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Last active May 30, 2023 19:52
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Compare quaternion interpolation methods
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{
"cells": [
{
"cell_type": "markdown",
"id": "5558daf0-2c2d-4bae-a3f0-95d0085c0f08",
"metadata": {},
"source": [
"This notebook compares various approaches to creating splines for quaternions, as discussed in [this issue](https://github.com/moble/quaternionic/issues/41). I have re-implemented `Squad` as @mgeier's [suggestion for `quaternionic`](https://github.com/moble/quaternionic/pull/42), because it was wrong in `splines`.\n",
"\n",
"Also, I strenuously object to automatically \"unflipping\" rotors, but for the sake of comparison, this is what I mostly do below."
]
},
{
"cell_type": "markdown",
"id": "7dbdae97-d7ce-4262-8e30-ad7211d18afb",
"metadata": {},
"source": [
"# Code loading"
]
},
{
"cell_type": "raw",
"id": "cc0806dd-22ec-4dac-a51b-c6444a6198ce",
"metadata": {
"execution": {
"iopub.execute_input": "2023-05-30T14:54:30.575382Z",
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"source": [
"# Highlight this cell and press \"y\" to convert to python code, then run it if\n",
"# you need to install `numpy`, `matplotlib`, `quaternionic`, and/or `splines`.\n",
"\n",
"import sys\n",
"from pathlib import Path\n",
"\n",
"executable = str(Path(sys.executable).resolve())\n",
"\n",
"! {executable} -m pip install numpy matplotlib quaternionic splines"
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "f81ec81d-1da8-4df5-b942-d6e1c262c0b0",
"metadata": {
"execution": {
"iopub.execute_input": "2023-05-30T18:45:54.174685Z",
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"outputs": [],
"source": [
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"from scipy.interpolate import CubicSpline\n",
"\n",
"import quaternionic\n",
"from splines.quaternion import CatmullRom, UnitQuaternion\n",
"from helper import angles2quat, animate_rotations, display_animation"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "a9874dc2-42ad-45e3-99f7-3ff40a33ea83",
"metadata": {
"execution": {
"iopub.execute_input": "2023-05-30T18:45:55.478484Z",
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"shell.execute_reply": "2023-05-30T18:45:55.485546Z",
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"tags": []
},
"outputs": [],
"source": [
"# Some helpful utilities for converting between `splines` and `quaternionic`\n",
"\n",
"def uq(q):\n",
" return UnitQuaternion.from_unit_xyzw(q.vector.tolist() + [q.w[()],])\n",
"\n",
"def q(uq):\n",
" try:\n",
" iterator = iter(uq)\n",
" except TypeError:\n",
" return quaternionic.array(uq.wxyz)\n",
" else:\n",
" return quaternionic.array([uqi.wxyz for uqi in uq])\n",
"\n",
"class QuaternionicSquad:\n",
" def __init__(self, Rs, t):\n",
" self.Rs = Rs\n",
" self.grid = t\n",
" def evaluate(self, t):\n",
" try:\n",
" iterator = iter(t)\n",
" except TypeError:\n",
" return uq(quaternionic.squad(self.Rs, self.grid, np.array([t])[0]))\n",
" else:\n",
" return list(map(uq, quaternionic.squad(self.Rs, self.grid, np.array(t))))"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "39708d61-b5d9-4d8b-a61e-e54a3b9542f7",
"metadata": {
"execution": {
"iopub.execute_input": "2023-05-30T18:45:55.487893Z",
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},
"tags": []
},
"outputs": [],
"source": [
"def evaluate(spline, frames=200):\n",
" times = np.linspace(\n",
" spline.grid[0], spline.grid[-1], frames, endpoint=False)\n",
" return spline.evaluate(times)"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "edd36df0-e4da-4c35-bae2-df3b5115f559",
"metadata": {
"execution": {
"iopub.execute_input": "2023-05-30T18:45:55.494709Z",
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},
"outputs": [],
"source": [
"def squad(R_in, t_in, t_out):\n",
" from functools import partial\n",
" from quaternionic import slerp\n",
"\n",
" roll = partial(np.roll, axis=0)\n",
"\n",
" if R_in.size == 0 or t_out.size == 0:\n",
" return quaternionic.array(np.zeros((0, 4)))\n",
"\n",
" R_in = quaternionic.array(R_in)\n",
" i_in_for_out = t_in.searchsorted(t_out, side='right')-1\n",
" t_in_broadcast_shape = t_in.shape + (1,)*len(R_in.shape[1:])\n",
" t_out_broadcast_shape = t_out.shape + (1,)*len(R_in.shape[1:])\n",
" A = R_in * np.exp(\n",
" (\n",
" - np.log((~R_in) * roll(R_in, -1))\n",
" + np.log((~roll(R_in, 1)) * R_in)\n",
" * np.reshape((roll(t_in, -1) - t_in) / (t_in - roll(t_in, 1)), t_in_broadcast_shape)[..., 0]\n",
" ) * (roll(t_in, -1) - t_in) / (2 * (roll(t_in, -1) - roll(t_in, 1)))\n",
" )\n",
" B = roll(R_in, -1) * np.exp(\n",
" (\n",
" np.log((~roll(R_in, -1)) * roll(R_in, -2))\n",
" * np.reshape((roll(t_in, -1) - t_in) / (roll(t_in, -2) - roll(t_in, -1)), t_in_broadcast_shape)[..., 0]\n",
" - np.log((~R_in) * roll(R_in, -1))\n",
" ) * -(roll(t_in, -1) - t_in) / (2 * (roll(t_in, -2) - t_in))\n",
" )\n",
" A[0] = R_in[0]\n",
" A[-1] = R_in[-1]\n",
" B[-2] = R_in[-1]\n",
" B[-1] = R_in[-1] * (~R_in[-2]) * R_in[-1]\n",
" R_ip1 = roll(R_in, -1)\n",
" R_ip1[-1] = R_in[-1]*(~R_in[-2])*R_in[-1]\n",
" R_ip1 = quaternionic.array(R_ip1[i_in_for_out])\n",
" t_inp1 = roll(t_in, -1)\n",
" t_inp1[-1] = t_in[-1] + (t_in[-1] - t_in[-2])\n",
" tau = np.reshape((t_out - t_in[i_in_for_out]) / ((t_inp1 - t_in)[i_in_for_out]), t_out_broadcast_shape)[..., 0]\n",
"\n",
" R_out = slerp(\n",
" slerp(R_in[i_in_for_out], R_ip1, tau),\n",
" slerp(A[i_in_for_out], B[i_in_for_out], tau),\n",
" 2*tau*(1-tau)\n",
" )\n",
"\n",
" return R_out\n",
"\n",
"class Squad:\n",
" def __init__(self, Rs, t):\n",
" self.Rs = quaternionic.unflip_rotors(Rs)\n",
" self.grid = t\n",
" def evaluate(self, t):\n",
" try:\n",
" iterator = iter(t)\n",
" except TypeError:\n",
" return uq(squad(self.Rs, self.grid, np.array([t])[0]))\n",
" else:\n",
" return list(map(uq, squad(self.Rs, self.grid, np.array(t))))"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "3df98096-a6cc-4843-841a-d0b11831ef19",
"metadata": {
"execution": {
"iopub.execute_input": "2023-05-30T18:45:55.581118Z",
"iopub.status.busy": "2023-05-30T18:45:55.580599Z",
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},
"tags": []
},
"outputs": [],
"source": [
"rotations = [\n",
" angles2quat(0, 0, 0),\n",
" angles2quat(90, 0, -45),\n",
" angles2quat(-45, 45, -90),\n",
" angles2quat(135, -35, 90),\n",
" angles2quat(90, 0, 0),\n",
"]\n",
"\n",
"# `splines` treats the rotations as periodic\n",
"rotations_periodic = rotations + rotations[:1]\n",
"\n",
"Rs = q(rotations_periodic) # quaternionic.array([np.array(q.wxyz) for q in rotations_periodic])\n",
"\n",
"## I'm gonna make these times a little more non-uniform to accentuate things\n",
"#times = 0, 0.75, 1.2, 2, 3.5, 4\n",
"times = 0, 0.15, 1.2, 2, 3.9, 4\n",
"t = np.array(times)\n",
"\n",
"#sq = Squad(rotations)\n",
"cr = CatmullRom(rotations, endconditions='closed')\n",
"\n",
"#sq2 = Squad(rotations, alpha=0.5)\n",
"cr2 = CatmullRom(rotations, alpha=0.5, endconditions='closed')\n",
"\n",
"sq3 = Squad(Rs, t)\n",
"cr3 = CatmullRom(rotations, times, endconditions='closed')"
]
},
{
"cell_type": "markdown",
"id": "777dbffb-29c0-4d9e-a0bf-8c414c7fd435",
"metadata": {},
"source": [
"# Flipped rotors"
]
},
{
"cell_type": "markdown",
"id": "a1b0153b-3a5e-4a13-81c5-f8b690cb207b",
"metadata": {},
"source": [
"The first thing I note is that `sq3` and `cr3` do not actually reproduce the input data:"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "2659ba34-6422-4233-a515-dff2fb5d20a5",
"metadata": {
"execution": {
"iopub.execute_input": "2023-05-30T18:45:55.591364Z",
"iopub.status.busy": "2023-05-30T18:45:55.590988Z",
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"shell.execute_reply": "2023-05-30T18:45:55.603288Z",
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},
"tags": []
},
"outputs": [
{
"data": {
"text/plain": [
"False"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sq3.evaluate(t) == rotations_periodic"
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "54db7c81-a014-4912-a906-009b398c9618",
"metadata": {
"execution": {
"iopub.execute_input": "2023-05-30T18:45:55.608074Z",
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},
"outputs": [
{
"data": {
"text/plain": [
"False"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"all(cr3.evaluate(times) == rotations_periodic)"
]
},
{
"cell_type": "markdown",
"id": "e64a9c60-8792-4da7-8e91-2a29271a3709",
"metadata": {},
"source": [
"The reason is that there is a sign flip in the middle."
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "d74c54c6-6997-4149-8021-6a31a3f26a61",
"metadata": {
"execution": {
"iopub.execute_input": "2023-05-30T18:45:55.619314Z",
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},
"outputs": [
{
"data": {
"text/plain": [
"[UnitQuaternion(scalar=1.0, vector=(0.0, 0.0, 0.0)),\n",
" UnitQuaternion(scalar=0.6532814824381883, vector=(0.27059805007309845, -0.2705980500730985, 0.6532814824381882)),\n",
" UnitQuaternion(scalar=0.5, vector=(5.551115123125783e-17, -0.7071067811865475, -0.5)),\n",
" UnitQuaternion(scalar=-0.4545194776720437, vector=(0.7044160264027586, -0.06162841671621935, -0.5416752204197018)),\n",
" UnitQuaternion(scalar=-0.7071067811865476, vector=(0.0, 0.0, -0.7071067811865475)),\n",
" UnitQuaternion(scalar=-1.0, vector=(0.0, 0.0, 0.0))]"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sq3.evaluate(times)"
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "adbee9ad-1167-4f9b-b104-b17aa43e8bac",
"metadata": {
"execution": {
"iopub.execute_input": "2023-05-30T18:45:55.630311Z",
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"tags": []
},
"outputs": [
{
"data": {
"text/plain": [
"[UnitQuaternion(scalar=1.0, vector=(0.0, 0.0, 0.0)),\n",
" UnitQuaternion(scalar=0.6532814824381883, vector=(0.27059805007309845, -0.2705980500730985, 0.6532814824381882)),\n",
" UnitQuaternion(scalar=0.5, vector=(5.551115123125783e-17, -0.7071067811865475, -0.5)),\n",
" UnitQuaternion(scalar=0.4545194776720437, vector=(-0.7044160264027586, 0.06162841671621935, 0.5416752204197018)),\n",
" UnitQuaternion(scalar=0.7071067811865476, vector=(0.0, 0.0, 0.7071067811865475)),\n",
" UnitQuaternion(scalar=1.0, vector=(0.0, 0.0, 0.0))]"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"rotations_periodic"
]
},
{
"cell_type": "markdown",
"id": "9deba2c1-8d9a-4cb9-9c3a-94b5cc9fa98b",
"metadata": {},
"source": [
"This looks a lot like what would happen if I applied [`unflip_rotors`](https://github.com/moble/quaternionic/blob/074b626d0c63aa78479ff04ed41638931ca6693a/quaternionic/interpolation.py#L28) to the input:"
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "63703b43-5a48-4ffc-9368-58a920cd9a10",
"metadata": {
"execution": {
"iopub.execute_input": "2023-05-30T18:45:55.639165Z",
"iopub.status.busy": "2023-05-30T18:45:55.638402Z",
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},
"tags": []
},
"outputs": [
{
"data": {
"text/plain": [
"quaternionic.array([[ 1.00000000e+00, 0.00000000e+00, 0.00000000e+00,\n",
" 0.00000000e+00],\n",
" [ 6.53281482e-01, 2.70598050e-01, -2.70598050e-01,\n",
" 6.53281482e-01],\n",
" [ 5.00000000e-01, 5.55111512e-17, -7.07106781e-01,\n",
" -5.00000000e-01],\n",
" [-4.54519478e-01, 7.04416026e-01, -6.16284167e-02,\n",
" -5.41675220e-01],\n",
" [-7.07106781e-01, -0.00000000e+00, -0.00000000e+00,\n",
" -7.07106781e-01],\n",
" [-1.00000000e+00, -0.00000000e+00, -0.00000000e+00,\n",
" -0.00000000e+00]])"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"quaternionic.unflip_rotors(Rs)"
]
},
{
"cell_type": "markdown",
"id": "96fbd40a-4234-4c10-b421-adfac9f0933a",
"metadata": {},
"source": [
"Let's take a look at what my code does."
]
},
{
"cell_type": "code",
"execution_count": 11,
"id": "53be7a68-d47a-454f-95d3-441925916f1e",
"metadata": {
"execution": {
"iopub.execute_input": "2023-05-30T18:45:55.646326Z",
"iopub.status.busy": "2023-05-30T18:45:55.645908Z",
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"shell.execute_reply": "2023-05-30T18:45:55.766536Z",
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},
"tags": []
},
"outputs": [],
"source": [
"qs = QuaternionicSquad(Rs, t)\n",
"qs2 = QuaternionicSquad(quaternionic.unflip_rotors(Rs), t)"
]
},
{
"cell_type": "markdown",
"id": "45976454-2913-4b90-bce5-3736c41e6178",
"metadata": {},
"source": [
"Here, `qs` reproduces the input data exactly"
]
},
{
"cell_type": "code",
"execution_count": 12,
"id": "ee503ceb-3776-4fc8-98ce-900e47818201",
"metadata": {
"execution": {
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"iopub.status.busy": "2023-05-30T18:45:55.769277Z",
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"shell.execute_reply": "2023-05-30T18:45:55.777497Z",
"shell.execute_reply.started": "2023-05-30T18:45:55.769634Z"
},
"tags": []
},
"outputs": [
{
"data": {
"text/plain": [
"True"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"qs.evaluate(times) == rotations_periodic"
]
},
{
"cell_type": "markdown",
"id": "26f31d7e-e071-4373-959d-f859320713d4",
"metadata": {},
"source": [
"whereas `qs2` reproduces the results of `sq3` exactly:"
]
},
{
"cell_type": "code",
"execution_count": 13,
"id": "d6cec628-cdf7-4d0c-b588-9778348babb9",
"metadata": {
"execution": {
"iopub.execute_input": "2023-05-30T18:45:55.779826Z",
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},
"outputs": [
{
"data": {
"text/plain": [
"True"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"qs2.evaluate(times) == sq3.evaluate(times)"
]
},
{
"cell_type": "markdown",
"id": "a8bafd98-f897-4be6-8db8-4a1e51e5eab5",
"metadata": {},
"source": [
"So it looks as though `splines` effectively unflips its rotors on input.\n",
"\n",
"You can actually see this difference quite clearly in the animation comparing `quaternionic` (on the left) to `splines` (on the right):"
]
},
{
"cell_type": "code",
"execution_count": 14,
"id": "b3ab58ca-aafd-4b3a-b565-e10a651f45b2",
"metadata": {
"execution": {
"iopub.execute_input": "2023-05-30T18:45:55.797094Z",
"iopub.status.busy": "2023-05-30T18:45:55.796666Z",
"iopub.status.idle": "2023-05-30T18:46:07.097462Z",
"shell.execute_reply": "2023-05-30T18:46:07.095939Z",
"shell.execute_reply.started": "2023-05-30T18:45:55.797051Z"
},
"tags": []
},
"outputs": [
{
"data": {
"text/html": [
"\n",
"<link rel=\"stylesheet\"\n",
"href=\"https://maxcdn.bootstrapcdn.com/font-awesome/4.4.0/css/font-awesome.min.css\">\n",
"<script language=\"javascript\">\n",
" function isInternetExplorer() {\n",
" ua = navigator.userAgent;\n",
" /* MSIE used to detect old browsers and Trident used to newer ones*/\n",
" return ua.indexOf(\"MSIE \") > -1 || ua.indexOf(\"Trident/\") > -1;\n",
" }\n",
"\n",
" /* Define the Animation class */\n",
" function Animation(frames, img_id, slider_id, interval, loop_select_id){\n",
" this.img_id = img_id;\n",
" this.slider_id = slider_id;\n",
" this.loop_select_id = loop_select_id;\n",
" this.interval = interval;\n",
" this.current_frame = 0;\n",
" this.direction = 0;\n",
" this.timer = null;\n",
" this.frames = new Array(frames.length);\n",
"\n",
" for (var i=0; i<frames.length; i++)\n",
" {\n",
" this.frames[i] = new Image();\n",
" this.frames[i].src = frames[i];\n",
" }\n",
" var slider = document.getElementById(this.slider_id);\n",
" slider.max = this.frames.length - 1;\n",
" if (isInternetExplorer()) {\n",
" // switch from oninput to onchange because IE <= 11 does not conform\n",
" // with W3C specification. It ignores oninput and onchange behaves\n",
" // like oninput. In contrast, Microsoft Edge behaves correctly.\n",
" slider.setAttribute('onchange', slider.getAttribute('oninput'));\n",
" slider.setAttribute('oninput', null);\n",
" }\n",
" this.set_frame(this.current_frame);\n",
" }\n",
"\n",
" Animation.prototype.get_loop_state = function(){\n",
" var button_group = document[this.loop_select_id].state;\n",
" for (var i = 0; i < button_group.length; i++) {\n",
" var button = button_group[i];\n",
" if (button.checked) {\n",
" return button.value;\n",
" }\n",
" }\n",
" return undefined;\n",
" }\n",
"\n",
" Animation.prototype.set_frame = function(frame){\n",
" this.current_frame = frame;\n",
" document.getElementById(this.img_id).src =\n",
" this.frames[this.current_frame].src;\n",
" document.getElementById(this.slider_id).value = this.current_frame;\n",
" }\n",
"\n",
" Animation.prototype.next_frame = function()\n",
" {\n",
" this.set_frame(Math.min(this.frames.length - 1, this.current_frame + 1));\n",
" }\n",
"\n",
" Animation.prototype.previous_frame = function()\n",
" {\n",
" this.set_frame(Math.max(0, this.current_frame - 1));\n",
" }\n",
"\n",
" Animation.prototype.first_frame = function()\n",
" {\n",
" this.set_frame(0);\n",
" }\n",
"\n",
" Animation.prototype.last_frame = function()\n",
" {\n",
" this.set_frame(this.frames.length - 1);\n",
" }\n",
"\n",
" Animation.prototype.slower = function()\n",
" {\n",
" this.interval /= 0.7;\n",
" if(this.direction > 0){this.play_animation();}\n",
" else if(this.direction < 0){this.reverse_animation();}\n",
" }\n",
"\n",
" Animation.prototype.faster = function()\n",
" {\n",
" this.interval *= 0.7;\n",
" if(this.direction > 0){this.play_animation();}\n",
" else if(this.direction < 0){this.reverse_animation();}\n",
" }\n",
"\n",
" Animation.prototype.anim_step_forward = function()\n",
" {\n",
" this.current_frame += 1;\n",
" if(this.current_frame < this.frames.length){\n",
" this.set_frame(this.current_frame);\n",
" }else{\n",
" var loop_state = this.get_loop_state();\n",
" if(loop_state == \"loop\"){\n",
" this.first_frame();\n",
" }else if(loop_state == \"reflect\"){\n",
" this.last_frame();\n",
" this.reverse_animation();\n",
" }else{\n",
" this.pause_animation();\n",
" this.last_frame();\n",
" }\n",
" }\n",
" }\n",
"\n",
" Animation.prototype.anim_step_reverse = function()\n",
" {\n",
" this.current_frame -= 1;\n",
" if(this.current_frame >= 0){\n",
" this.set_frame(this.current_frame);\n",
" }else{\n",
" var loop_state = this.get_loop_state();\n",
" if(loop_state == \"loop\"){\n",
" this.last_frame();\n",
" }else if(loop_state == \"reflect\"){\n",
" this.first_frame();\n",
" this.play_animation();\n",
" }else{\n",
" this.pause_animation();\n",
" this.first_frame();\n",
" }\n",
" }\n",
" }\n",
"\n",
" Animation.prototype.pause_animation = function()\n",
" {\n",
" this.direction = 0;\n",
" if (this.timer){\n",
" clearInterval(this.timer);\n",
" this.timer = null;\n",
" }\n",
" }\n",
"\n",
" Animation.prototype.play_animation = function()\n",
" {\n",
" this.pause_animation();\n",
" this.direction = 1;\n",
" var t = this;\n",
" if (!this.timer) this.timer = setInterval(function() {\n",
" t.anim_step_forward();\n",
" }, this.interval);\n",
" }\n",
"\n",
" Animation.prototype.reverse_animation = function()\n",
" {\n",
" this.pause_animation();\n",
" this.direction = -1;\n",
" var t = this;\n",
" if (!this.timer) this.timer = setInterval(function() {\n",
" t.anim_step_reverse();\n",
" }, this.interval);\n",
" }\n",
"</script>\n",
"\n",
"<style>\n",
".animation {\n",
" display: inline-block;\n",
" text-align: center;\n",
"}\n",
"input[type=range].anim-slider {\n",
" width: 374px;\n",
" margin-left: auto;\n",
" margin-right: auto;\n",
"}\n",
".anim-buttons {\n",
" margin: 8px 0px;\n",
"}\n",
".anim-buttons button {\n",
" padding: 0;\n",
" width: 36px;\n",
"}\n",
".anim-state label {\n",
" margin-right: 8px;\n",
"}\n",
".anim-state input {\n",
" margin: 0;\n",
" vertical-align: middle;\n",
"}\n",
"</style>\n",
"\n",
"<div class=\"animation\">\n",
" <img id=\"_anim_imge0a79e3a836d4164899f9abb7b4eca2c\">\n",
" <div class=\"anim-controls\">\n",
" <input id=\"_anim_slidere0a79e3a836d4164899f9abb7b4eca2c\" type=\"range\" class=\"anim-slider\"\n",
" name=\"points\" min=\"0\" max=\"1\" step=\"1\" value=\"0\"\n",
" oninput=\"anime0a79e3a836d4164899f9abb7b4eca2c.set_frame(parseInt(this.value));\">\n",
" <div class=\"anim-buttons\">\n",
" <button title=\"Decrease speed\" aria-label=\"Decrease speed\" onclick=\"anime0a79e3a836d4164899f9abb7b4eca2c.slower()\">\n",
" <i class=\"fa fa-minus\"></i></button>\n",
" <button title=\"First frame\" aria-label=\"First frame\" onclick=\"anime0a79e3a836d4164899f9abb7b4eca2c.first_frame()\">\n",
" <i class=\"fa fa-fast-backward\"></i></button>\n",
" <button title=\"Previous frame\" aria-label=\"Previous frame\" onclick=\"anime0a79e3a836d4164899f9abb7b4eca2c.previous_frame()\">\n",
" <i class=\"fa fa-step-backward\"></i></button>\n",
" <button title=\"Play backwards\" aria-label=\"Play backwards\" onclick=\"anime0a79e3a836d4164899f9abb7b4eca2c.reverse_animation()\">\n",
" <i class=\"fa fa-play fa-flip-horizontal\"></i></button>\n",
" <button title=\"Pause\" aria-label=\"Pause\" onclick=\"anime0a79e3a836d4164899f9abb7b4eca2c.pause_animation()\">\n",
" <i class=\"fa fa-pause\"></i></button>\n",
" <button title=\"Play\" aria-label=\"Play\" onclick=\"anime0a79e3a836d4164899f9abb7b4eca2c.play_animation()\">\n",
" <i class=\"fa fa-play\"></i></button>\n",
" <button title=\"Next frame\" aria-label=\"Next frame\" onclick=\"anime0a79e3a836d4164899f9abb7b4eca2c.next_frame()\">\n",
" <i class=\"fa fa-step-forward\"></i></button>\n",
" <button title=\"Last frame\" aria-label=\"Last frame\" onclick=\"anime0a79e3a836d4164899f9abb7b4eca2c.last_frame()\">\n",
" <i class=\"fa fa-fast-forward\"></i></button>\n",
" <button title=\"Increase speed\" aria-label=\"Increase speed\" onclick=\"anime0a79e3a836d4164899f9abb7b4eca2c.faster()\">\n",
" <i class=\"fa fa-plus\"></i></button>\n",
" </div>\n",
" <form title=\"Repetition mode\" aria-label=\"Repetition mode\" action=\"#n\" name=\"_anim_loop_selecte0a79e3a836d4164899f9abb7b4eca2c\"\n",
" class=\"anim-state\">\n",
" <input type=\"radio\" name=\"state\" value=\"once\" id=\"_anim_radio1_e0a79e3a836d4164899f9abb7b4eca2c\"\n",
" >\n",
" <label for=\"_anim_radio1_e0a79e3a836d4164899f9abb7b4eca2c\">Once</label>\n",
" <input type=\"radio\" name=\"state\" value=\"loop\" id=\"_anim_radio2_e0a79e3a836d4164899f9abb7b4eca2c\"\n",
" checked>\n",
" <label for=\"_anim_radio2_e0a79e3a836d4164899f9abb7b4eca2c\">Loop</label>\n",
" <input type=\"radio\" name=\"state\" value=\"reflect\" id=\"_anim_radio3_e0a79e3a836d4164899f9abb7b4eca2c\"\n",
" >\n",
" <label for=\"_anim_radio3_e0a79e3a836d4164899f9abb7b4eca2c\">Reflect</label>\n",
" </form>\n",
" </div>\n",
"</div>\n",
"\n",
"\n",
"<script language=\"javascript\">\n",
" /* Instantiate the Animation class. */\n",
" /* The IDs given should match those used in the template above. */\n",
" (function() {\n",
" var img_id = \"_anim_imge0a79e3a836d4164899f9abb7b4eca2c\";\n",
" var slider_id = \"_anim_slidere0a79e3a836d4164899f9abb7b4eca2c\";\n",
" var loop_select_id = \"_anim_loop_selecte0a79e3a836d4164899f9abb7b4eca2c\";\n",
" var frames = new Array(200);\n",
" \n",
" frames[0] = \"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAA0gAAAC0CAYAAACuXwwkAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90\\\n",
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