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: October 20, 2009
Top |
Introduction |
Intersections |
Sphere and Dome Models |
Contact |
2005.10.25.
I've now added the data type sphere to the 3DLDF language.
It corresponds to the type class Sphere in the C++ code.
There are currently two types of spheres:
Globe-Type Spheres and
Panel-Type Spheres (illustrated below).
The 3DLDF code for the following images is in the file
sphrsmp0.ldf.
The file
sphrsmp0.tex contains
TeX code for including the images generated from
sphrsmp0.ldf.
The file
sphrsmp0.txt is the same as
sphrsmp0.tex. It can be viewed if your
browser is as stupid as mine, and won't display the latter as plain text.
Both types of spheres are created with their centers at the origin by default. Assuming they haven't been transformed, globe-type spheres have horizontal circles (i.e., circles parallel to the x-z plane) of different sizes, with their centers at different positions along the the y-axis. The vertical circles, however, are all the same size, with their centers at the origin, but rotated at equally spaced angles about the y-axis. Globe-type spheres thus have north and south poles only.
Again, assuming they haven't been transformed, panel-type spheres
have horizontal circles like globe-type spheres.
Unlike globe-type spheres, however,
the vertical circles are similar to the horizontal circles, but
rotated by 90°. That is, they are of different sizes and their
centers are at different positions along the x-axis.
Panel-type spheres thus have east and west poles, as well as
north and south ones. They also have an additional vertical circle
that lies in the x-y plane. Without this circle, the series of
steps
formed by the horizontal circles makes the sphere look
less spherical. However, this may not help if the sphere is
transformed. This is a general problem with wire-frame
representations of spheres.
I did try adding a third set of circles, parallel to the x-y plane, with their centers along the z-axis. Such spheres have front and back poles, too. However, while indisputably spherical (in the sense that all of the points on the circles lie on the sphere), they were rather confusing to look at. I therefore decided not to use them.
spheres can be constructed, drawn, and transformed, but I haven't yet implemented a full set of parser rules for operations on them. I plan to work on parser rules and functions for the following features in the near future:
There are other features that I'd like to implement, but they will probably take much more time:
The 3DLDF code and TeX code for the examples in this section can be found sphrint_0.ldf and sphrint_0.txt , respectively.
If the plane is merely tangent to the sphere, the intersection is a point. I haven't accounted for this case yet. However, if it actually passes through the sphere, it is a circle.
In the following sequence of images, the sphere has a radius of 3cm. In the first image, its center is located at (0, -2.75cm, 0). The center of the rectangle is at the origin, and it's rotated by 5° about the x and z-axes. In each of the subsequent images, the sphere is shifted upwards (along the positive y-axis) by 1cm, and the rectangle is rotated by an additional 5° about the x and z-axes.