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On the Applicability and Deformation of Surfaces

Published online by Cambridge University Press:  24 October 2008

B. M. Sen
Affiliation:
King's College

Extract

Beltrami's discovery that corresponding to any ruled surface there is another applicable on it having corresponding generators parallel but with the parameter of distribution of opposite signs, has brought into prominence the distinction between the applicability of two surfaces and the continuous deformation of one into the other. It is pointed out that since the parameter of distribution differs in sign, the tangent plane rotates in opposite directions as the point of contact proceeds along the two corresponding generators on the respective surfaces. One surface cannot, therefore, be deformed into the other. It is proposed in the present paper to investigate the distinction between applicability and deformability of two surfaces in general, and to connect this isolated fact with the general theory of deformation.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1924

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References

* Ann. di Mat., t. 7 (1865), pp. 139–150, or Forsyth, Differential Geometry, p. 387.

See, for instance, Forsyth, loc. cit., p. 376.

* These commonplaces of Differential Geometry have been reproduced to emphasize their analytical bearing on the problem of deformation.

* Forsyth, loc. cit., p. 50.

* Forsyth, loc. cit., p. 388.