3D Reflections with Vectors and Quaternions

In addition to rotating vectors, there is a way to reflect vectors through planes using quaternions. We'll derive how to do this transformation from the point of view of standard vector analysis and then from the point of view of quaternions. The quaternion formula is actually quite compact and can help give us further insight into the nature of reflections beyond standard vectors.

[@programmer1379]

Need to show that v * -1 = -1 * v but that's minor detail and is very easy to prove. Great series - enjoyed watching. Thank!

[@MrPetoria33]

For the curious, the composition of two reflections is a rotation (see his Geometric Algebra series for the proof).

[@josuemanuelparejacontreras7975]

just thank you man, thank you very much, I am learning a lot during the quarantine.

[@giannisanna5462]

can you explain the concept of spinor like this lesson? thank you!

[@rahulpatil5397]

awesome video series on quaternions

[@adamkhan1613]

I just want to check if I'm making an error here, at 17:50 on you make 2n dot v = n dot v + v dot n. Although n dot v and v dot n are equal shouldn't we make the object on the left a conjugate so that v' = nvn now becomes v' = n*vn?

[@thegamingtaco6283]

I have a question, is there a particular matrix that will allow you to reflect a vector across this plane (x+y+z=1)

[@TwirlySocrates]

I'm interested in learning about dual-quaternions, and how they encode translation

[@AssassinGrudge]

shoudn't the reflected vector be the same side as the view vector i mean we see what's behind us when we look to the mirror and not throw it

[@zbotpoint]

This video was good, it's really interesting to explore the uses of quaternions like this and I learned a lot. Very well-done video.

[@Math_oma]

+Zachary LaPointe Thanks! There's going to be at least one more video on quaternions, so hopefully people aren't getting too sick of this topic.

[@pronounjow]

Not at all! Very awesome to have more accessible exposure to these topics!

[@Math_oma]

+Jo Reven Well, I'm glad people are enjoying and hopefully getting some use out of these videos. I think the next one I'll make will be about how that question I asked about two reflections at the end leads naturally to the formula derived for rotations.

[@SlipperyTeeth]

You only talked about the cases in which you reflect an nth-dimensional vector across a (n-1)th dimensional plane. You also could've talked about reflecting a three-dimensional vector across a one-dimensional line (or the arbitrary cases of across a point or across the entire nth-dimensional space itself). This isn't much of a critique being as we've limited ourselves to three dimensions, so there's not that many different types of reflections that we can play around with, but a three-dimensional vector across a line is at least worth noting.

[@Math_oma]

+Pheonix Fire Sure, those are all worthy topics however I can't do it all in one video. However, I think the viewer should be able to figure those out after having seen this video.

[@TheDistributist]

I love your matrix algebra videos, what application is this for?

[@Math_oma]

+The Distributist Thanks! This whole topic of using quaternions for doing transformations (e.g. rotations) in 3D is apparently used by some game designers instead of standard vector analysis (with inner products and cross products). However, I don't have any experience with that, so I can't tell you too much about how easy it is to implement or if there's any conceptual benefit to it for a programmer. There's also a long history to these quaternions and it's a fascinating story filled with politics and bickering over whether physics should be done with quaternions or the vector analysis championed by Josiah Gibbs and Oliver Heaviside. Eventually, the vector analysis won out and that's why people generally don't know about quaternions or geometric algebra but do know about cross products, dot products, divergence, curl, etc. The original intention behind these quaternions was to do rotations and other transformations in 3D similar to how complex numbers can do rotations in 2D but unfortunately their weird 4-dimensional nature turns people off. Fortunately, there's a way to make sense of quaternions in 3D using the geometric algebra of R^3 (which will actually be 8-dimensional), where the quaternions get interpreted as bivectors (basically oriented areas).

[@Math_oma]

+The Distributist By the way, have you seen 3blue1brown's new series on linear algebra? It's a great series of videos, as it teaches the proper abstract interpretation of matrices, and I think it's really going to get more people into linear algebra.

[@TheDistributist]

Thanks, I am interested.

[@cdch10]

Just as a quick note: Besides the computer graphics community, quaternions have been used in the aerospace industry for decades now. Fewer computations and no gimbal lock FTW.

[@Math_oma]

+Christian Caballero Although as a wannabe mathematician I'm interested in them from the abstract point of view, they do seem like a more natural way of doing rotations than using matrices and vectors (using the Rodrigues formula). The latter more so because it has a cross product and cross products are the work of the devil.

[@TheUnorthodoxGears]

Great video, when is the next one out?!

[@Math_oma]

+TheUnorthodoxGears I might make one last video on this reflection business using quaternions however the topic will be nicely subsumed under "geometric algebra" which is the series I'm working on now. If you like this quaternion stuff, I encourage you to check out those geometric algebra videos, and as a spoiler alert, quaternions will be "scalar plus bivectors" in R^3 with all formulas looking nearly identical to the ones I have in the quaternion videos.

[@TheUnorthodoxGears]

ahhhh okay that sounds very interesting! For my A-Level Computing Project I am going to create a game engine in which I can rotate and manipulate 3D objects and have shading on them etc. Do you have any advice on where I can start? I honestly have no idea, I understand all of this theory but have no idea where to go with it all to get my project going and actually creating an image! Once again, many thanks!

[@Math_oma]

+TheUnorthodoxGears Unfortunately, I'm not the right guy to ask when it comes to implementing this stuff in a game engine. However, perhaps you could check out Jorge Rodriguez's channel, as he has a "math for game developers" series, if I remember correctly. On the theoretical side, although the many ways of manipulating 3D objects will get the job done, some are almost self-evidently superior on the theoretical side. For rotations, I find Euler angles and rotation matrices to be repugnant ways of thinking of rotations whereas quaternions and geometric algebra are mathematically natural. Even the vector analysis I have in my videos (using dot products and cross products) is butt-ugly. Same goes for reflections, where quaternions and geometric algebra are superior to vector analysis. If I could be even more polemical, I would say one has to be trained to use Euler angles, whereas common sense leads one to thinking of either axis-angle or motion in a plane; Euler angles are garbage. So, there's my monologue in favor of either quaternions or geometric algebra as the superior conceptual apparatus. The others are like pounding a square peg through a round hole; it can be done, but it's butt-ugly.

[@TheUnorthodoxGears]

haha, I love your analysis on the topic, thank you very much, I will probably message you again if I get stuck!

[@Math_oma]

+TheUnorthodoxGears Anytime. But remember my position is but one side of the discussion and probably asinine in some places.