New Insight For Rotations In 4D Using Animations

The following paragraphs and animations will attempt to provide new insight into how Rotations behave in the 4th Dimension (4D) using animation techniques. The open source software packages, OpenGL and the OpenGL Utility Toolkit (GLUT), were extensively used to create the animations. The OpenGL and GLUT software is configured to display a Perspective World View (PWV) on the monitor screen. The PWV shows closer objects as larger than more distant objects and this makes it easier to visualize what is going on in the animations. First an array of points representing the objects that are being studied is generated and displayed in the PWV. This array of points is then processed using standard Matrix Rotation methods for an incremental angle to calculate new locations for all the points and these are displayed in the PWV. The processing and displaying is repeated for a series of incremental angles to animate the objects being studied.

 

Introduction To Point World

The Animations take place in a simulated world called Point World. There are no continuous objects in Point World only individual points. Objects such as Lines, Circles, Spheres, and even Hyper Spheres will be constructed from individual points. The goal was to find a way to see how each point in a more complex object behaves during a 4D Rotation.

Point World is drawn in the PWV and consists of a 100 by 100 Unit grid in the xz plane with a 100 Unit y axis perpendicular to and passing through the center point of the xz plane. The y axis extends 50 Units up (+direction) and 50 Units down (-direction) from the the intersection with the xz plane. The y axis can be hidden for 2D animations. Also from this y axis intersection, the x axis extends 50 Units to the right (+direction) and 50 Units to the left (-direction), and the z axis extends 50 Units to the front (+direction) and 50 Units to the back (-direction). Animations for 2D will take place on the horizontal xz plane.

In addition to the object points that are drawn there is a special point in Point World called Point Man. This is a single point that is used as the obsrver in Point World. Point Man is a point observer that can scan accross a field of view to look at objects in Point World. Point Man can only see one point at a time but can remember what has been seen to construct an image for himself. In the animations Point Man scans the scene from his perspective and then sets the points that he sees to the color Red. Point Man also knows which points are closer and sets those closer points to Red first progessiveley setting more distant points to Red until the field of view is complete.

It is also good at this time to talk about the Eye Location concept. The Eye Location is effectively your location when viewing the PWV. The Eye Location can be moved to any location in the PWV. Moving the Eye Location can help visualize what is happening in an Animation. For most Animations the Eye Location will be above and to the right of the z axis.

The following Animation will show a Sphere at the origin of Point World and Point Man with a 2D scan field of view looking at the Sphere. Point Man will scan the Spere from several locations. At the same time the Eye Location will Zoom in from a distance, go around Point World, and then go back out to where it started.

Point World Introduction Animation