30:Taf2,md`## The Complex Conjugate Before we begin I want to define the complex conjugate. In fact, this entire notebook is really about the complex conjugate, which plays a prominent role in the equations of lines and circles in the complex plane. Let's get the definitions out of the way. The complex conjugate ${tex`\tilde z`} of a complex number ${tex`z = x + yi \in \mathbb C`} is defined as the complex number ${tex`\tilde z = x - yi`}. So the complex conjugate is found by negating the imaginary part. It turns out to not just be a very useful number, it also generalizes effortlessly to higher dimensional spaces (e.g. quaternions, Clifford algebras). To see why it is indeed such a good friend, lets calculate the sum, difference and product of a complex number with its own conjugate : ${tex`z + \tilde z = (x + yi) + (x - yi) = x \color{red} + yi \color{black}+ x\color{red} - yi\color{black} = 2x`} ${tex`\Leftrightarrow`} ${tex`x = \cfrac{z+\tilde z}{2}`}(1) So, the sum of a complex number and its conjugate is twice the real part. Let's see what the difference gives us : ${tex`z - \tilde z = (x+yi)-(x-yi) = \color{red}x \color{black} + yi \color{red}- x\color{black} + yi = 2yi`} Similarly, the difference of a complex number and its conjugate is twice the imaginary part. To simplify (using ${tex`\cfrac 1 i = i^{-1} = -i`}, aka the inverse of a rotation is the same rotation, negated.) ${tex`\Leftrightarrow y = \cfrac{z-\tilde z}{2i} = \cfrac {z-\tilde z}{2}*(-i) =`} ${tex`\Leftrightarrow`} ${tex`y = \cfrac{\tilde z - z }{2}i`}(2) So in short, using the complex conjugate we can easily express the real and imaginary parts of a complex number. As you will soon see, that makes it easy to convert formulas from ${tex`\mathbb R^2`} to ${tex`\mathbb C`} There's another identity involving the complex conjugate that will turn out good to have : the product of a complex number with its own conjugate. ${tex`z \tilde z = (x + yi) * (x - yi) = x^2 \color{red} - xyi + yix\color{black} + y^2 = x^2 + y^2`} ${tex`\Leftrightarrow`}${tex`z\tilde z = |z|^2 = x^2 + y^2`}(3) This product is always real-valued, and you should recognize it as the squared distance from the origin in the complex plane. (i.e. the Euclidean distance function.). The square root of this product is called the modulus. (${tex`|z|=\sqrt {z \tilde z}`}) Armed with these three, the sum, difference and product of a complex number with its conjugate, we are now ready to tackle today's subject - the equations of lines and circles in the complex plane ${tex`\mathbb C`} --- `31:T7e9,md`## Lines in ${tex`\mathbb C`} When working in the complex plane, we'll identify a point ${tex`[x,y]`} in ${tex`\mathbb R^2`} with the complex number ${tex`z=x + yi`} in ${tex`\mathbb C`}. We now want to write equation (4) in terms of the complex number ${tex`z`}. To do that, we plug (1) and (2), the complex form of ${tex`x`} and ${tex`y`} into (4), the equation of our line in ${tex`\mathbb R^2`}. ${tex`ax + by + c = 0`} (4) ${tex`a\cfrac {(z+\tilde z)} 2 + bi\cfrac {(\tilde z- z)}{2} + c = 0`}using (1) and (2) We'll simplify this step by step : ${tex`\Leftrightarrow \cfrac{az + a\tilde z}{2}+\cfrac{bi\tilde z - biz}{2} + c = 0`}bring ${tex`a`} and ${tex`bi`} into the fraction ${tex`\Leftrightarrow \cfrac{az - biz}{2}+\cfrac{a\tilde z + bi\tilde z}{2} + c = 0`}group ${tex`z`} and ${tex`\tilde z`} terms ${tex`\Leftrightarrow \cfrac{a - bi}{2}z+\cfrac{a + bi}{2}\tilde z + c = 0`}take ${tex`z`} and ${tex`\tilde z`} out of the fraction Looking at this we recognize that the coefficient of ${tex`z`} and ${tex`\tilde z`} is itself a complex number (and its conjugate). So we define beta ${tex`\beta = \frac{a - bi} 2`} and can now write the equation of a line in the complex plane : ${tex`\beta z + \tilde\beta \tilde z + c = 0`}(5) Again, this is a good time to get a feel for what these numbers mean. We'll make a picture similar to the one above, this time the horizontal axis represents the real number line and the vertical axis the imaginary line. Notice how vertical lines have a pure real ${tex`\beta`}, while horizontal lines have a pure imaginary ${tex`\beta`}. I've scaled ${tex`\beta`} to be of length ${tex`0.5`}, which keeps the meaning of ${tex`c`} the same as in the graph above. (the shortest distance from the origin to the line.) ` 32:T48b,md`## Circles in ${tex`\mathbb C`} For circles in the complex plane, we note that (3), the product of a complex number with its conjugate directly represents the equation of a circle around the origin. ${tex`|z|^2 = r^2`} ${tex`\Leftrightarrow z\tilde z = r^2`} Similarly for a circle around a given point ${tex`z_0`} we find : ${tex`(z-z_0)\tilde{(z-z_0)}=r^2`} Simplifying gives us : ${tex`\Leftrightarrow (z-z_0)\tilde {(z-z_0)} - r^2 = 0`} ${tex`\Leftrightarrow z\tilde z - \tilde {z_0}z - z_0\tilde z + z_0\tilde{z_0} - r^2 = 0`} Now set ${tex`\beta=-\tilde z_0`} and ${tex`\gamma = z_0\tilde{z_0} - r^2`} so we can write : ${tex`\Leftrightarrow`}${tex`z\tilde z + \beta z + \tilde \beta \tilde z + \gamma = 0`} That looks almost exactly like the equation of a line in ${tex`\mathbb C`}! Only the first term is different, so we introduce alpha ${tex`\alpha`} and write : ${tex`\Leftrightarrow`}${tex`\alpha z\tilde z + \beta z + \tilde \beta \tilde z + \gamma = 0`} (7) with ${tex`\alpha,\gamma \in \mathbb R`} and ${tex`\beta \in \mathbb C`}. `33:T54d,md` Note how a real β means the circle's center is on the horizontal axis, while a pure imaginary β means its on the vertical axis. And of course - the point of this notebook, if alpha is zero, we have a line instead of a circle. (I've drawn in light blue, the line you would get with alpha set to zero.) When β is zero, our line escapes to infinity and (assuming α is one), you can easily read ${tex`z\tilde z`} as the squared distance to the origin and γ as the negative squared radius.

## Conclusion Just to state the obvious. In ${tex`\mathbb C`}, equations of lines and circles have the same form. (and if β is a real number, they represent either vertical lines or circles with their center on the horizontal axis. These will turn out to be of significance in some future lunch break.) ${tex`\alpha z\tilde z + \beta z + \tilde \beta \tilde z + \gamma = 0`} (7) ${tex`\alpha,\gamma \in \mathbb R`} and ${tex`z, \beta \in \mathbb C`} Also notice how decreasing α but keeping β and γ constant will grow the radius of the sphere. As α approaches zero, the radius approaches ∞. This implies that one could view **lines as circles with an infinite radius**. (a concept we'll use again in a future math break) `34:[] 2e:{"title":"Lunchbreak Math 1","cells":[{"id":0,"value":"md`# Lunchbreak Math 1\n## Lines & Circles in ${tex`\\mathbb C`}\n The first notebook in a series aimed at exploring some of the math needed to appreciate non-euclidean geometry, specifically targeted at my lunch break programmer friends.\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":42,"value":"md`Non euclidean geometry is a fascinating subject, and a lot of fun to explore and contemplate on, but a bit of math background is needed to appreciate what's going on.\n\nIn this notebook we'll make the first (easy) steps on a path towards hyperbolic geometry. Today, we'll examine the equations of lines and circles in ${tex`\\mathbb C`}, fascinating in its own right, and they will turn out to be useful tools for what is to come.

`","pinned":false,"hidden":false,"mode":"ojs"},{"id":409,"value":"$30","pinned":false,"hidden":false,"mode":"ojs"},{"id":49,"value":"md`## Lines in ${tex`\\mathbb R^2`}\n\nWe'll start with the equation of a line in ${tex`\\mathbb R^2`}.\n\n ${tex`ax + by + c = 0`} (4)\n\nFor any triplet ${tex`a,b,c`} a non-zero multiple ${tex`xa,xb,xc\\space |\\space x \\in \\mathbb R_0`} determines the same line. In the graph below we fix the length of the vector ${tex`[a,b]`} to one. (and will call such a line normalized).\n\nTo get a feel, drag the yellow and blue points below to set the ${tex`a,b,c`} parameters. Try to figure out what happens when you make either of these numbers zero.\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":61,"value":"Algebra(2,0,1,()=>{\n var ab = 1e12 - 1e02, c = 1e12 - 0.5e01, line=1e1, l=1;\n return this.graph(\n [ ()=>{\n l = (!ab).Length;\n ab.e01=ab.e01/l; ab.e02=ab.e02/l; c.e02=0;\n line = ab.e02*1e1 - ab.e01*1e2 + c.e01*1e0;\n },\n ()=>(`line : `+((ab.e02.toFixed(1)!=0?((-ab.e02).toFixed(1))+\"x\":\"\").replace(\"1.0x\",\"x\"))\n +((ab.e01.toFixed(1)!=0?(ab.e02.toFixed(1)!=0?\" + \":\"\")+((-ab.e01).toFixed(1))+\"y\":\"\").replace(\"1.0y\",\"y\"))\n +((c.e01.toFixed(1)!=0?\" + \"+((-c.e01).toFixed(1)):\"\").replace(\"+ -\",\"- \"))\n +\" = 0\").replace(\"+ -\",\"- \"),\n 0xCC4444,()=>`a=`+((-ab.e02).toFixed(1)),\n 0x448844,()=>`b=`+((-ab.e01).toFixed(1)),\n 0x4444AA,()=>`c=`+((-c.e01).toFixed(1)),\n 0x999999,1e1,1e2,\n 0,()=>line,\"line\",\n 0x448844,[1e12,ab],ab, \"a,b\",0xCC4444, ab, \"a\",0xCCCC44,ab,\n 0x4444AA,[1e12,c],c, \"c\",\n ]\n ,{grid:true}) \n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":461,"value":"md`When ${tex`c=0`}, the line passes through the origin. When ${tex`a=0`} the line is horizontal and when ${tex`b=0`} the line is vertical.

`","pinned":false,"hidden":false,"mode":"ojs"},{"id":213,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":310,"value":"Algebra(2,0,1,()=>{\n var ab = 1e12 - 1e02, c = 1e12 - 0.5e01, line=1e1, l=1, _a, _b, _c;\n return this.graph(\n [ ()=>{\n l = (!ab).Length*2;\n ab.e01=ab.e01/l; ab.e02=ab.e02/l; c.e02=0;\n line = ab.e02*2e1 - ab.e01*2e2 + c.e01*1e0;\n _a = (-ab.e02).toFixed(1); _b = (-ab.e01).toFixed(1); _c = (-c.e01).toFixed(1);\n },\n 0x000000,()=>`line (${_a}-${_b}i)z + (${_a}+${_b}i)z̄ + ${_c} = 0`\n .replace(new RegExp(\"[ +-]+0.0i*|0.0+\",\"g\"),\"\")\n .replace(new RegExp(\"\\\\+[ ]*\\\\-\",\"g\"),\"-\")\n .replace(new RegExp(\"\\\\-\\\\-\",\"g\"),\"+\"),\n 0x884444,()=>(`ß = `+_a+\" - \"+_b+\"i\").replace(\"- -\",\"+\").replace(\"+ -\",\"- \"),\n 0x4444AA,()=>\"c =\"+_c,\n 0x999999,1e1,1e2,\n 0,()=>line,\"line\",\n 0x884444,[1e12,ab],ab, \"ß\",\n 0x4444AA,[1e12,c],c, \"c\",\n ]\n ,{grid:true}) \n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":641,"value":"md`

`","pinned":false,"hidden":false,"mode":"ojs"},{"id":362,"value":"md`## Circles in ${tex`\\mathbb R^2`}\n\nFor circles, we'll be doing the same thing. Lets start with the equation of a circle in ${tex`\\mathbb R^2`}\n\nA circle around the origin is the collection of points the same distance from the origin. \n\n ${tex`x^2 + y^2 = r^2`}\n\nFor a circle around a given point, we first translate to the origin :\n\n ${tex`(x-x_0)^2 + (y-y_0)^2 = r^2`}(6)\n\nWhere ${tex`(x_0,y_0)`} is the circle's center and ${tex`r`} is its radius. For completeness sake we'll graph it but this should feel trivial :\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":369,"value":"Algebra(3,1,()=>{ \n var ei = 1e4+1e3, // e-infinite\n eo = 0.5*(1e4-1e3); // e-origin\n \n var point = (x,y)=>eo + x*1e1 + y*1e2 + 0.5*(x*x+y*y)*ei,\n line = (a,b,c)=>!(a*1e1 + b*1e2 + c*ei),\n circle = (x,y,r)=>!(point(x,y) - r**2/2*ei);\n\n var p1 = point(0.5,-0.5), r = point(1.4,-0.5);\n \n// Graph these items.\n return this.graph([\n ()=>{\n return 0;\n },\n ()=>\"c : (\"+p1.e1.toFixed(1)+\",\"+p1.e2.toFixed(1)+\")\",\n ()=>\"r : \"+((Math.sqrt((r-p1).e1**2+(r-p1).e2**2)).toFixed(2)),\n 0x999999, line(1,0,0), line(0,1,0),0,\n ()=>!(p1 - ((r-p1).e1**2+(r-p1).e2**2)/2*ei),\n p1,\"c\",\n [p1,r],\"r\",r,\n ],{conformal:true,grid:true});\n\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":643,"value":"md`

`","pinned":false,"hidden":false,"mode":"ojs"},{"id":482,"value":"$32","pinned":false,"hidden":false,"mode":"ojs"},{"id":616,"value":"Algebra(3,1,()=>{ \n var ei = 1e4+1e3, // e-infinite\n eo = 0.5*(1e4-1e3); // e-origin\n \n var point = (x,y)=>eo + x*1e1 + y*1e2 + 0.5*(x*x+y*y)*ei,\n line = (a,b,c)=>!(a*1e1 + b*1e2 + c*ei),\n circle = (x,y,r)=>!(point(x,y) - r**2/2*ei);\n\n var p1 = point(0.5,0.0), p2 = point(0.0,0.5), r = point(0.0,0.5);\n var g = (x)=>x.e1**2+x.e2**2;\n var l;\n// Graph these items.\n return this.graph([\n ()=>{\n l = Math.sqrt(g(p1));\n p2.set(point(0,p2.e2));\n return 0;\n },\n \"α : 1\",\n ()=>(\"β : \"+((p1.e1).toFixed(1))+\" + \"+(p1.e2.toFixed(1))+\"i\").replace(\"+ -\",\"- \"),\n ()=>(\"γ : \"+ ((g(p1)-((r-p1).e1**2+(r-p1).e2**2)).toFixed(2)) ),\n 0x999999, line(1,0,0), line(0,1,0),0,\n 0xaaaaff,()=>line(p1.e1*2,p1.e2*2,(g(p1)-((r-p1).e1**2+(r-p1).e2**2))),0,\n ()=>!(p1 - ((r-p1).e1**2+(r-p1).e2**2)/2*ei),\n [point(0,0),p1],p1,\"β\",\n r,\n ],{conformal:true,grid:true});\n\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":543,"value":"$33","pinned":false,"hidden":false,"mode":"ojs"},{"id":594,"value":"md`---`","pinned":false,"hidden":false,"mode":"ojs"},{"id":51,"value":"Algebra = require('ganja.js')","pinned":false,"hidden":false,"mode":"ojs"},{"id":11,"value":"html``","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W34","stdlib":"1","titling":"auto"}