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 3D Geometry Primer: Chapter 1 - Issue 05 - 3D Space: Righthanded Rules, And More... by (11 September 2000) Return to The Archives
 3D Space: Righthanded Rules, And More...
 Today we're going to examine what the benefits are of a righthanded base, what the rule of the corkscrew is, and how you should draw 3D bases on paper.

 (I) Two Ways To Arrange Those 3D Bases

 (II) That Cross Product Again
 You remember this from last week?I×J = K J×K = I K×I = JNow, look again at the cross product (issue I.02, (vi)), take your righthanded base {I,J,K}, and try to construct each cross product I×J, J×K, K×I. Pay attention to that hand! Notice that the magnitudes are correct too: all magnitudes are 1, and sin(90°)=1 too.Put this in your head. Now!

 (III) The Rule Of The Righthanded Corkscrew

 (IV) How To Draw A 3D Base On 2D Paper Correctly?
 Basically you can lay your base vectors in any way you want. The only thing you should keep in mind, is that you draw the base vectors from I to K in counter-clockwise order (and again...). That will be the most natural way. Of course, if you want to draw a lefthanded space, you do it clockwise. There's also another way if you're only interested in 2 of the 3 base vectors. For example, when you're currently working with a situation in which the z-value of all points is 0 (zero), then you're not interested in the K vector. The z-value is 0 anyway. So, you only need the I and J vectors. The way to do that, is to draw the K vector perpendicular to your paper. *Yoink?* How do you that? Well, you can't draw a vector really perpendicular to your paper, so you project it. Just like you would do in other cases. But the projection of a perpendicular vector is just 1 point! How do you see if the vector sticks in the paper, or comes out of it? Simple. If you point a real arrow (of an archer) into your paper, what do you see? Correct, the tail of the arrow. So you draw the tail (with those feathers). That is a little cross in a circle. And what if you point the arrow in the other sense? You see the head of the arrow, the sharp point of it. So you draw that, and this is a little ball (the sharp point...) in a circle. Voila, if you use your imagination, everything comes clear...

 (V) Why Is The Origin Of The Screen In The Upper-Left Corner?
 Have you ever wondered why the heck the origin of the screen is in the upper-left corner, while normally it should be in the lower-left corner like in math? I did. I don't have a real answer to it, but I did notice one nice thing. If you take a righthanded space and lay it's origin in that one of the screen, lay the I and J vectors aside those ones of the screen, then you will find out the the K vector will point into the screen. Just like we want in 3D-engines: the camera in the origin and lays aside K. Isn't that very handy? (Or maybe we want to point the camera that way, because the screen is made that weird. I don't know...)So, if you should have a weird screen with its origin in the lower-left corner, you could decide to use a lefthanded 3D space, because then the K vector will point into the screen again. But beware! If you do that, you need to turn around the whole thing: every righthanded becomes lefthanded and every counter-clockwise because clockwise, and vice versa (or something ...).