Plücker’S Conoid, Hyperboloids of Revolution, and Orthogonal Hyperbolic Paraboloids

EasyChair Preprint 3706

Abstract

Plücker’s conoid C , also known under the name cylindroid, is a ruled surface of degree three with a finite double line and a director line at infinity. The following two properties of C play a major role in the geometric literature: The bisector of two skew lines ℓ₁ , ℓ ₂ in the Euclidean 3-space, i.e., the locus of points at equal distance to ℓ ₁ and ℓ ₂ , is an orthogonal hyperbolic paraboloid P . All generators of P are axes of one-sheeted hyperboloids of revolution H which pass through ℓ ₁ and ℓ ₂ . Conversely, the locus of pairs of skew lines ℓ ₁ , ℓ ₂ for which a given orthogonal hyperbolic paraboloid P is the bisector, is a Plücker conoid C . In spatial kinematics, Plücker’s conoid C is well-known as the locus of axes ℓ₁₂ of the relative screw motion for two wheels which rotate about fixed skew axes ℓ ₁ and ℓ ₂ with constant velocities. The axodes of the relative screw motion are one-sheeted hyperboloids of revolution H₁ , H₂ with mutual contact along ℓ ₁₂ . The common surface normals along ℓ ₁₂ form an orthogonal hyperboloid paraboloid P passing through the axes ℓ ₁ and ℓ ₂ . The underlying paper aims to discuss these two main properties. It seems that there is no close relation between them though both deal with Plücker’s conoid, orthogonal hyperbolic paraboloids, and hyperboloids of revolution – however in different ways.

Keyphrases: Plücker’s conoid, bisector, cylindroids, one-sheeted hyperboloid of revolution, orthogonal hyperbolic paraboloid

@booklet{EasyChair:3706,
  author    = {Hellmuth Stachel},
  title     = {Plücker’S Conoid, Hyperboloids of Revolution, and Orthogonal Hyperbolic Paraboloids},
  howpublished = {EasyChair Preprint 3706},
  year      = {EasyChair, 2020}}