LESSON

# Right-Handed 3D Coordinate Frame

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In the previous lecture when we were discussing 2-dimensional worlds, we introduced the concept of a right-handed coordinate frame. And, just to remind you, the right-handed coordinate frame is constructed like this. We draw the x-axis and then we swing the y-axis up by 90 degrees. A three-dimensional coordinate frame is actually very similar to the two-dimensional coordinate frame. So, here is the two-dimensional right-handed coordinate frame that we just introduced. And, what I’m going to do do now is to rotate that in three dimensions. And, what we see is that the z-axis was previously hidden. Previously, it was sticking up at us out of the screen.

To create a 3-dimensional right-handed coordinate frame, we start in a similar way. We draw the x-axis. We swing the y-axis so that it makes an angle of 90 degrees to the x-axis. And then, we swing the z-axis upwards and it also makes an angle of 90 degrees to the x-axis and also makes an angle of 90 degrees to the y-axis.

There’s a very close relationship between the 2-dimensional and the 3-dimensional coordinate frames. Let me put down the two-dimensional coordinate frame and here is three-dimensional coordinate frame. And, I can overlay the x-axis and the y-axis and we see that the z-axis points upwards. We refer to this as a right-handed coordinate frame. And, the reason we call it a right-handed frame is the orientation of the axes can be defined very simply using my right hand. So, this is the x-axis. This is the y-axis. And, this is the z-axis pointing upwards. So, whenever we’re doing work with 3-dimensional coordinate frames, they are always right-handed coordinate frames.

This particular coordinate frame that I built, we have an x-axis and a y-axis and this one, the z-axis pointing downwards. This is a left-handed coordinate frame. So it’s best not to use these, they are going to cause you all sorts of grief. So, don’t use left-handed coordinate frames. Go with the right-handed coordinate frame.

So, just to recap. When we create a right-handed coordinate frame, we use this rule what’s called the right-handed rule, and I take my right hand. The x-axis is parallel to my thumb. The y-axis is parallel to my index finger. And, the z-axis is parallel to my middle finger.

We discuss the structure of a right-handed 3D coordinate frame and the spatial relationship between its axes which is encoded in the right-hand rule.

### Skill level

Undergraduate mathematics

This content assumes high school level mathematics and requires an understanding of undergraduate-level mathematics; for example, linear algebra - matrices, vectors, complex numbers, vector calculus and MATLAB programming.

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Is it really true, that the third coordinate frame is righthanded? I’m pretty sure one would have to switch the x-axis with the y- or z- axis to fulfill that (or switch the direction of the z axis)

It’s a good question. While it looks RH to me, I have become aware that different people have different interpretations since there are very few perspective cues in the 2-dimensional picture. These were created using the Toolbox trplot() function and the ‘perspective’ option. Here’s the same frame from a different viewpoint, and with the world-frame coordinates shown. Does that make it easier?

Why is the second frame wrong?

See the comment above your question.