+5*Math.sin(t)];}; board.create('curve',[r, 0, Math.PI],{strokeColor:'red'}); board.create('functiongraph',[r1, 0, Math.PI],{strokeColor:'red'}); Let0,0OP and are Bertrand curves then
(i)
Let be a point on and be its corresponding on with
Then,
is principal normal to C
or
where is distance between corresponding points and
or
or
or (ii)
Differentiating (ii) w. r. to. s , we get
or
or
Taking dot product by , we get
or
or
It shows that, distance between corresponding points of Bertrand curves is constant
The first property established.
- Tangent at corresponding points of Bertrand curves inclined at constant angle
Let is tangent to and is tangent to ,
and also let, is angle between and , then
or
Differentiating w. r. to s, we get
or
or
In Bertrand curves, , so
Replacing by and also replacing by in the right, we get
or
or
It shows that,
derivative of is zero
so
is constant
or is constant
Hence, yangent at corresponding points of Bertrand curves inclined at constant angle
The second property established.
- Curvature and torsion of either curves are connected by linear relation
Let and are Bertrand curves then using (A), we write
or (iii)
Also, we know that,0,0az = 1.00el = 0.30rotate = 2.00t_1b_1tbn=n_1- principal normals to
and are same - tangents to
and are inclined at constant angle - binormals to
and also inclined at same constant angle
Hence,
or (iv)
Comparing (iii) and (iv), we get
We have to show relation between and , so
Taking, second and third part, we get
or
or
This shows that and are connected by linear relation in the form
where are constants given by
Next,
Since relation between and are reverse in terms of and we have
will be and will be for
Thus, while writting expression for and , we get
or
This shows that and are connected by linear relation in the form
where are constants given by
The third property established.
- principal normals to
-
Torsion of Bertrand curves have same sign and their product is constant
Let and are Bertrand curves then using (B), we get
We have to show relation between and , so
Taking, first and third part, we get
(v)
Since relation between and are reverse in terms of and . Thus, while writing expression for , we get
(vi)
Multiplying (v) and (vi), we get
or
or
or
Torsion of Bertrand curves have same sign and their product is constant
The forth property established.
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