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I think matrices are easier for people to visualize because they can think about them as a vector frame and coordinate transformations. Quaternions are seen as

I think matrices are easier for people to visualize because they can think about them as a vector frame and coordinate transformations. Quaternions are seen as this abstract 4D space somehow containing 3D space transformations, and people have a much harder time visualizing 4D space. I do not think spending a lot of time trying to visualize this 4D space, like this video does, is the right way to go. Rotors make that method obsolete.

jacobolus 8y ago
To be honest the hardest part is not understanding single rotations, but understanding how rotations compose. Thinking about a scalar + bivector as the quotient of two vectors or as the composition of two reflections is easy enough. But that still doesn’t yield a solid understanding of how rotation works – 3D rotation is subtle and tricky, even for people who have pretty good spatial reasoning and a lot of experience working with solid geometry / 3D design. I have been thinking about the best interactive diagram(s) for giving people a good intuition about that for a long time, and I have some ideas but I still haven’t implemented anything and I’m not sure how well it will work.
gleed 8y ago
I mean, the math of quaternions just sucks for intuitive understanding. It's not so much about how they 'magically contain 3d transformations', because you can (and I was taught that way) derive generalized quaternion rotation by writing down the math to rotate a vector around an axis by a certain angle, and then given quaternions and their computational rules as a black box, you can bend that math you wrote down into a shape that fits exactly into quaternions. In that sense how they contain 3D transformations can be made completely intuitive via just mathematical derivation. You don't really need to visualize quaternions that way because you can visualize it using the basic tools - axis and angle - and then just saying that Quaternions are just a way to hammer that into a compact mathematical shape with certain neat properties. Imo it's almost exactly the same as gaining an understanding of why you plug translations in matrices into a fourth column and all the accompanying shenanigans with homogeneous coordinates where the other basic operations only require 3 columns when you can just translate stuff by adding the translation vector onto your existing vector: Well, it's a neat mathematical tool with extra neat properties like being able to concatenate a wild series of translations intermingling with other transformations, and putting it all into a single matrix. The problem with quaternions itself is that you always have that black box that is completely sufficient to working with them on an advanced level, but you will have that uneasiness of a looming black box of nonunderstanding unless you go on a voluntary tangent into a different mathematical field and its accompanying history of how and why they were created in the first place.

jacobolus 8y ago

To be honest the hardest part is not understanding single rotations, but understanding how rotations compose. Thinking about a scalar + bivector as the quotient of two vectors or as the composition of two reflections is easy enough. But that still doesn’t yield a solid understanding of how rotation works – 3D rotation is subtle and tricky, even for people who have pretty good spatial reasoning and a lot of experience working with solid geometry / 3D design. I have been thinking about the best interactive diagram(s) for giving people a good intuition about that for a long time, and I have some ideas but I still haven’t implemented anything and I’m not sure how well it will work.

gleed 8y ago

I mean, the math of quaternions just sucks for intuitive understanding. It's not so much about how they 'magically contain 3d transformations', because you can (and I was taught that way) derive generalized quaternion rotation by writing down the math to rotate a vector around an axis by a certain angle, and then given quaternions and their computational rules as a black box, you can bend that math you wrote down into a shape that fits exactly into quaternions. In that sense how they contain 3D transformations can be made completely intuitive via just mathematical derivation. You don't really need to visualize quaternions that way because you can visualize it using the basic tools - axis and angle - and then just saying that Quaternions are just a way to hammer that into a compact mathematical shape with certain neat properties. Imo it's almost exactly the same as gaining an understanding of why you plug translations in matrices into a fourth column and all the accompanying shenanigans with homogeneous coordinates where the other basic operations only require 3 columns when you can just translate stuff by adding the translation vector onto your existing vector: Well, it's a neat mathematical tool with extra neat properties like being able to concatenate a wild series of translations intermingling with other transformations, and putting it all into a single matrix. The problem with quaternions itself is that you always have that black box that is completely sufficient to working with them on an advanced level, but you will have that uneasiness of a looming black box of nonunderstanding unless you go on a voluntary tangent into a different mathematical field and its accompanying history of how and why they were created in the first place.