Abstract
In Chap. 5 a novel kind of characteristic curve for the purposes of analytical description of the contact geometry of two smooth regular part surfaces in the first order of tangency is discussed in detail. The discussion begins with preliminary remarks and follows with introduction and with derivation of an equation of the indicatrix of conformity at point of contact of two parts surfaces. Then, directions of extremum degree of conformity of two part surfaces in contact are specified and are described analytically. This analysis is followed by determination and by derivation of corresponding equations of asymptotes of the indicatrix of conformity. Capabilities of the indicatrix of conformity R point of contact of two smooth regular part surfaces in the first order of tangency are compared with the corresponding capabilities of “Dupin indicatrix” of the surface of relative curvature. Important properties of the indicatrix of conformity of two smooth regular part surfaces are outlined. Ultimately, the converse indicatrix of conformity at point of contact of two regular part surfaces in the first order of tangency is introduced and is briefly discussed as an alternative to the regular indicatrix of conformity.
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Notes
- 1.
Corresponding points of the “Dupin indicatrices” \( {\text{Dup}}(P_{1} ) \) and \( {\text{Dup}}(P_{2} ) \) share the same straight line through the contact point K of the surfaces \( P_{1} \) and \( P_{2} \) and are located at the same side of the point K.
- 2.
Equation of this characteristic curve is known from:
(a) Pat. No.1249787, USSR, A Method of Sculptured Part Surface Machining on a Multi-Axis NC Machine, S.P. Radzevich, B23C 3/16, Filed: December 27, 1984, [1] and (in a hidden form) from:
(b) Pat. No.1185749, USSR, A Method of Sculptured Part Surface Machining on a Multi-Axis NC Machine, S.P. Radzevich, B23C 3/16, Filed: October 24, 1983, [2].
- 3.
The diameter of a centro-symmetrical curve can be defined as a distance between two points of the curve, measured along the corresponding straight line through the center of symmetry of the planar curve.
References
Pat. No. 1249787. (1984). A method of sculptured surface machining on multi-axis NC machine. S.P. Radzevich, Int. Cl. B23c 3/16, Filed: December 27, 1984.
Pat. No. 1185749. (1983). A Method of sculptured surface machining on multi-axis NC machine. S.P. Radzevich, Int. Cl. B23c 3/16, Filed: October 24, 1983.
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Radzevich, S.P. (2020). Indicatrix of Conformity at Point of Contact of Two Smooth Regular Part Surfaces in the First Order of Tangency. In: Geometry of Surfaces. Springer, Cham. https://doi.org/10.1007/978-3-030-22184-3_5
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DOI: https://doi.org/10.1007/978-3-030-22184-3_5
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