The GNU 3DLDF Ellipsoids Page

Author: Laurence D. Finston

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Last updated: April 29, 2006

Top
Introduction
Related_Cuboids
Classifying Points with Respect to an Ellipsoid
Intersections
	Ellipsoid–Linear Path
	Ellipsoid–Plane
Contact

Introduction

2005.10.31.
I've now added the data type ellipsoid to the 3DLDF language. It corresponds to the type class Ellipsoid in the C++ code.

[Ellipsoid 1]

Ellipsoid

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Related Cuboids

2005.12.07.
I've now added parser rules and functions for finding the enclosing cuboid of an ellipsoid, i.e., the smallest cuboid that contains the ellipsoid. Finding the inscribed cuboid of an ellipsoid, i.e., the largest cuboid that fits into an ellipsoid, is more difficult. I don't yet know how to do this.

The 3DLDF code for generating the following four images is in elpsd_02.ldf.

[Ellipsoid 2]

Perspective Projection

[Ellipsoid 3]

Parallel Projection, X-Z Plane

[Ellipsoid 4]

Parallel Projection, X-Y Plane

[Ellipsoid 5]

Parallel Projection, Z-Y Plane

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Classifying Points with Respect to an Ellipsoid

points can be classified according to their position with respect to a ellipsoid by using the location operator:

ellipsoid E;
set E with_center origin with_axis_x 24 with_axis_y 16 with_axis_z 10
      with_divisions_x 8 with_divisions_y 8 with_divisions_z 2
      with_point_count 64;
point p[];
p0 := (1, 2, 1);  %% Inside.
p1 := (4, 9, 0);  %% Outside.

ellipse e;
e := get_ellipse 3 E;

p2 := get_point 60 e; %% On the surface.

message "p0 location E:";
show p0 location E;

message "p1 location E:";
show p1 location E;

message "p2 location E:";
show p2 location E;

location returns one of the following numerical values:

0:		The `point` lies on the surface of the `ellipsoid`.
1:		The `point` lies within the `ellipsoid`.
-1:		The `point` lies outside the `ellipsoid`.
`INVALID_NUMERIC`:		An error occurred.

In the following image, p₀ lies inside the ellipsoid, p₁ outside it, and p₂ on its surface. The complete 3DLDF code for generating this image can be found in elpsd_06.ldf.

[Ellipsoid 6]

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Intersections

The Intersection Points of an Ellipsoid and a Linear Path

2005.12.09.
The following four images illustrate the intersections of an Ellipsoid and a linear Path. The 3DLDF code for generating these images is in elpsd_13.ldf.

[Ellipsoid-Linear Path Intersection 1]