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How would you define rotations using only the wedge product? It is also terribly useful being able to invert these rotations. You could make it work in some way

by yiyus 8y ago

How would you define rotations using only the wedge product? It is also terribly useful being able to invert these rotations. You could make it work in some way, but it is not clear to me that the result will be simpler than a GA (or quaternion) based approach.

ajkjk 8y ago
The GA approach is in there; I just think the 'geometric product' itself is not that fundamental. A bivector x^y can be used as a generator of rotations, by contracting vectors with it: x.(x^y) = y, y.(x^y) = -x. If you define R(v) = v.(x^y), then general rotation is via the exponential map: e^(Rt)x = (cos t) x + (sin t) Rx. (To be clear: exterior algebra as the algebra of the wedge product operation alone is not enough to do physics. The contraction operation is also necessary (and Hodge star, which is contraction with a particular volume element). But I don't think the geometric product as a standalone operation is useful compared to these components.)
- yiyus 8y ago
  I am having problems to follow your argument. I mean, I understand what you are saying, but not how you arrive to your conclusions. You say the geometric product complicates things, but while you can derive any geometric algebra result simply defining this product (which does not have a difficult definition at all), in the approach you are suggesting you need to define not only the wedge product (which I do not find conceptually simpler than the geometric product) but also contractions. Both approaches are valid, and in certain contexts the differential forms one may be better, but I do not think you can claim it is simpler (when you need to define more concepts) or more fundamental (if you can define wedge product and contraction as a special case of the geometric product, I would say the geometric product is more fundamental). Nevertheless, as I said I would be glad of learning some simpler tool than geometric algebra, and I am not an expert, so I will give you more details about my use case to see if you can come up with a good suggestion (it may happen that although differential forms are a better tool for physics in general, they are not for my particular problems). I work in the field of crystallography. We have to constantly deal with crystal orientations (calculate misorientations between them, distributions of orientations in polycrystals, define properties and behavior in basis to these distributions, and so on). The field has traditionally used Euler angles (ugh!), but when new PhD students start developing computer models, obviously they run into problems, and then someone (usually me) suggests them to learn about quaternions. The students gladly start looking into the subject, but reading definitions that talk about generalizations of complex numbers to four dimensions of stories about mathematical vandalism in bridges do not really help them to build an intuitive idea of how quaternions work, so I usually give them a brief and very informal lecture on geometric algebra (in a very simplified way, I avoid introducing the wedge product for example). After learning the geometric product, they see quaternions as a quite simple concept. We then do a few exercises, like rotating some vectors, calculating misorientations, defining symmetry operators,... Now, you claim the geometric product complicates things, so I am intrigued. However, I do not see how the life of the students could be made any easier without it. I would have to explain what is the wedge product and contractions, rotations expressed as an exponential have the problem of trigonometric functions (and consequent rounding problems), and some simple operations like calculating the rotation between two vectors become problematic. The connection with quaternions is also much less obvious. I have a very basic understanding of differential forms. I found fascinating the Discrete Differential Geometry introduction by Keenan Crane and have read some other introductory material, but that's about it. If you can point me in the right direction to learn how to best use it for my problems, I would appreciate it.
  - ajkjk 8y ago
    My stance is: the wedge product is the fantastic mathematical tool you are looking for, to make sense of rotations and generalizations of complex numbers to higher dimensions, and the "geometric product" (of Hestenes) is not that useful, and almost every time it is useful it's useful because it's got the wedge product as a special case. I think that the vector calculus that should be used in physics should consist of "wedge products and interior products" as the fundamental operations, and the 'geometric product' should probably not be mentioned. That said, this is only my current judgment of the situation, which may change someday. I'm waiting to be convinced that the geometric product is so useful. I just find it so very cludgy when, for instance, GA texts are filled with use of 'grade projection" : take a geometric product with all its terms and then project off the grade you're interested in. It seems like the wrong tool is being used, and almost every time there's a multivector of 'mixed grade', it's regrettable. (Indeed, wedge products are used everywhere already, in hiding -- integration, determinants, matrix minors, commutators, cross products, rotation matrices, projective geometry, grassmann algebra, ...)

by yiyus 8y ago

ajkjk 8y ago

The GA approach is in there; I just think the 'geometric product' itself is not that fundamental. A bivector x^y can be used as a generator of rotations, by contracting vectors with it: x.(x^y) = y, y.(x^y) = -x. If you define R(v) = v.(x^y), then general rotation is via the exponential map: e^(Rt)x = (cos t) x + (sin t) Rx. (To be clear: exterior algebra as the algebra of the wedge product operation alone is not enough to do physics. The contraction operation is also necessary (and Hodge star, which is contraction with a particular volume element). But I don't think the geometric product as a standalone operation is useful compared to these components.)

yiyus 8y ago
I am having problems to follow your argument. I mean, I understand what you are saying, but not how you arrive to your conclusions. You say the geometric product complicates things, but while you can derive any geometric algebra result simply defining this product (which does not have a difficult definition at all), in the approach you are suggesting you need to define not only the wedge product (which I do not find conceptually simpler than the geometric product) but also contractions. Both approaches are valid, and in certain contexts the differential forms one may be better, but I do not think you can claim it is simpler (when you need to define more concepts) or more fundamental (if you can define wedge product and contraction as a special case of the geometric product, I would say the geometric product is more fundamental). Nevertheless, as I said I would be glad of learning some simpler tool than geometric algebra, and I am not an expert, so I will give you more details about my use case to see if you can come up with a good suggestion (it may happen that although differential forms are a better tool for physics in general, they are not for my particular problems). I work in the field of crystallography. We have to constantly deal with crystal orientations (calculate misorientations between them, distributions of orientations in polycrystals, define properties and behavior in basis to these distributions, and so on). The field has traditionally used Euler angles (ugh!), but when new PhD students start developing computer models, obviously they run into problems, and then someone (usually me) suggests them to learn about quaternions. The students gladly start looking into the subject, but reading definitions that talk about generalizations of complex numbers to four dimensions of stories about mathematical vandalism in bridges do not really help them to build an intuitive idea of how quaternions work, so I usually give them a brief and very informal lecture on geometric algebra (in a very simplified way, I avoid introducing the wedge product for example). After learning the geometric product, they see quaternions as a quite simple concept. We then do a few exercises, like rotating some vectors, calculating misorientations, defining symmetry operators,... Now, you claim the geometric product complicates things, so I am intrigued. However, I do not see how the life of the students could be made any easier without it. I would have to explain what is the wedge product and contractions, rotations expressed as an exponential have the problem of trigonometric functions (and consequent rounding problems), and some simple operations like calculating the rotation between two vectors become problematic. The connection with quaternions is also much less obvious. I have a very basic understanding of differential forms. I found fascinating the Discrete Differential Geometry introduction by Keenan Crane and have read some other introductory material, but that's about it. If you can point me in the right direction to learn how to best use it for my problems, I would appreciate it.
- ajkjk 8y ago
  My stance is: the wedge product is the fantastic mathematical tool you are looking for, to make sense of rotations and generalizations of complex numbers to higher dimensions, and the "geometric product" (of Hestenes) is not that useful, and almost every time it is useful it's useful because it's got the wedge product as a special case. I think that the vector calculus that should be used in physics should consist of "wedge products and interior products" as the fundamental operations, and the 'geometric product' should probably not be mentioned. That said, this is only my current judgment of the situation, which may change someday. I'm waiting to be convinced that the geometric product is so useful. I just find it so very cludgy when, for instance, GA texts are filled with use of 'grade projection" : take a geometric product with all its terms and then project off the grade you're interested in. It seems like the wrong tool is being used, and almost every time there's a multivector of 'mixed grade', it's regrettable. (Indeed, wedge products are used everywhere already, in hiding -- integration, determinants, matrix minors, commutators, cross products, rotation matrices, projective geometry, grassmann algebra, ...)