Author: Laurence D. Finston

This copyright notice applies to the text and source code of this web site, and the graphics that appear on it. The software described in this text has its own copyright notice and license, which can be found in the distribution itself.

Copyright (C) 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018 The Free Software Foundation

Permission is granted to copy, distribute, and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, with no Front-Cover Texts, and with no Back-Cover Texts. A copy of this license is included in the file COPYING.TXT

Last updated: August 7, 2007

Top |

Introduction |

Intersections |

Pseudo-Paraboloid |

Standardizing |

Classifying Points with Respect to a Parabola |

Contact |

Back to top

Back to main page

2005.11.09.

I've now added the data type **parabola** to the 3DLDF language.
It corresponds to the type **class Parabola** in the C++ code.

The 3DLDF code for generating the following image is in prbla_00.ldf. TeX code for the including it is in prbla_00.txt.

Back to contents

Back to top

Back to main page

The 3DLDF code for generating the following image is in prbla_05.ldf. TeX code for the including it is in prbla_05.txt.

The intersection points of a parabola **p** and a line seqment
**l** such that **p** and **l** are coplanar.

The 3DLDF code for generating the following image is in prbla_12.ldf. TeX code for the including it is in prbla_12.txt.

The intersection points of a parabola **p** and two line seqments
**l** and **m**, such that **p** and
**l**, and **p** and **m** are non-coplanar.

The 3DLDF code for generating the following image can be found in prbla_11.ldf. TeX code for the including it can be found in prbla_11.txt.

The intersection points of a parabola and a plane.

Back to contents

Back to top

Back to main page

The following four images represent a paraboloid generated by rotating
a parabola about the x-axis. The circles parallel to the axis of the
parabola (the x-axis) are **paths** created by taking corresponding
**points** from the **parabola** each time it's rotated.

The 3DLDF code for generating these images is in prbla_01.ldf. TeX code for including it is in prbla_01.txt.

Parabola: | |

vertex at origin | |

parameter = 3cm | |

axis = positive x-axis | |

Focus: | |

position: (0, 10cm, -20cm) | |

direction: (0, 10cm, 100cm) | |

distance: 15cm | |

Parallel Projection, X-Y Plane

Parallel Projection, X-Z Plane

(Similar to the last one, isn't it?)

Parallel Projection, Z-Y Plane

Back to contents

Back to top

Back to main page

A **parabola** can be standardized or placed in
standard position, i.e., in the x-z plane, with its vertex
at the origin and its focus on the positive x-axis.

This is done using the `standardize`

operator, which
returns a `transform`

.
`standardize`

leaves the `parabola`

unchanged.
To actually put it into standard position, you must
multiply it by the `transform`

that was returned.

The following is the gist of the code for creating the following four images. The complete code can be found in prbla_06.ldf. TeX code for including them is in prbla_06.txt.

parabola p; set p with_parameter 3 with_extent 7; rotate p (75, 50); shift p (3.5, 8, -1.75); transform t; t := standardize p; p *= t;

Perspective Projection

Focus: | |

position: (-5cm, 10cm, -20cm) | |

direction: (-5, 10cm, 100cm) | |

distance: 15cm | |

Parallel Projection, X-Y Plane

Parallel Projection, X-Z Plane

Parallel Projection, Z-Y Plane

Back to contents

Back to top

Back to main page

`points`

can be classified according to their position with respect to a
`parabola`

by using the `location`

operator:

parabola q; set q with_parameter 3 with_extent 7; point p; p := (2, 3, 4.5); r := p location q;

`location`

returns one of the following numerical values:

0:
The `point`

lies on the segment of the parabola represented
by the `parabola`

.

1:
The `point`

lies on the parabola, but not the segment, represented
by the `parabola`

object.

2:
The `point`

lies in the region enclosed by the branches of the parabola and the
line connecting the end points of the segment.

3:
The `point`

lies between the branches of the parabola, but
outside the region enclosed by them and the line connecting the
end points of the segment.

-1:
The `point`

is
coplanar with the `parabola`

,
but does not lie on the curve or between the branches.

-2:
The `point`

is not coplanar with the `parabola`

.

-3:
The `parabola`

is not parabolic.

-4:
The `point`

is invalid.

-5:
Something has gone terribly wrong.

The complete 3DLDF code for generating the following image can be found in prbla_10.ldf. TeX code for including it can be found in prbla_10.txt.

Back to contents

Back to top

Back to main page