### Abstract

Although offset surfaces are widely used in various engineering applications, their degenerating mechanism is not well known in a quantitative manner. Offset surfaces are functionally more complex than their progenitor surfaces and may degenerate even if the progenitor surfaces are regular. Self-intersections of the offsets of regular surfaces may be induced by concave regions of surface where the positive offset distance exceeds the maximum absolute value of the negative minimum principal curvature or the absolute value of the negative offset distance exceeds the maximum value of the positive maximum principal curvature. It is well known that any regular surface can be locally approximated in the neighborhood of a point p by the explicit quadratic surface of the form r(x, y) = [x, y,1/2(αx^{2} + βy^{2})]^{T} to the second order where -α and -βare the principal curvatures at point p. Therefore investigations of the selfintersecting mechanisms of the offsets of explicit quadratic surfaces due to differential geometry properties lead to an understanding of the self-intersecting mechanisms of offsets of regular parametric surfaces. In this paper, we develop the equations of the self-intersection curves of an offset of an explicit quadratic surface. We also develop an algorithm to detect and trace a small loop of a self-inter section curve of an offset of a regular parametric surface based on our analysis of offsets of explicit quadratic surfaces. Examples illustrate our method.

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

Pages (from-to) | 1-13 |

Number of pages | 13 |

Journal | Engineering with Computers |

Volume | 14 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1998 Jan 1 |

Externally published | Yes |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

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

### Cite this

**Self-intersections of offsets of quadratic surfaces : Part I, Explicit surfaces.** / Maekawa, Takashi.

Research output: Contribution to journal › Article

*Engineering with Computers*, vol. 14, no. 1, pp. 1-13. https://doi.org/10.1007/BF01198970

}

TY - JOUR

T1 - Self-intersections of offsets of quadratic surfaces

T2 - Part I, Explicit surfaces

AU - Maekawa, Takashi

PY - 1998/1/1

Y1 - 1998/1/1

N2 - Although offset surfaces are widely used in various engineering applications, their degenerating mechanism is not well known in a quantitative manner. Offset surfaces are functionally more complex than their progenitor surfaces and may degenerate even if the progenitor surfaces are regular. Self-intersections of the offsets of regular surfaces may be induced by concave regions of surface where the positive offset distance exceeds the maximum absolute value of the negative minimum principal curvature or the absolute value of the negative offset distance exceeds the maximum value of the positive maximum principal curvature. It is well known that any regular surface can be locally approximated in the neighborhood of a point p by the explicit quadratic surface of the form r(x, y) = [x, y,1/2(αx2 + βy2)]T to the second order where -α and -βare the principal curvatures at point p. Therefore investigations of the selfintersecting mechanisms of the offsets of explicit quadratic surfaces due to differential geometry properties lead to an understanding of the self-intersecting mechanisms of offsets of regular parametric surfaces. In this paper, we develop the equations of the self-intersection curves of an offset of an explicit quadratic surface. We also develop an algorithm to detect and trace a small loop of a self-inter section curve of an offset of a regular parametric surface based on our analysis of offsets of explicit quadratic surfaces. Examples illustrate our method.

AB - Although offset surfaces are widely used in various engineering applications, their degenerating mechanism is not well known in a quantitative manner. Offset surfaces are functionally more complex than their progenitor surfaces and may degenerate even if the progenitor surfaces are regular. Self-intersections of the offsets of regular surfaces may be induced by concave regions of surface where the positive offset distance exceeds the maximum absolute value of the negative minimum principal curvature or the absolute value of the negative offset distance exceeds the maximum value of the positive maximum principal curvature. It is well known that any regular surface can be locally approximated in the neighborhood of a point p by the explicit quadratic surface of the form r(x, y) = [x, y,1/2(αx2 + βy2)]T to the second order where -α and -βare the principal curvatures at point p. Therefore investigations of the selfintersecting mechanisms of the offsets of explicit quadratic surfaces due to differential geometry properties lead to an understanding of the self-intersecting mechanisms of offsets of regular parametric surfaces. In this paper, we develop the equations of the self-intersection curves of an offset of an explicit quadratic surface. We also develop an algorithm to detect and trace a small loop of a self-inter section curve of an offset of a regular parametric surface based on our analysis of offsets of explicit quadratic surfaces. Examples illustrate our method.

KW - NC machining

KW - Offset

KW - Self-intersections

KW - Trimmed offset

UR - http://www.scopus.com/inward/record.url?scp=0031701374&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0031701374&partnerID=8YFLogxK

U2 - 10.1007/BF01198970

DO - 10.1007/BF01198970

M3 - Article

AN - SCOPUS:0031701374

VL - 14

SP - 1

EP - 13

JO - Engineering with Computers

JF - Engineering with Computers

SN - 0177-0667

IS - 1

ER -