The Elementary Differential Geometry of Plane Curves
The Elementary Differential Geometry of Plane Curves
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For proofs of many of these assertions and for further developments in the region of Solid Geometry the reader is referred to de la Valine Poussin, Picard, and the paper (already quoted) by Bromwich and Hudson.
EXAMPLES V (1) The edge of regression has, in general, second order contact with any surface of the family on the characteristic.
[Use Theorem 4-61. ] (2) Discuss the problem of the envelope of a moving sphere, whose centre lies on a twisted curve, (i) when the radius is constant {uiie s...urface canal), (ii) when the radius varies.
[Extend § 5*50 from circles to spheres. ] * Fr. Varete de rebi'oussement. Plane sections of the envelope have in general a cusp {point de rebroussement) where the plane cuts this curve. See Picard, vol. I, p. 322.
THE THEORY OF ENVELOPES 79 (3) Tke wave surface. The envelope of the plane where a^+/3^ + y^=l a2 j82 72 is the surface ^. +, 1:4^ + ^=0. ('•^=.^+/+^^).
(4) The direction cosines of the normal to the surface obtained by eliminating a between /(^, y, 2;, a)=0 and g {x, y, 0, a) = are given by d {x, a) * 8 (y, a) * a (a, a)' Obtain the corresponding result for a siuface obtained by elimination of two parameters between three equations.
What to read after The Elementary Differential Geometry of Plane Curves?
You can find similar books in the "Read Also" column, or choose other free books by R H Ralph Howard Fowler to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
For proofs of many of these assertions and for further developments in the region of Solid Geometry the reader is referred to de la Valine Poussin, Picard, and the paper (already quoted) by Bromwich and Hudson.
EXAMPLES V (1) The edge of regression has, in general, second order contact with any surface of the family on the characteristic.
[Use Theorem 4-61. ] (2) Discuss the problem of the envelope of a moving sphere, whose centre lies on a twisted curve, (i) when the radius is constant {uiie s...urface canal), (ii) when the radius varies.
[Extend § 5*50 from circles to spheres. ] * Fr. Varete de rebi'oussement. Plane sections of the envelope have in general a cusp {point de rebroussement) where the plane cuts this curve. See Picard, vol. I, p. 322.
THE THEORY OF ENVELOPES 79 (3) Tke wave surface. The envelope of the plane where a^+/3^ + y^=l a2 j82 72 is the surface ^. +, 1:4^ + ^=0. ('•^=.^+/+^^).
(4) The direction cosines of the normal to the surface obtained by eliminating a between /(^, y, 2;, a)=0 and g {x, y, 0, a) = are given by d {x, a) * 8 (y, a) * a (a, a)' Obtain the corresponding result for a siuface obtained by elimination of two parameters between three equations.
What to read after The Elementary Differential Geometry of Plane Curves?
You can find similar books in the "Read Also" column, or choose other free books by R H Ralph Howard Fowler to read onlineMoreLess
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