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tBuLi committed Jul 25, 2024
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286 changes: 286 additions & 0 deletions content/3DPGA/ex_3dpga_thinlens.ipynb
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{
"cells": [
{
"cell_type": "markdown",
"id": "bc700e90-af8f-446e-a3e2-a5623567113e",
"metadata": {},
"source": [
"# Thin Lens\n",
"\n",
"This notebook shows the power of dimension agnostic thinking, by using the thin lens example first presented by Steven De Keninck in his [SIBGRAPI2021 talk](https://www.youtube.com/watch?v=2DgxeizE3E8). (Highly recommended viewing!)\n",
"The code below is agnostic to the dimension of space as well as to the dimension of the object being imaged and the dimension of the properties of the lens."
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "4bca94dc-d8d9-40bf-99cf-563387a08807",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Note: you may need to restart the kernel to use updated packages.\n"
]
}
],
"source": [
"%pip install -q kingdon anywidget==0.9.9"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "2e501c13-9a4f-45f5-8d46-d2e824be9bfe",
"metadata": {
"tags": []
},
"outputs": [],
"source": [
"from kingdon import Algebra\n",
"\n",
"clrs = [0xff9900, 0xfed290, 0x009977, 0x000000]"
]
},
{
"cell_type": "markdown",
"id": "ce86a33b-3c8f-4543-ae49-e9a5b06ebe14",
"metadata": {},
"source": [
"This code was written to be dimension agnostic: try changing to $d=3$ and rerun the notebook!"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "e65f0c8f-900b-4b51-bc92-91f5ca1550ee",
"metadata": {},
"outputs": [],
"source": [
"d = 2"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "ee8df3eb-90c8-4574-8c62-f52876568ecf",
"metadata": {},
"outputs": [],
"source": [
"alg = Algebra(d, 0, 1, graded=True)\n",
"globals().update(alg.blades)"
]
},
{
"cell_type": "markdown",
"id": "c81d2934-e455-455e-ae70-967bad4345d3",
"metadata": {},
"source": [
"Immutable properties of the lens:"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "9581e782-5c36-4dfc-98be-02730fb12072",
"metadata": {},
"outputs": [],
"source": [
"lens = e1\n",
"center_point = e0.dual() "
]
},
{
"cell_type": "markdown",
"id": "aa24a8eb-4129-4834-8a3f-30d20965c2d7",
"metadata": {},
"source": [
"A thin lens also has a focal point and a center point. This is not true for a cylindrical lens; see the next cell."
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "b895a022-e588-4497-9988-4a6f40c536c5",
"metadata": {},
"outputs": [],
"source": [
"focal = (e0 - 0.8*e1).dual() # Focal point of the lens\n",
"center = e0.dual() # Center point of the lens"
]
},
{
"cell_type": "markdown",
"id": "7de46a6c-169e-492c-849c-94df36db9d7a",
"metadata": {},
"source": [
"In order to switch the lens to a cylindrical lens, uncomment the lines below. This will turn the focal point and optical center of the lens into lines instead of points, and has the result that the lens now maps points to lines."
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "98404b8f-8649-4732-9f54-60db59521dee",
"metadata": {},
"outputs": [],
"source": [
"# focal = focal | e3\n",
"# center = center | e3"
]
},
{
"cell_type": "markdown",
"id": "5f43a813-bc2a-4737-80eb-c4b3eef58b0e",
"metadata": {},
"source": [
"The object to be imaged. By default, this is a point."
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "fd7c9afb-16ae-49b1-a498-95d4ea833e3e",
"metadata": {},
"outputs": [],
"source": [
"world = (e0 - 2*e1 - e2).dual()"
]
},
{
"cell_type": "markdown",
"id": "d7a23ddc-b4be-4013-8ac4-5df22206e924",
"metadata": {},
"source": [
"Alternativelly we can switch the input from a point to a line. To do so, uncomment the code below to turn the object into a line."
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "ff7f5a33-d88d-4380-99d7-d649ef1b3661",
"metadata": {},
"outputs": [],
"source": [
"# world = world | e3"
]
},
{
"cell_type": "markdown",
"id": "29ab6083-f8e1-4e94-8e8f-c53854973d05",
"metadata": {},
"source": [
"Now all we need to find the image of the object through our lens is a ruler to draw some straight lines/planes! In essence, this makes the light ray trough the center of the lens, and intersects it with the light ray that travels through the focal point."
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "36d8b3ac-88b4-41e0-8fbc-7aad067029cf",
"metadata": {},
"outputs": [],
"source": [
"wf = lambda: world & focal\n",
"wc = lambda: world & center\n",
"wfl = lambda: wf ^ lens # world on lens through focal\n",
"wfl_dot_l = lambda: wfl | (center_point & wfl)\n",
"img = lambda: alg.op(wfl_dot_l, wc).normalized()"
]
},
{
"cell_type": "code",
"execution_count": 11,
"id": "79daf7b0-401f-4201-9416-dc6fb8185610",
"metadata": {},
"outputs": [
{
"data": {
"application/vnd.jupyter.widget-view+json": {
"model_id": "4f88a5ce3f58471d9f8637d44d4d7595",
"version_major": 2,
"version_minor": 1
},
"text/plain": [
"GraphWidget(cayley=[['1', 'e0', 'e1', 'e2', 'e01', 'e02', 'e12', 'e012'], ['e0', '0', 'e01', 'e02', '0', '0', …"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"axis = lens | center_point # Optical axis of the line.\n",
"\n",
"def lens_graph_func():\n",
" return [\n",
" '<G stroke-opacity=\"0.5\">',\n",
" axis, \n",
" '</G>',\n",
" clrs[2],\n",
" world,\n",
" clrs[0],\n",
" center,\n",
" focal, 'f',\n",
" lens,\n",
" '<G stroke-dasharray=\"0.1 0.1\" fill-opacity=\"0.4\">', clrs[1],\n",
" wf,\n",
" wc,\n",
" wfl,\n",
" lens | wfl,\n",
" '</G>',\n",
" '<G fill-opacity=\"0.2\">', clrs[1],\n",
" [world, img, wfl],\n",
" '</G>',\n",
" clrs[3],\n",
" img,\n",
" ]\n",
"\n",
"alg.graph(\n",
" lens_graph_func,\n",
" lineWidth=4,\n",
" pointRadius=2.5,\n",
" fontSize=3,\n",
" animate=True\n",
")"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "1902da41-ef1a-4ea5-843d-001dff341d7b",
"metadata": {},
"outputs": [],
"source": []
},
{
"cell_type": "code",
"execution_count": null,
"id": "82f88290-c21b-4316-a20d-308a598d1606",
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3 (ipykernel)",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.10.14"
}
},
"nbformat": 4,
"nbformat_minor": 5
}
5 changes: 5 additions & 0 deletions jp_app_launcher_config.yaml
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type: notebook
catalog: 3DPGA

- title: Thin Lens
source: content/3DPGA/ex_3dpga_thinlens.ipynb
type: notebook
catalog: 3DPGA

# 2DCGA

- title: Distances and Angles
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