3f:null 40:["$","title",null,{"children":"Perfect partnerships | skydog23 | Observable"}] 46:T4b2,md`One of the foundation stones of plane projective geometry is the relationship between a point and a line that are not incident. There is a perfect partnerships between the point range of the line and the line pencil of the point defined as follows: the perfect partnership ${tex`{P\barwedge m}`} associates to each line of ${tex`P`} its intersection point with ${tex`m`}, which is the same as pairing a point of ${tex`m`} with its joining line with ${tex`P`}. Such a perfect pairing is called a perspectivity. A chain of two such perspectivities ${tex`P\barwedge m \barwedge Q`} gives a perfect partnership between the line pencils ${tex`P`} and ${tex`Q`}. It's also called a perspectivity. To distinguish between them we call the former unlike perspectivities and the latter, like ones. You can build chains of more than two unlike perspectivites, such a map (ignoring the intermediate links) is called a *projectivity* and also denoted with ${tex`P \doublebarwedge Q`}. A chain with an even (odd) number of unlike perspectivities is called a like (unlike) projectivity. We will typically study like projectivities since they map points to points and lines to lines.`47:T464,// Thanks to Steven De Keninck for making the ganja.js library chainOfPerspectivities = Algebra(2,0,1,()=>{ // Using Geometric/Clifford algebra with signature (2,0,1), based on the work of Charles Gunn var pt = (x,y)=>1e12-x*1e02+y*1e01, line = (a,b,c)=>c*1e0+a*1e1+b*1e2; // bring the pencil in A in perspective with the point range m var circle = [...Array(4)].map((x,k)=>pt(Math.cos(Math.PI*2*k/4), Math.sin(Math.PI*2*k/4))), P=circle[0], U=circle[1], Q=circle[2], W=circle[3], m=()=>U&W, n=50; var X = (a) =>(Math.cos(Math.PI*a)*1e01 + Math.sin(Math.PI*a)*1e02); var pencilP =()=> [...Array(n)].map((x,k)=> () =>P&X(k/n)), rangem =()=> [...Array(n)].map((x,k)=> ()=> (pencilP()[k])^m), pencilM =()=> [...Array(n)].map((x,k)=> () =>(rangem()[k])&Q); return this.graph(()=>{ return [ 0x00ff00, ...pencilP(), 0xaa6600, ...rangem(), 0x00aaaa, ...pencilM(), 0x0000ff, m, "m", 0xff0000, P,"P", Q,"Q"] }, {animate:false}) })48:[] 44:{"id":"d3944023beef00fd","slug":"perfect-partnerships","title":"Perfect partnerships","thumbnail":"5b83d428ff5b265b5901017839fb59a60a3a77a543470a06bb71260bffcc6b64","access_level":"public","create_time":"$D2018-08-28T07:22:56.433Z","update_time":"$D2018-08-29T14:19:41.390Z","view_time":null,"first_public_time":"$D2018-08-28T09:57:11.482Z","has_unpublished_edits":null,"fork":false,"stars":2,"starred":false,"owner":{"type":"team","id":"c12ededdbf97bde4","login":"skydog23","name":"","avatar_url":"https://avatars-next.observableusercontent.com/57d55b329e0de79c4f1a8c1447c6cb3689b05343c2cde413e6c55225ab16ea26","tier":"starter_2024","bio":"","following":false},"version":49,"publish_version":49,"role":null,"authored":false,"authors":[{"type":"individual","id":"8ebee8d27ef26f03","login":"skydog23","name":"","avatar_url":"https://avatars-next.observableusercontent.com/57d55b329e0de79c4f1a8c1447c6cb3689b05343c2cde413e6c55225ab16ea26","tier":null,"bio":"","following":false}],"members":[{"type":"individual","id":"8ebee8d27ef26f03","login":"skydog23","name":"","avatar_url":"https://avatars-next.observableusercontent.com/57d55b329e0de79c4f1a8c1447c6cb3689b05343c2cde413e6c55225ab16ea26","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":1,"collection":null,"type":"notebook","body":{"titling":"auto","title":"Perfect partnerships","cells":[{"id":0,"value":"md`# Perfect partnerships`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":9,"value":"$46","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":17,"value":"md`The following construction shows the chain of unlike perspectivities ${tex`P \\barwedge m \\barwedge Q`} that produces the like perspectivity ${tex`P \\doublebarwedge Q`}. By dragging the red points you can change the resulting perspectivities. Note that although\nthe relationship between ${tex`P`} and ${tex`Q`} is mathematically symmetric, the configuration is defined in terms of a uniform sampling of lines in the pencil P so that the lines chosen from Q depend on P but not vice-versa. Notice also how the configuration degenerates when the defining elements become incident, for example P crosses the line m.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":25,"value":"$47","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":44,"value":"md`[Next](https://beta.observablehq.com/@skydog23/projective-generation-of-conics): what happens when we lengthen the chain of unlike perspectivities to have length 4.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":19,"value":"PGA = Algebra(2,0,1)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":20,"value":"Algebra = require('ganja.js')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":23,"value":"checkbox = (label = '', value = false) => {\n let wrapper = html``\n let check = html``\n wrapper.append(check)\n wrapper.append(label || '')\n check.checked = value\n check.addEventListener('change', (e) => {\n wrapper.value = check.checked\n wrapper.dispatchEvent(new CustomEvent('input'))\n })\n wrapper.value = check.checked\n return wrapper\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W48","stdlib":"1"}} 45:"$44:body"