Vectors and Planes 16.6

The "Vectors and Planes" application offers a comprehensive tool for visualizing and understanding three-dimensional linear and quadratic structures, as well as their intersections.

Linear structures and their combinations include:

• Two vector combination
• Single plane
• Line intersecting with a plane
• Two planes intersecting
• Three planes intersecting

The current catalog of quadratic (quadric) surfaces comprises:

• Two spheres in combination
• Standard ellipsoid
• Ellipsoid expressed through a positive-definite matrix
• Standard cone
• Cone represented via an indefinite matrix
• Single-sheet hyperboloid
• Single-sheet hyperboloid in matrix form with an indefinite matrix
• Two-sheet hyperboloid
• Two-sheet hyperboloid in matrix form with an indefinite matrix
• Standard cylinder
• Cylinder represented by a positive-semidefinite matrix
• Hyperbolic paraboloid
• Elliptic paraboloid

The application also details combinations of quadric surfaces with linear structures, including:

• Matrix ellipsoid with a line
• Standard ellipsoid with a plane
• Matrix ellipsoid with a plane
• Matrix cone with a line
• Standard cone with a plane
• Single-sheet hyperboloid in matrix form with a line
• Two-sheet hyperboloid in matrix form with a line
• Matrix cylinder with a line
• Elliptic paraboloid with a line
• Hyperbolic paraboloid with a line

All of these structures and their intersections are visualized using interactive 3D graphs, facilitating intuitive exploration.

The application provides step-by-step algorithms for calculating intersection points, many of which are grounded in Linear Algebra principles. These algorithms are complemented by explanations available within the "Matrix Solver Step by Step" application for enhanced understanding.

The latest release, Version 10, has been fully redesigned to improve usability and functionality.

The developers invite user feedback via email at [email protected] to further refine and enhance the application's capabilities.