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Interpolating solid orientations with circular blending quaternion curves

https://doi.org/10.1016/0010-4485(95)96802-SGet rights and content

Abstract

The paper presents a method to smoothly interpolate a given sequence of solid orientations using circular blending quaternion curves. Given three solid orientations, a circular quaternion curve is constructed that interpolates the three orientations. Therefore, given four orientations qi − 1, qi, qi + 1, qi + 2, there are two circular quaternion curves Ci and Ci + 1 which interpolate the triples of orientations (qi − 1, qi, qi + 1) and (qi, qi + 1, qi + 2), respectively; thus, both Ci and Ci + 1 interpolate the two orientations qi and qi + 1.

Using a method similar to the parabolic blending of Overhauser, quaternion curve Qi(t) is generated which interpolates two orientations qi and qi + 1 while smoothly blending the two circular quaternion curves Ci(t) and Ci + 1(t) with a blending function f(t) of degree (2k − 1). The quaternion curve Qi has the same derivatives (up to order k) with Ci at qi and with Ci + 1 at qi + 1, respectively. By connecting the quaternion curve segments Qi in a connected sequence, a Ck-continuous quaternion path is generated which smoothly interpolates a given sequence of solid orientations.

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