 _ Tutorial 3: The Geometry of 3-Dimensions  When the graphic box is clicked on a three dimensional grid will appear. You will not be able to see the z-axis because it is coming out at you. Go to view 2 and the axis will be rotated so you may view the grid in 3-Dimensions. Click on any point and notice the lower left hand corner; a set of three coordinates appear (x,y,z).

What do you think the z now represents?

To get a better feel for 3-D geometry click on cube at the right side of the menu box. Click on "label cube" , the letters A,B,C,D will appear.

**What are the x,y,z coordinates for A______ B______ C_____ D____
**What is the measure of line segment AB? _____

When you have completed click off the cube and label cube

 To rotate object hold left mouse button down and slide cursor arrow over object

In tutorial II you used Kinemage to measure angles and distances in two dimensions. Because the real world is 3- Dimensions, lets take a look at measuring angles and distances in 3-D space. When measuring angles in space we will need to click four consecutive points. Can you explain why three point will not be enough?

(Optional: In solid geometry the dihedral angle is defined as he angle formed by the intersection of two planes. A dihedral angle may be regarded as formed by the rotation of a plane about any line in the plane of an axis. Thus the value of a dihedral angle depends upon the amount of rotation about the edge, and not really upon the extent of the planes.)

Click on two planes, and "label plane". Go to measure under the Other pulldown. Just as you measured angles in Tutorial II, click point A, then B, then point C. This will give you the measure of <ABC. Now click on point D you will have measured the dihedral angle ABCD (see dhr= on the lower right of the black Mage box.

Much like angles shown in tutorial 2 required three points located on a two dimensional plane, the dihedral angles demonstrated here requires four points in three dimensional space.

**What is the measure of <ABC?_____
**What is the measure of the dihedral angle ABCD formed by the intersection of the two planes?______

Clear the screen of the planes and labels. Return to View 2. Choose dihedral 1, click on label angle. Click consecutively points A,B,C.D notice the lower right gives the angle between the last three points and that dhr= gives the dihedral angle for ABCD.

Go back to View 1, it shows the dihedral angle face on. Measure the angle as you see it, face on. What is this telling you? Rotate the image about several axis to gain a better understanding of a dihedral angle.

Clear the screen of dihedral angle 1. Repeat using the other sample dihedral angles (no letters will appear). Try to rotate the axis until you gain a clear understanding of dihedral angles. You may also use this module to form three dimensional images. These images may be saved and used on your own databases. (To save images will require the use of text editor, so check with your teacher first).

Activities for Students:

1) In tutorial II we used the Pythagorean theorem to find the diagonal of a square (a^2+b^2=c^2), and then checked the results using Kinemage. What equation would represent the diagonal of a cube? Why? Test out your theory and them use Kinemage to check you reasoning.

2)Try creating several 3-Dimensional figures using the 3-D grid. It may be helpful to be in either view2 or view3 when this is done.