We present our 3-dimensional constructions in graphics objects on the story pages of the Microworld. These objects are easy to navigate, either in the third person (moving the scene) or in the first person (moving the camera).
In fact, you will want to change your viewpoint from time to time while viewing our simulations, if only to get a better idea of the geometry that the computer is creating. For this, the Navigation Bars above and below the object are handy. You will be able to move the scene using these bars:
Initially, a Graph3D Object comes up in the Scene Mode. In what follows, I assume either that it is in that mode or that you have clicked Scene (Move the Scene) on the lower Navigation bar. A Graph3D object may also be put in Camera (Move the Camera) mode, and I will discuss that later.
We therefore consider first navigation by active transformations, which move the entire scene (as a whole with reference to an imaginary local or internal reference frame) in one of two ways:
The six dimensional group of motions generated by these elementary motions (on composition) is sometimes called the Galilean group. These motions are the only way to move space in a "rigid" manner.
Now if we use the upper Navigation Bar,
we have six "generators" of that group. These are the translations in the x,y, and z directions, or rotations (using the right hand rule) about the y, x, and z axes. Note that the z axis points out of the board towards you. These axes are color-coded:
These translations and rotations are multiples of a predetermined "step" that you may set using the "navigation timer" button on the Graph3D menu. To get that menu, simply right-click on the Graph3D object. The shorter the delay, the larger the step, the faster the scene moves.
You may also restore the picture to its original state, or simply stop its motion, by using the "Restore" button or the "Stop" button on the lower Navigation Bar:
Using the navigation bar, the 6 arrows on the left describe the 6 ways of translating the scene: ±x, ±y, and ±z. By themselves, these are easy to understand. But once rotations are admitted, using the next six buttons, things become a bit more complicated, because the internal x, y, and z directions have now become mixed together. And new motions are with respect to these changed axes. Here is a great opportunity to explore the geometric consequences of the non-commutativity of the Galileo group.
In any case, if you click, say a rotation button, the scene begins to rotate. You may stop the rotation be clicking again, or by pressing "Stop". It is the same for translation buttons. If you get lost, you may always bring things back to the original position by pressing the "Restore" button. And you may slow things down by changing the delay to a number larger than 10 in the Timer menu. These are the local transformations. We now move to the more intuitive global transformations.
When you select Camera (Move the camera), navigation becomes a little easier. The reason for this is probably that (as Poincaré observed) we first learn about motion by moving ourselves. The changes in our field of view that we consider geometric may always be undone by changing the attitude of our bodies, by turning our heads, walking towards an object, or away from it, and so on. So the camera represents our eyes, our field of view.
The 6 translations on the navigation bar are now always motions of the camera in terms of an imaginary fixed global reference frame (Earth Coordinates). So "up" is always up, forward is always straight ahead, and so on. The rotations are also with respect to the camera coordinates. Counterclockwise rotation about the y axis always has the effect of "turning the camera to the left" so the scene drifts to the right. This is an important point. As the camera moves "up" the scene moves "down." In general, the actions of the camera have the opposite effect (other things being equal) of local transformations of the scene.