Differential calculus Radius of curvature in Cartesian ...
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Differential calculus Radius of curvature in Cartesian coordinates ( ) () Differentiating (1) with respect to , we get ( ) ( ( ) ) ( ) ( ) ( ) ( ())
Transcript of Differential calculus Radius of curvature in Cartesian ...
Differential calculus
Radius of curvature in Cartesian coordinates
(
)
( )
Differentiating (1) with respect to , we get
(
)
(
(
)
)
(
)
(
)
(
)
( ( ))
(
)
( (
)
)
(
)
(
)
( )
( ( ))
( )
(
)
( )
Differentiating (1) with respect to , we get
(
)
⇒ ( )
(
)(
)
( ) (
)
( )
( )
( ) (
) ( ) (
)
( )
(
)
(
)
( ( )
( )) ( ( ) (
)) ( (
) (
)
) ( ( ) ( ) )
( ( )
( ))
(
) ( ) (
) ( )
(
)
(
) ( )( ( ))
(
)
(
)
(
)
(
)
(
)
(
)
( ( ) )
( )
√
√
√
( )
(
)
( )
Differentiating (1) with respect to , we get
(
)( )
( )
( )
(
)
(
)
( )
(( )
(
)
( ) (
)
(
)
)
( )
(
)
( )
(
)
( ( ) )
√
√
( )
( )
Differentiating (1) with respect to , we get
(
)
( ( ) )
(
)
( ( )
)
( )
(
)
( )
( )
( )
( )
( )
( )
( )
( )
Differentiating (1) with respect to , we get
(
)( )
(
)
( )
( )
(
)
( ( )
)
(
)
( )
(
)
( ) ( )
(
)
(
)
( )
(
)
(
)
( )
Differentiating (1) with respect to , we get
√
√
√
√
√
√
√
√
√
√ [
(
)
]
√
√
[ (
(
√
√
)
)
]
√
√
[ (
(
√
√ )
√
√ )
]
√
√ [(√ √ )
√ ]
(
)
( ( √
√
)
)
√
√ *(√ √ )
√ +
( ( ))
√
√ *(√ √ )
√ +
( )
( ) √
√ *(√ √ )
√ +
( )
( ) √
√ *(√ √ )
√ +
( ( ))
( )
Radius of curvature in Parametric form
IF ( ) ( )
(( ) ( ) )
The parametric form is ( )
( )
( )
( )
(( ) ( ) )
(( ) ( ) )
( )( ) ( )( )
( )
( )( )
( )
( )
( )( )
( )( )
( )( )
( )
(
)
(
)
( )
The parametric form is ( )
(( ) ( ) )
(( ) ( ) )
( )( ) ( )( )
( ) ( )
( )
( ) ( )
The parametric form is ( )
(( ) ( ) )
(( ) ( ) )
( )( ) ( )( )
( ) ( )
( )
( )
Centre of Curvature in Cartesian form:
( ( )
)
( ( )
)
Circle of curvature
( ) ( )
Find the circle of the curve ( )
( )
Differentiating (1) with respect to , we get
⇒
( )( )
( )( ) ( )
( ( )
)
( ( ) )
( )
( ( )
)
( ( ) )
( )
The center of curvature is ( )
(
)
( ( ) )
( )
√
The circle of curvature is given by
( ) ( )
( ) ( )
Centre of Curvature in parametric form:
IF ( ) ( )
(( ) ( ) )
(( ) ( ) )
Find the equation of the evolute of the curve
Solution:
The parametric form is ( )
( )
( )
( )
(( ) ( ) )
(( ) ( ) )
( )( ) ( )( )
( )
( )( )
( ) ( )
( )( )
( )
( )
(( ) ( ) )
(( ) ( ) )
( )( ) ( )( )
( )
( )( )
( ) ( )
( )( )
( )
( )
( ) ⇒( )
( ) ( )
( ) ⇒( )
( ) ( )
( ) ( )
( ) ( )
( )
( )
( )
( ) ( )
( )
The locus of ( ) the evolute of the given curve
( ) ( )
Find the equation of the evolute of the curve
Solution:
The parametric form is ( )
(( ) ( ) )
(( ) ( ) )
( )( ) ( )( )
(( ) ( ))
( )
( )
⇒
⇒ (
)
( )
(( ) ( ) )
(( ) ( ) )
( )( ) ( )( )
(( ) ( ))
( )
( )
⇒
⇒ (
)
( )
The center of curvature is ( )
From (2) and (3), we get
(
)
(
)
Raising the powers with 6 on both sides, we get
(
)
(
)
⇒
( )
( )
The locus of ( ) the evolute of the given curve
( )
Find the equation of the evolute of the curve
Solution:
The parametric form is ( )
(( ) ( ) )
(( ) ( ) )
( )( ) ( )( )
(( ) ( ))
( )
( )
⇒
⇒ (
)
( )
(( ) ( ) )
(( ) ( ) )
( )( ) ( )( )
(( ) ( ))
( )
( )
⇒
⇒ (
)
( )
The center of curvature is ( )
From (2) and (3), we get
(
)
(
)
Raising the powers with 6 on both sides, we get
(
)
(
)
⇒
( )
( )
The locus of ( ) the evolute of the given curve
( )
Find the equation of the evolute of the curve
Solution:
( )
(( ) ( ) )
(
) (( ) (
)
)
( ) ( ) (
) ( )
(
) ( )
( ) ( )
(
)
(
)
(( ) ( ) )
( ) (( ) ( )
)
( ) ( ) (
) ( )
( )
(
)
The center of curvature is (
(
)
(
) )
(
)
(
)
(
)
( ) (
)
(
) ( )
(
)
(
)
(
)
( ) (
)
(
) ( )
( ) ( )
( ) ( )
(
)
(
)
(
)
(
)
( ) ( )
(
)
*(
)
(
)
+ (
)
(
(
))
( ) ( )
(
)
( ) ( )
( )
The locus of ( ) the evolute of the given curve
( ) ( )
( )
Solution:
The parametric form is ( )
(( ) ( ) )
(( ) ( ) )
( )( ) ( )( )
( )
( )
( )
( )
( ( ) )
( )
( )
( ) ⇒
⇒(
)
( )
(( ) ( ) )
(( ) ( ) )
( )( ) ( )( )
( )
( )
( )
( )
( ( ))
( )
( ) ⇒
⇒(
)
( )
( ) ( )
(
)
(
)
( ) ( )
( )
( )
The locus of ( ) the evolute of the given curve
( ) ( )
( )
( ) ( )
Solution:
( ) ( ) ( )
( )
(( ) ( ) )
( ) (( ( ))
( ) )
( )( ) ( )( )
( ) ( )
( )
( )
( )
( )
(( ) ( ) )
( ) ( ) (( ( ))
( ) )
( )( ) ( )( )
( ) ( )( )
( ) ( )( )
( ) ( )( )
( ) ( )( )
( ) ( )
( ) ( )
The locus of ( ) ( ) the evolute of the given curve
( )
Envelope
Solution:
( )
( )
Differentiating (2) with respect to , we get
⇒
⇒
⇒
⇒
( )
( )
√
( )
√( ) ( )
( )
√
( )
√( ) ( )
( )
Substituting (3) and (4) in (1), we get
( )
√( ) ( )
( )
√( ) ( )
( ) √( )
( )
( )
√( )
( )
(( ) ( )
)√( )
( )
(( ) ( )
)
⇒ ( ) ( )
( )
( ) ( )
( )
Solution:
( )
⇒ ( )
Substituting (2) in (1), we get
( ) ( ) ⇒ ( )
The above equation is quadratic in
Here ( )
( ) ⇒ ( ) ⇒ ( ) √
√ √ ⇒ (√ √ ) ⇒ √ √ √
√ √ √ which gives the envelope of the family of curves.
Solution:
( )
⇒
( )
Substituting (2) in (1), we get
(
)
⇒
The above equation is quadratic in
Here
( ) ⇒ ⇒
which gives the envelope of the family of curves.
Solution:
( )
( )
Differentiating (1) and (2) with respect to we get
⇒
⇒
( )
⇒
( )
From (3) and (4), we get
⇒
⇒
( ( ) ( ))
⇒ ⇒ ( )
⇒ ⇒ ( )
Substituting (5) and (6) in (1), we get
⇒
which gives the envelope of the family of curves.
Evolutes as the envelope of the normal
Solution:
The parametric form is ( )
Equation of normal is given is by
( )
( )
( )
( )
( ) ( )
( )
( ) ( )
To find the envelope:
Differentiating (2) partially with respect to we get
( )
( )
( ) ( ) we get
( )
( ) ( ) we get
( )
( ) ⇒( )
( ) ( )
( ) ⇒( )
( ) ( )
( ) ( )
( ) ( )
( )
( )
( )
( ) ( )
Find the evolute of the parabola considering it as the envelope of its normals.
Solution:
The parametric form is ( )
Equation of normal is given is by
( )
( )
( )
( ) ⇒ ( )
To find the envelope:
Differentiating (2) partially with respect to we get
⇒
⇒ (
)
( )
Substituting in (2)
( )
⇒
⇒ (
)
( )
From (3) and (4)
(
)
(
)
Raising the powers with 6 on both sides, we get
(
)
(
)
⇒
( )
( )
Find the evolute of the parabola considering it as the envelope of its normals.
Solution:
The parametric form is ( )
Equation of normal is given is by
( )
( )
( ) ⇒ ( )
To find the envelope:
Differentiating (2) partially with respect to we get
⇒
⇒
⇒ (
)
( )
Substituting in (2)
( )
⇒
⇒ (
)
( )
From (3) and (4)
(
)
(
)
Raising the powers with 6 on both sides, we get
(
)
(
)
⇒
( )
( )
Solution:
The parametric form is ( )
Equation of normal is given is by
( )
( )
( )
( ) ( )
To find the envelope:
Differentiating (2) partially with respect to we get
( )( ) ( )
( ) ( ) we get
( ) ( ) ( )
( ) ( ) ( )
( ) ⇒ ( ) ⇒
(
)
( )
( ) ( ) we get
( ) ( ) ( )
( ) ( )
( ) ⇒ ( ) ⇒
(
)
( )
( ) ( )
(
)
(
)
( ) ( )
( )
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