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Latest revision as of 01:40, 28 August 2024
Conformal Geometric Algebra
This wiki is a repository of information about Conformal Geometric Algebra, and specifically the five-dimensional Clifford algebra $$\mathcal G_{4,1}$$. This wiki is associated with the following websites:
Conformal geometric algebra is an area of active research, and new information is frequently being added to this wiki.
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Introduction
Conformal geometric algebra is constructed by adding two projective basis vectors called $$\mathbf e_-$$ and $$\mathbf e_+$$ to the set of ordinary basis vectors $$\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n$$ of n-dimensional Euclidean space. The new vectors are named this way because their squares under the dot product are
- $$\mathbf e_- \cdot \mathbf e_- = -1 \qquad \mathrm{and} \qquad \mathbf e_+ \cdot \mathbf e_+ = +1$$ .
This by itself is enough to build the entire algebra and observe all of its emergent properties. What follows is the standard way to interpret what's going on when we talk about specific vectors, bivectors, etc., and start multiplying them together with the exterior product and geometric product. Using 3D Euclidean space as an example, we begin by considering a homogeneous point
- $$\mathbf p = x \mathbf e_1 + y \mathbf e_2 + z \mathbf e_3 + w \mathbf e_-$$ .
As in rigid geometric algebra, any nonzero scalar multiple of this point belongs to the same equivalence class, so we can just assume $$w = 1$$. Now we perform a stereographic projection onto the four-dimensional unit hypersphere centered at $$\mathbf e_-$$ and extending into the $${\mathbf e_1, \mathbf e_2, \mathbf e_3, \mathbf e_+}$$ subspace. The north pole of this hypersphere toward which points are projected is $$\mathbf e_- + \mathbf e_+$$. This projection transforms the point $$\mathbf p$$ into
- $$\mathbf p = \dfrac{2x}{x^2 + y^2 + z^2 + 1} \mathbf e_1 + \dfrac{2y}{x^2 + y^2 + z^2 + 1} \mathbf e_2 + \dfrac{2z}{x^2 + y^2 + z^2 + 1} \mathbf e_3 + \mathbf e_- + \dfrac{x^2 + y^2 + z^2 - 1}{x^2 + y^2 + z^2 + 1} \mathbf e_+$$ .
We can homogeneously multiply this by $$(x^2 + y^2 + z^2 + 1)/2$$ without changing the meaning of $$\mathbf p$$ to obtain
- $$\mathbf p = x \mathbf e_1 + y \mathbf e_2 + z \mathbf e_3 + \dfrac{1}{2}(x^2 + y^2 + z^2 + 1) \mathbf e_- + \dfrac{1}{2}(x^2 + y^2 + z^2 - 1) \mathbf e_+$$ .
Regrouping the coefficients of the $$\mathbf e_-$$ and $$\mathbf e_+$$ terms lets us write
- $$\mathbf p = x \mathbf e_1 + y \mathbf e_2 + z \mathbf e_3 + \dfrac{1}{2}(\mathbf e_- - \mathbf e_+) + \dfrac{1}{2}(x^2 + y^2 + z^2)(\mathbf e_- + \mathbf e_+)$$ .
It is convenient to define two new vectors $$\mathbf e_4$$ and $$\mathbf e_5$$ as
- $$\mathbf e_4 = \dfrac{1}{2}(\mathbf e_- - \mathbf e_+) \qquad \mathrm{and} \qquad \mathbf e_5 = \mathbf e_- + \mathbf e_+$$ .
Using these two vectors, the point $$\mathbf p$$ is finally expressed as
- $$\mathbf p = x \mathbf e_1 + y \mathbf e_2 + z \mathbf e_3 + \mathbf e_4 + \dfrac{1}{2}(x^2 + y^2 + z^2) \mathbf e_5$$ .
Numerous alternate definitions for $$\mathbf e_4$$ and $$\mathbf e_5$$ in which $$\mathbf e_4 = a(\mathbf e_- - \mathbf e_+)$$ and $$\mathbf e_5 = b(\mathbf e_- + \mathbf e_+)$$ are possible, but the definitions above where $$a = 1/2$$ and $$b = 1$$ tend to produce the cleanest formulation of the entire algebra. In particular, any definitions in which $$ab = 1/2$$ make the bivectors $$\mathbf e_- \wedge \mathbf e_+$$ and $$\mathbf e_4 \wedge \mathbf e_5$$ equal to each other.
When we consider the case that $$(x, y, z) = (0, 0, 0)$$, we can clearly identify the vector $$\mathbf e_4$$ as the origin. If we homogeneously rescale $$\mathbf p$$ as
- $$\mathbf p = \dfrac{2x}{x^2 + y^2 + z^2} \mathbf e_1 + \dfrac{2y}{x^2 + y^2 + z^2} \mathbf e_2 + \dfrac{2z}{x^2 + y^2 + z^2} \mathbf e_3 + \dfrac{2}{x^2 + y^2 + z^2} \mathbf e_4 + \mathbf e_5$$ ,
and allow the magnitude of $$(x, y, z)$$ to become arbitrarily large, then it's apparent that $$\mathbf e_5$$ represents the point at infinity.
The following diagram illustrates the image of the x axis under the homogeneous stereographic projection that we defined $$\mathbf p$$ with. Euclidean space becomes a parabolic surface called the horosphere. The x-y plane is mapped to a paraboloid, and the full three-dimensional Euclidean space is mapped to a higher-dimensional parabolic volume.
