G1 motion interpolation using cubic PH biarcs with prescribed length

Knez, Marjeta

doi:10.1016/j.cagd.2018.09.004

Cited by 5 publications

(3 citation statements)

References 28 publications

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“…We substitute (25)-(26) into the third of equations (16), and into equations (27)- (28). For equation (29), we multiply both sides by exp(i ψ), substitute (25)- (26), and separate real and imaginary parts. On clearing denominators, this yields a system of five real equations of the form…”

Section: Curve Construction Schemementioning

confidence: 99%

“…The corresponding complex coefficients (25)-(26) are α 0 = 1.531804 + 0.000000 i , β 0 = 0.000000 + 0.000000 i ,α 1 = 0.957587 + 0.000000 i , β 1 = 0.777819 − 0.468314 i , α 2 = 0.745894 + 0.000000 i , β 2 = 0.129891 − 0.617865 i , α 3 = 1.546118 + 0.822085 i , β 3 = −0.892652 − 0.474631 i ,and from the quaternion coefficients A i = α i + k β i the curve control points are determined using (36) as p 0 = (0.000000, 0.000000, 0.000000) , p 1 = (0.335203, 0.000000, 0.000000) , p 2 = (0.544751, 0.170210, 0.102481) , p 3 = (0.617982, 0.309264, 0.233441) , p 4 = (0.693456, 0.380316, 0.364809) , p 5 = (0.818552, 0.394800, 0.547700) , p 6 = (0.957971, 0.255809, 0.750000) , p 7 = (1.250000, −0.250000, 0.750000) .…”

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confidence: 99%

“…The ability to impose a prescribed arc length on a polynomial free-form curve segment is unique to the PH curves -for example, see[10,11,26].…”

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confidence: 99%

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Rational minimal-twist motions on curves with rotation-minimizing Euler–Rodrigues frames

Farouki

Giannelli

Sestini

2019

Journal of Computational and Applied Mathematics

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A minimal twist frame (f 1 (ξ), f 2 (ξ), f 3 (ξ)) on a polynomial space curve r(ξ), ξ ∈ [ 0, 1 ] is an orthonormal frame, where f 1 (ξ) is the tangent and the normal-plane vectors f 2 (ξ), f 3 (ξ) have the least variation between given initial and final instances f 2 (0), f 3 (0) and f 2 (1), f 3 (1). Namely, if ω = ω 1 f 1 +ω 2 f 2 +ω 3 f 3 is the frame angular velocity, the component ω 1 does not change sign, and its arc length integral has the smallest value consistent with the boundary conditions. We consider construction of curves with rational minimal twist frames, based on the Pythagoreanhodograph curves of degree 7 that have rational rotation-minimizing Euler-Rodrigues frames (e 1 (ξ), e 2 (ξ), e 3 (ξ)) -i.e., the normal-plane vectors e 2 (ξ), e 3 (ξ) have no rotation about the tangent e 1 (ξ). A set of equations that govern the construction of such curves with prescribed initial/final points and tangents, and total arc length, is derived. For the resulting curves f 2 (ξ), f 3 (ξ) are then obtained from e 2 (ξ), e 3 (ξ) by a monotone rational normal-plane rotation, subject to the boundary conditions. A selection of computed examples is included to illustrate the construction, and it is shown that the desirable feature of a uniform rotation rate (i.e., ω 1 = constant) can be accurately approximated.

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Section: Curve Construction Schemementioning

confidence: 99%

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confidence: 99%