### Abstract

The paper investigates self-intersections of offsets of implicit quadratic surfaces. The quadratic surfaces are the simplest curved objects, referred to as quadrics, and are widely used in mechanical design. In an earlier paper, we have investigated the self-intersections of offsets of explicit quadratic surfaces, such as elliptic paraboloid, hyperbolic paraboloid and parabolic cylinder, since not only are they used in mechanical design, but also any regular surface can be locally approximated by such explicit quadratic surfaces. In this paper, we investigate the rest of the quadrics whose offsets may degenerate, i.e. the implicit quadratic-surfaces (ellipsoid, hyperboloid, elliptic cone, elliptic cylinder and hyperbolic cylinder). We found that self-intersection curves of offsets of all the implicit quadratic surfaces are planar implicit conics and their corresponding curve on the progenitor surface can be expressed as the intersection curve between an ellipsoid, whose semi-axes are proportional to the offset distance, and the implicit quadratic surfaces themselves.

Original language | English |
---|---|

Pages (from-to) | 14-22 |

Number of pages | 9 |

Journal | Engineering with Computers |

Volume | 14 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1998 Jan 1 |

Externally published | Yes |

### Fingerprint

### Keywords

- NC machining
- Offset surfaces
- Self-intersections
- Trimmed offset

### ASJC Scopus subject areas

- Software
- Modelling and Simulation
- Engineering(all)
- Computer Science Applications

### Cite this

*Engineering with Computers*,

*14*(1), 14-22. https://doi.org/10.1007/BF01198971

**Self-intersections of offsets of quadratic surfaces : Part II, Implicit surfaces.** / Maekawa, Takashi.

Research output: Contribution to journal › Article

*Engineering with Computers*, vol. 14, no. 1, pp. 14-22. https://doi.org/10.1007/BF01198971

}

TY - JOUR

T1 - Self-intersections of offsets of quadratic surfaces

T2 - Part II, Implicit surfaces

AU - Maekawa, Takashi

PY - 1998/1/1

Y1 - 1998/1/1

N2 - The paper investigates self-intersections of offsets of implicit quadratic surfaces. The quadratic surfaces are the simplest curved objects, referred to as quadrics, and are widely used in mechanical design. In an earlier paper, we have investigated the self-intersections of offsets of explicit quadratic surfaces, such as elliptic paraboloid, hyperbolic paraboloid and parabolic cylinder, since not only are they used in mechanical design, but also any regular surface can be locally approximated by such explicit quadratic surfaces. In this paper, we investigate the rest of the quadrics whose offsets may degenerate, i.e. the implicit quadratic-surfaces (ellipsoid, hyperboloid, elliptic cone, elliptic cylinder and hyperbolic cylinder). We found that self-intersection curves of offsets of all the implicit quadratic surfaces are planar implicit conics and their corresponding curve on the progenitor surface can be expressed as the intersection curve between an ellipsoid, whose semi-axes are proportional to the offset distance, and the implicit quadratic surfaces themselves.

AB - The paper investigates self-intersections of offsets of implicit quadratic surfaces. The quadratic surfaces are the simplest curved objects, referred to as quadrics, and are widely used in mechanical design. In an earlier paper, we have investigated the self-intersections of offsets of explicit quadratic surfaces, such as elliptic paraboloid, hyperbolic paraboloid and parabolic cylinder, since not only are they used in mechanical design, but also any regular surface can be locally approximated by such explicit quadratic surfaces. In this paper, we investigate the rest of the quadrics whose offsets may degenerate, i.e. the implicit quadratic-surfaces (ellipsoid, hyperboloid, elliptic cone, elliptic cylinder and hyperbolic cylinder). We found that self-intersection curves of offsets of all the implicit quadratic surfaces are planar implicit conics and their corresponding curve on the progenitor surface can be expressed as the intersection curve between an ellipsoid, whose semi-axes are proportional to the offset distance, and the implicit quadratic surfaces themselves.

KW - NC machining

KW - Offset surfaces

KW - Self-intersections

KW - Trimmed offset

UR - http://www.scopus.com/inward/record.url?scp=0031699070&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0031699070&partnerID=8YFLogxK

U2 - 10.1007/BF01198971

DO - 10.1007/BF01198971

M3 - Article

AN - SCOPUS:0031699070

VL - 14

SP - 14

EP - 22

JO - Engineering with Computers

JF - Engineering with Computers

SN - 0177-0667

IS - 1

ER -