Page 262 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 21: New Spherical Indicatrices and their Characterizations

21.5 Summary Notes

If the tangent vector of this curve forms a constant angle with a fixed constant vector U,

then this curve is called a general helix or an inclined curve.
A regular curve  : I  E is called a slant helix according to Bishop frame provided the unit
3

vector M (s) of  has constant angle  with some fixed unit vector u; that is,
1
M , u = cos 
1
for all s  I.
Let  = (s) be a regular curve in E . If we translate of the first (tangent) vector field of
3

Bishop frame to the center O of the unit sphere S , we obtain a spherical image  = (s ). This
2

curve is called tangent Bishop spherical image or indicatrix of the curve  = (s).
Let  = (s) be a regular curve in E . If we translate of the second vector field of Bishop frame
3

to the center O of the unit sphere S , we obtain a spherical image  = (s ). This curve is
2

called M Bishop spherical image or indicatrix of the curve  = (s).
1
Let  = (s) be a regular curve in E . If we translate of the third vector field of Bishop frame
3

to the center O of the unit sphere S , we obtain a spherical image of  = (s ). This curve is
2

called the M Bishop spherical image or the indicatrix of the curve  = (s).
2
21.6 Keywords
General helix: If the tangent vector of this curve forms a constant angle with a fixed constant
vector U, then this curve is called a general helix or an inclined curve.
Slant helix: A regular curve  : I  E is called a slant helix according to Bishop frame provided
3
the unit vector M (s) of  has constant angle  with some fixed unit vector u; that is,
1
M , u = cos 
1
for all s  I.
Tangent Bishop spherical image: Let  = (s) be a regular curve in E . If we translate of the first
3
(tangent) vector field of Bishop frame to the center O of the unit sphere S , we obtain a spherical
2
image  = (s ). This curve is called tangent Bishop spherical image or indicatrix of the curve

 = (s).
M Bishop spherical image: Let  = (s) be a regular curve in E . If we translate of the second vector
3
1
field of Bishop frame to the center O of the unit sphere S , we obtain a spherical image  = (s ).
2

This curve is called M Bishop spherical image or indicatrix of the curve  = (s).
1
M Bishop spherical image: Let  = (s) be a regular curve in E . If we translate of the third vector
3
2
field of Bishop frame to the center O of the unit sphere S , we obtain a spherical image of  =
2
(s ). This curve is called the M Bishop spherical image or the indicatrix of the curve  = (s).
 2
21.7 Self Assessment
1. A regular curve  : I  E is called a .................. according to Bishop frame provided the unit
3
vector M (s) of  has constant angle  with some fixed unit vector u; that is,
1
M , u = cos 
1
for all s  I.

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