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- A ''quaternion'' is an operator that performs a rotation about the origin in 3D space. Conventionally, a quaternion $$\mathbf q$$ is ...ector representing the axis of rotation, and $$\phi$$ is half the angle of rotation. ...3 KB (494 words) - 06:23, 15 July 2023
- A ''rotation'' is a proper isometry of Euclidean space. ...the case that $$R_{mw} = 0$$. The line $$\boldsymbol l$$ and its [[weight dual]] $$\boldsymbol l^\unicode["segoe ui symbol"]{x2606}$$ are invariant under ...3 KB (414 words) - 07:11, 8 August 2024
- '''4.''' Let $$\mathbf m$$ be a $$4 \times 4$$ matrix that performs a rotation about the $$z$$ axis in homogeneous coordinates. Calculate the $$16 \times ...e ui symbol"]{x2605}\unicode["segoe ui symbol"]{x2605}}$$, the double bulk dual of $$\mathbf u$$, that uses only $$\operatorname{gr}(\mathbf u)$$, $$\opera ...3 KB (522 words) - 04:49, 4 May 2024
- ...represents a proper Euclidean isometry, which can always be regarded as a rotation about a line $$\boldsymbol l$$ and a displacement along the same line.]] ...ion operator'' or ''moment vector''. Motors are equivalent to the set of ''dual quaternions'' used in conventional theories, and the functionality is prope ...19 KB (3,294 words) - 01:19, 8 July 2024
- ...Plücker coordinates, [[quaternions]], and screw theory (which makes use of dual quaternions). This makes rigid geometric algebra a natural fit for areas of ...de of its inputs and outputs, we can make the same statement about how the dual operation relates to the antigrade of its inputs and outputs. ...8 KB (1,184 words) - 03:02, 10 August 2024