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.

#### Code

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.

The answer to the first question is showing incorrect even if I chose the option “Out of the page”

The correct answer is “into the page”.

If you put your thumb (X axis) pointing upwards in the page, and the next finger (Y axis) pointing to the right, you will inmediately see that your z finger points down and into the paper.

in a different way, if you start with your fingers in the way mentioned in the video in 1:34, you will see that you need to rotate your hand 90° around Z in order to put X pointing upwards, and then you will need to rotate 180° around X for Y to point to the right. Z will be then pointing down.

I hope this helps.

Regards