Reflection in Geometric Algebra

Question

When I reflect $v$ through the hyperplane orthogonal to $v-w$, where $w = f(v)$ and $f$ is an isometry on $\mathbb{R}^n$ that preserves the origin. I believe I am supposed to get $w$, however whenever I do the calculation I get: $$\text{Ref}_{v-w}(v) = -(v-w)v(v-w)^{-1} = \frac{-(v-w)v(v-w)}{(v-w)^2} = $$ $$= \frac{(w-v)v(v-w)}{(v-w)^2} = \frac{(wv-v^2)(v-w)}{(v-w)^2} = \frac{wv - v^2}{v-w} = -v$$ No matter what I do I can't seem to get away from this conclusion. Can anybody help me understand what I'm doing wrong?

Answer 1

I found the answer in Guide to Geometric Algebra in Practice by Dorst and Lasenby. As celtschk explains, my use of fraction notation messed me up. This is the correct derivation.

$$-(v-w)v(v-w)^{-1} = (-(v-w)v)(v-w)^{-1} = (-(v-w)((v+w)-w))(v-w)^{-1}=(-(v^2 - w^2) + w(v-w))(v-w)^{-1} = (w(v-w) - (v^2-w^2))(v-w)^{-1} = w - (v^2 - w^2)(v-w)^{-1} = w$$ Because $v^2 = |v|^2 = |w|^2 = w^2$ since $f(v) = w$ is an isometry.

Answer 2

Here's a less convoluted version of your calculation that should make your error more immediately obvious: $$\mathrm{Ref}_{v-w}(v) = −(v−w)v(v−w)^{−1} = -\frac{(v−w)v}{v−w} = -v$$ In short, you've tricked yourself by using fraction notation in a non-commutative algebra. Since geometric algebra is non-commutative, there's a difference between $ab^{-1}$ and $b^{-1}a$ (if that were not the case, you could simply exchange the last two terms in the reflection formula, and then cancel $(v-w)$ with $(v-w)^{-1}$).

Now if you choose to use fraction notation $\frac{a}{b}$ (which is a very bad idea for non-commutative algebra to begin with), you have to decide on a meaning for it. Now in the step $$-(v-w)v(v-w)^{-1} = \frac{-(v-w)v(v-w)}{(v-w)^2}$$ you used the definition $\frac{a}{b} = ab^{-1}$ (it's the only definition under which that equation would hold), but in the step $$\frac{wv - v^2}{v-w} = -v$$ you used the definition $\frac{a}{b} = b^{-1}a$ (again, that's the only definition where that step would be valid). So essentially, in those two steps together, you exchanged the factors in $v(v-w)^{-1}$; all the other stuff only serves to obfuscate this. And thus your result is easily explained, as indeed $$-(v-w)(v-w)^{-1}v = -v,$$ it's just no longer the reflection of $v$ on a hyperplane.

Bottom line: When dealing with non-commutative algebra, don't use fraction notation. It only invites errors such as this one.