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78 lines
3.2 KiB
Go
78 lines
3.2 KiB
Go
// Copyright ©2021 The Gonum Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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package r3_test
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import (
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"fmt"
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"math"
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"gonum.org/v1/gonum/num/quat"
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"gonum.org/v1/gonum/spatial/r3"
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)
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// euler returns an r3.Rotation that corresponds to the Euler
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// angles alpha, beta and gamma which are rotations around the x,
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// y and z axes respectively. The order of rotations is x, y, z;
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// there are many conventions for this ordering.
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func euler(alpha, beta, gamma float64) r3.Rotation {
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// Note that this function can be algebraically simplified
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// to reduce floating point operations, but is left in this
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// form for clarity.
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var rot1, rot2, rot3 quat.Number
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rot1.Imag, rot1.Real = math.Sincos(alpha / 2) // x-axis rotation
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rot2.Jmag, rot2.Real = math.Sincos(beta / 2) // y-axis rotation
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rot3.Kmag, rot3.Real = math.Sincos(gamma / 2) // z-axis rotation
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return r3.Rotation(quat.Mul(rot3, quat.Mul(rot2, rot1))) // order of rotations
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}
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func ExampleRotation_eulerAngles() {
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// It is possible to interconvert between the quaternion representation
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// of a rotation and Euler angles, but this leads to problems.
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//
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// The first of these is that there are a variety of conventions for
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// application of the rotations.
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//
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// The more serious consequence of using Euler angles is that it is
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// possible to put the rotation system into a singularity which results
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// in loss of degrees of freedom and so causes gimbal lock. This happens
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// when the second axis to be rotated around is rotated to 𝝿/2.
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//
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// See https://en.wikipedia.org/wiki/Euler_angles for more details.
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pt := r3.Vec{1, 0, 0}
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// For the Euler conversion function in this example, the second rotation
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// is around the y-axis.
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const singularY = math.Pi / 2
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arb := math.Pi / 4
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fmt.Printf("rotate around x-axis: %.2f\n", euler(arb, 0, 0).Rotate(pt))
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fmt.Printf("rotate around y-axis: %.2f\n", euler(0, arb, 0).Rotate(pt))
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fmt.Printf("rotate around z-axis: %.2f\n", euler(0, 0, arb).Rotate(pt))
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fmt.Printf("rotate around x+y-axes: %.2f\n", euler(arb, arb, 0).Rotate(pt))
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fmt.Printf("rotate around x+z-axes: %.2f\n", euler(arb, 0, arb).Rotate(pt))
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fmt.Printf("rotate around y+z-axes: %.2f\n", euler(0, arb, arb).Rotate(pt))
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fmt.Printf("rotate around y-axis to singularity: %.2f\n", euler(0, singularY, 0).Rotate(pt))
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fmt.Printf("rotate around x+y-axes with singularity → gimbal lock: %.2f\n", euler(arb, singularY, 0).Rotate(pt))
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fmt.Printf("rotate around z+y-axes with singularity → gimbal lock: %.2f\n", euler(0, singularY, arb).Rotate(pt))
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fmt.Printf("rotate around all-axes with singularity → gimbal lock: %.2f\n", euler(arb, singularY, arb).Rotate(pt))
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// Output:
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//
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// rotate around x-axis: {1.00 0.00 0.00}
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// rotate around y-axis: {0.71 0.00 -0.71}
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// rotate around z-axis: {0.71 0.71 0.00}
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// rotate around x+y-axes: {0.71 0.00 -0.71}
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// rotate around x+z-axes: {0.71 0.71 0.00}
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// rotate around y+z-axes: {0.50 0.50 -0.71}
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// rotate around y-axis to singularity: {0.00 0.00 -1.00}
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// rotate around x+y-axes with singularity → gimbal lock: {0.00 0.00 -1.00}
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// rotate around z+y-axes with singularity → gimbal lock: {0.00 0.00 -1.00}
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// rotate around all-axes with singularity → gimbal lock: {0.00 0.00 -1.00}
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}
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