The GNU 3DLDF Parabolæ Page

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, 2019, 2020, 2021, 2022, 2023, 2024 The Free Software Foundation, Inc.

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

Table of Contents

Introduction

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.

Intersections

The Intersection of a Parabola and a Linear Path

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 Intersection of a Parabola and a Plane

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.

Pseudo-Paraboloid

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.

Perspective Projection

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

Standardizing

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

Classifying Points with Respect to a Parabola

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.