Pythagorean-hodograph space curves

The relationships between certain families of special curves, including the general helices, slant helices, rectifying curves, Salkowski curves, spherical curves, and centrodes, are analyzed. First, characterizations of proper slant helices and Salkowski curves are developed, and it is shown that, for any given proper slant helix with principal normal n, one may associate a unique general helix whose binormal b coincides with n. It is also shown that centrodes of Salkowski curves are proper slant helices. Moreover, with each unit-speed non-helical Frenet curve in the Euclidean space E3, one may associate a unique circular helix, and characterizations of the slant helices, rectifying curves, Salkowski curves, and spherical curves are presented in terms of their associated circular helices. Finally, these families of special curves are studied in the context of general polynomial/rational parameterizations, and it is observed that several of them are intimately related to the families of polynomial/rational Pythagorean-hodograph curves.

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Algebraic-Trigonometric Pythagorean-Hodograph space curves

Advances in Computational Mathematics ◽

10.1007/s10444-018-9606-8 ◽

2018 ◽

Vol 45 (1) ◽

pp. 75-98 ◽

Cited By ~ 4

Author(s):

Lucia Romani ◽

Francesca Montagner

Keyword(s):

Space Curves ◽

Pythagorean Hodograph

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$C^2$ Hermite interpolation by Pythagorean Hodograph space curves

Mathematics of Computation ◽

10.1090/s0025-5718-07-01925-4 ◽

2007 ◽

Vol 76 (259) ◽

pp. 1373-1392 ◽

Cited By ~ 32

Author(s):

Zbyněk Šír ◽

Bert Jüttler

Keyword(s):

Hermite Interpolation ◽

Space Curves ◽

Pythagorean Hodograph

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Rational Pythagorean-hodograph space curves

Computer Aided Geometric Design ◽

10.1016/j.cagd.2011.01.002 ◽

2011 ◽

Vol 28 (2) ◽

pp. 75-88 ◽

Cited By ~ 22

Author(s):

Rida T. Farouki ◽

Zbyněk Šír

Keyword(s):

Space Curves ◽

Pythagorean Hodograph

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Research on Well Trajectory Deduction Method Based on Pythagorean-Hodograph Quintic Space Curves

IOP Conference Series Materials Science and Engineering ◽

10.1088/1757-899x/688/3/033068 ◽

2019 ◽

Vol 688 ◽

pp. 033068

Author(s):

Yufei Li ◽

Yinping Cao ◽

Yuxue Liu ◽

Mingfei Li ◽

Dou Yihua ◽

...

Keyword(s):

Space Curves ◽

Pythagorean Hodograph ◽

Well Trajectory

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Particle and Astro-physics Challenge Kant’s Phenomenolism

The Paideia Archive: Twentieth World Congress of Philosophy ◽

10.5840/wcp20-paideia199837686 ◽

1998 ◽

pp. 323-333

Author(s):

Lawrence H. Starkey

Keyword(s):

General Relativity ◽

Black Hole ◽

Physical Basis ◽

Primary Sources ◽

One Dimension ◽

Space Curves ◽

Slowing Down ◽

Parity Conservation ◽

Charge Parity ◽

Einstein’S General Relativity

For two centuries Kant's first Critique has nourished various turns against transcendent metaphysics and realism. Kant was scandalized by reason's impotence in confronting infinity (or finitude) as seen in the divisibility of particles and in spatial extension and time. Therefore, he had to regard the latter as subjective and reality as imponderable. In what follows, I review various efforts to rationalize Kant's antinomies-efforts that could only flounder before the rise of Einstein's general relativity and Hawking's blackhole cosmology. Both have undercut the entire Kantian tradition by spawning highly probable theories for suppressing infinities and actually resolving these perplexities on a purely physical basis by positing curvatures of space and even of time that make them reëntrant to themselves. Heavily documented from primary sources in physics, this paper displays time’s curvature as its slowing down near very massive bodies and even freezing in a black hole from which it can reëmerge on the far side, where a new universe can open up. I argue that space curves into a double Möbius strip until it loses one dimension in exchange for another in the twin universe. It shows how 10-dimensional GUTs and the triple Universe, time/charge/parity conservation, and strange and bottom particle families and antiparticle universes, all fit together.

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