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Quaternion: Difference between revisions
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===Geodesic arc-length distance=== | ===Geodesic arc-length distance=== | ||
For two unit quaternions <math>q_1</math> and <math>q_2</math>, the geodesic arc length is <math>\alpha</math> where <math>q_1 \cdot q_2 = \cos \alpha</math>. | For two unit quaternions <math>q_1</math> and <math>q_2</math>, the geodesic arc length is <math>\alpha</math> where <math>q_1 \cdot q_2 = \cos \alpha</math>. | ||
Because of sign ambiguity | Because of sign ambiguity, <math>R(q_2) = R(-q_2)</math>, we take the minimum value: | ||
<math display="block"> | <math display="block"> | ||
d_{geodesic}(q_1, q_2) = \min(\alpha, \pi-\alpha) \text{ where } | d_{geodesic}(q_1, q_2) = \min(\alpha, \pi-\alpha) \text{ where } | ||
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==Combined Point + Frame Alignment Problem== | ==Combined Point + Frame Alignment Problem== | ||
Given a spatial profile matrix <math>M(E)</math> and orientation frame profile matrix <math>U(S)</math>, we can normalize the scale to have unit-eigenvalue and | Given a spatial profile matrix <math>M(E)</math> and orientation frame profile matrix <math>U(S)</math>, we can normalize the scale to have unit-eigenvalue and then apply linear interpolation with dimensional constant <math>\sigma</math>: | ||
<math display="block"> | <math display="block"> | ||
\Delta_{xf}(t, \sigma) = q \cdot \left[ (1-t) \frac{M(E)}{\epsilon_{x}} + t \sigma \frac{U(S)}{\epsilon_{f}} \right] \cdot q | \Delta_{xf}(t, \sigma) = q \cdot \left[ (1-t) \frac{M(E)}{\epsilon_{x}} + t \sigma \frac{U(S)}{\epsilon_{f}} \right] \cdot q | ||