{"rowid": 260, "title": "The Art of Mathematics: A Mandala Maker Tutorial", "contents": "In front-end development, there\u2019s often a great deal of focus on tools that aim to make our work more efficient. But what if you\u2019re new to web development? When you\u2019re just starting out, the amount of new material can be overwhelming, particularly if you don\u2019t have a solid background in Computer Science. But the truth is, once you\u2019ve learned a little bit of JavaScript, you can already make some pretty impressive things.\nA couple of years back, when I was learning to code, I started working on a side project. I wanted to make something colorful and fun to share with my friends. This is what my app looks like these days:\nMandala Maker user interface\nThe coolest part about it is the fact that it\u2019s a tool: anyone can use it to create something original and brand new. \nIn this tutorial, we\u2019ll build a smaller version of this app \u2013 a symmetrical drawing tool in ES5, JavaScript and HTML5. The tutorial app will have eight reflections, a color picker and a Clear button. Once we\u2019re done, you\u2019re on your own and can tweak it as you please. Be creative!\nPreparations: a blank canvas\nThe first thing you\u2019ll need for this project is a designated drawing space. We\u2019ll use the HTML5 canvas element and give it a width and a height of 600px (you can set the dimensions to anything else if you like).\nFiles\nCreate 3 files: index.html, styles.css, main.js. Don\u2019t forget to include your JS and CSS files in your HTML. \n\n\n\n \n \n \n\n\n \n\n\nI\u2019ll ask you to update your HTML file at a later point, but the CSS file we\u2019ll start with will stay the same throughout the project. This is the full CSS we are going to use:\nbody {\n background-color: #ccc;\n text-align: center;\n}\n\ncanvas {\n touch-action: none;\n background-color: #fff;\n}\n\nbutton {\n font-size: 110%;\n}\nNext steps\nWe are done with our preparations and ready to move on to the actual tutorial, which is made up of 4 parts:\n\nBuilding a simple drawing app with one line and one color \nAdding a Clear button and a color picker\nAdding more functionality: 2 line drawing (add the first reflection)\nAdding more functionality: 8 line drawing (add 6 more reflections!)\n\nInteractive demos\nThis tutorial will be accompanied by four CodePens, one at the end of each section. In my own app I originally used mouse events, and only added touch events when I realized mobile device support was (A) possible, and (B) going to make my app way more accessible. For the sake of code simplicity, I decided that in this tutorial app I will only use one event type, so I picked a third option: pointer events. These are supported by some desktop browsers and some mobile browsers. An up-to-date version of Chrome is probably your best bet.\nPart 1: A simple drawing app\nLet\u2019s get started with our main.js file. Our basic drawing app will be made up of 6 functions: init, drawLine, stopDrawing, recordPointerLocation, handlePointerMove, handlePointerDown. It also has nine variables:\nvar canvas, context, w, h,\n prevX = 0, currX = 0, prevY = 0, currY = 0,\n draw = false;\nThe variables canvas and context let us manipulate the canvas. w is the canvas width and h is the canvas height. The four coordinates are used for tracking the current and previous location of the pointer. A short line is drawn between (prevX, prevY) and (currX, currY) repeatedly many times while we move the pointer upon the canvas. For your drawing to appear, three conditions must be met: the pointer (be it a finger, a trackpad or a mouse) must be down, it must be moving and the movement has to be on the canvas. If these three conditions are met, the boolean draw is set to true. \n1. init\nResponsible for canvas set up, this listens to pointer events and the location of their coordinates and sets everything in motion by calling other functions, which in turn handle touch and movement events. \nfunction init() {\n canvas = document.querySelector(\"canvas\");\n context = canvas.getContext(\"2d\");\n w = canvas.width;\n h = canvas.height;\n\n canvas.onpointermove = handlePointerMove;\n canvas.onpointerdown = handlePointerDown;\n canvas.onpointerup = stopDrawing;\n canvas.onpointerout = stopDrawing;\n}\n2. drawLine\nThis is called to action by handlePointerMove() and draws the pointer path. It only runs if draw = true. It uses canvas methods you can read about in the canvas API documentation. You can also learn to use the canvas element in this tutorial.\nlineWidth and linecap set the properties of our paint brush, or digital pen, but pay attention to beginPath and closePath. Between those two is where the magic happens: moveTo and lineTo take canvas coordinates as arguments and draw from (a,b) to (c,d), which is to say from (prevX,prevY) to (currX,currY).\nfunction drawLine() {\n var a = prevX,\n b = prevY,\n c = currX,\n d = currY;\n\n context.lineWidth = 4;\n context.lineCap = \"round\";\n\n context.beginPath();\n context.moveTo(a, b);\n context.lineTo(c, d);\n context.stroke();\n context.closePath();\n}\n3. stopDrawing\nThis is used by init when the pointer is not down (onpointerup) or is out of bounds (onpointerout).\nfunction stopDrawing() {\n draw = false;\n}\n4. recordPointerLocation\nThis tracks the pointer\u2019s location and stores its coordinates. Also, you need to know that in computer graphics the origin of the coordinate space (0,0) is at the top left corner, and all elements are positioned relative to it. When we use canvas we are dealing with two coordinate spaces: the browser window and the canvas itself. This function converts between the two: it subtracts the canvas offsetLeft and offsetTop so we can later treat the canvas as the only coordinate space. If you are confused, read more about it.\nfunction recordPointerLocation(e) {\n prevX = currX;\n prevY = currY;\n currX = e.clientX - canvas.offsetLeft;\n currY = e.clientY - canvas.offsetTop;\n}\n5. handlePointerMove\nThis is set by init to run when the pointer moves. It checks if draw = true. If so, it calls recordPointerLocation to get the path and drawLine to draw it.\nfunction handlePointerMove(e) {\n if (draw) {\n recordPointerLocation(e);\n drawLine();\n }\n}\n6. handlePointerDown\nThis is set by init to run when the pointer is down (finger is on touchscreen or mouse it clicked). If it is, calls recordPointerLocation to get the path and sets draw to true. That\u2019s because we only want movement events from handlePointerMove to cause drawing if the pointer is down.\nfunction handlePointerDown(e) {\n recordPointerLocation(e);\n draw = true;\n}\nFinally, we have a working drawing app. But that\u2019s just the beginning!\nSee the Pen Mandala Maker Tutorial: Part 1 by Hagar Shilo (@hagarsh) on CodePen.\n\nPart 2: Add a Clear button and a color picker\nNow we\u2019ll update our HTML file, adding a menu div with an input of the type and class color and a button of the class clear.\n\n \n
\n \n \n
\n\nColor picker\nThis is our new color picker function. It targets the input element by its class and gets its value. \nfunction getColor() {\n return document.querySelector(\".color\").value;\n}\nUp until now, the app used a default color (black) for the paint brush/digital pen. If we want to change the color we need to use the canvas property strokeStyle. We\u2019ll update drawLine by adding strokeStyle to it and setting it to the input value by calling getColor.\nfunction drawLine() {\n //...code... \n context.strokeStyle = getColor();\n context.lineWidth = 4;\n context.lineCap = \"round\";\n\n //...code... \n}\nClear button\nThis is our new Clear function. It responds to a button click and displays a dialog asking the user if she really wants to delete the drawing.\nfunction clearCanvas() {\n if (confirm(\"Want to clear?\")) {\n context.clearRect(0, 0, w, h);\n }\n}\nThe method clearRect takes four arguments. The first two (0,0) mark the origin, which is actually the top left corner of the canvas. The other two (w,h) mark the full width and height of the canvas. This means the entire canvas will be erased, from the top left corner to the bottom right corner. \nIf we were to give clearRect a slightly different set of arguments, say (0,0,w/2,h), the result would be different. In this case, only the left side of the canvas would clear up.\nLet\u2019s add this event handler to init:\nfunction init() {\n //...code...\n canvas.onpointermove = handleMouseMove;\n canvas.onpointerdown = handleMouseDown;\n canvas.onpointerup = stopDrawing;\n canvas.onpointerout = stopDrawing;\n document.querySelector(\".clear\").onclick = clearCanvas;\n}\nSee the Pen Mandala Maker Tutorial: Part 2 by Hagar Shilo (@hagarsh) on CodePen.\n\nPart 3: Draw with 2 lines\nIt\u2019s time to make a line appear where no pointer has gone before. A ghost line! \nFor that we are going to need four new coordinates: a', b', c' and d' (marked in the code as a_, b_, c_ and d_). In order for us to be able to add the first reflection, first we must decide if it\u2019s going to go over the y-axis or the x-axis. Since this is an arbitrary decision, it doesn\u2019t matter which one we choose. Let\u2019s go with the x-axis. \nHere is a sketch to help you grasp the mathematics of reflecting a point across the x-axis. The coordinate space in my sketch is different from my explanation earlier about the way the coordinate space works in computer graphics (more about that in a bit!). \nNow, look at A. It shows a point drawn where the pointer hits, and B shows the additional point we want to appear: a reflection of the point across the x-axis. This is our goal.\nA sketch illustrating the mathematics of reflecting a point.\nWhat happens to the x coordinates?\nThe variables a/a' and c/c' correspond to prevX and currX respectively, so we can call them \u201cthe x coordinates\u201d. We are reflecting across x, so their values remain the same, and therefore a' = a and c' = c. \nWhat happens to the y coordinates?\nWhat about b' and d'? Those are the ones that have to change, but in what way? Thanks to the slightly misleading sketch I showed you just now (of A and B), you probably think that the y coordinates b' and d' should get the negative values of b and d respectively, but nope. This is computer graphics, remember? The origin is at the top left corner and not at the canvas center, and therefore we get the following values: b = h - b, d' = h - d, where h is the canvas height.\nThis is the new code for the app\u2019s variables and the two lines: the one that fills the pointer\u2019s path and the one mirroring it across the x-axis.\nfunction drawLine() {\n var a = prevX, a_ = a,\n b = prevY, b_ = h-b,\n c = currX, c_ = c,\n d = currY, d_ = h-d;\n\n //... code ...\n\n // Draw line #1, at the pointer's location\n context.moveTo(a, b);\n context.lineTo(c, d);\n\n // Draw line #2, mirroring the line #1\n context.moveTo(a_, b_);\n context.lineTo(c_, d_);\n\n //... code ...\n}\nIn case this was too abstract for you, let\u2019s look at some actual numbers to see how this works.\nLet\u2019s say we have a tiny canvas of w = h = 10. Now let a = 3, b = 2, c = 4 and d = 3.\nSo b' = 10 - 2 = 8 and d' = 10 - 3 = 7.\nWe use the top and the left as references. For the y coordinates this means we count from the top, and 8 from the top is also 2 from the bottom. Similarly, 7 from the top is 3 from the bottom of the canvas. That\u2019s it, really. This is how the single point, and a line (not necessarily a straight one, by the way) is made up of many, many small segments that are similar to point in behavior.\nIf you are still confused, I don\u2019t blame you. \nHere is the result. Draw something and see what happens.\nSee the Pen Mandala Maker Tutorial: Part 3 by Hagar Shilo (@hagarsh) on CodePen.\n\nPart 4: Draw with 8 lines\nI have made yet another confusing sketch, with points C and D, so you understand what we\u2019re trying to do. Later on we\u2019ll look at points E, F, G and H as well. The circled point is the one we\u2019re adding at each particular step. The circled point at C has the coordinates (-3,2) and the circled point at D has the coordinates (-3,-2). Once again, keep in mind that the origin in the sketches is not the same as the origin of the canvas. \nA sketch illustrating points C and D.\nThis is the part where the math gets a bit mathier, as our drawLine function evolves further. We\u2019ll keep using the four new coordinates: a', b', c' and d', and reassign their values for each new location/line. Let\u2019s add two more lines in two new locations on the canvas. Their locations relative to the first two lines are exactly what you see in the sketch above, though the calculation required is different (because of the origin points being different).\nfunction drawLine() {\n\n //... code ... \n\n // Reassign values\n a_ = w-a; b_ = b;\n c_ = w-c; d_ = d;\n\n // Draw the 3rd line\n context.moveTo(a_, b_);\n context.lineTo(c_, d_);\n\n // Reassign values\n a_ = w-a; b_ = h-b;\n c_ = w-c; d_ = h-d;\n\n // Draw the 4th line\n context.moveTo(a_, b_);\n context.lineTo(c_, d_);\n\n //... code ... \nWhat is happening?\nYou might be wondering why we use w and h as separate variables, even though we know they have the same value. Why complicate the code this way for no apparent reason? That\u2019s because we want the symmetry to hold for a rectangular canvas as well, and this way it will. \nAlso, you may have noticed that the values of a' and c' are not reassigned when the fourth line is created. Why write their value assignments twice? It\u2019s for readability, documentation and communication. Maintaining the quadruple structure in the code is meant to help you remember that all the while we are dealing with two y coordinates (current and previous) and two x coordinates (current and previous). \nWhat happens to the x coordinates?\nAs you recall, our x coordinates are a (prevX) and c (currX).\nFor the third line we are adding, a' = w - a and c' = w - c, which means\u2026\nFor the fourth line, the same thing happens to our x coordinates a and c.\nWhat happens to the y coordinates?\nAs you recall, our y coordinates are b (prevY) and d (currY).\nFor the third line we are adding, b' = b and d' = d, which means the y coordinates are the ones not changing this time, making this is a reflection across the y-axis. \nFor the fourth line, b' = h - b and d' = h - d, which we\u2019ve seen before: that\u2019s a reflection across the x-axis.\nWe have four more lines, or locations, to define. Note: the part of the code that\u2019s responsible for drawing a micro-line between the newly calculated coordinates is always the same:\n context.moveTo(a_, b_);\n context.lineTo(c_, d_);\nWe can leave it out of the next code snippets and just focus on the calculations, i.e, the reassignments. \nOnce again, we need some concrete examples to see where we\u2019re going, so here\u2019s another sketch! The circled point E has the coordinates (2,3) and the circled point F has the coordinates (2,-3). The ability to draw at A but also make the drawing appear at E and F (in addition to B, C and D that we already dealt with) is the functionality we are about to add to out code.\nA sketch illustrating points E and F.\nThis is the code for E and F:\n // Reassign for 5\n a_ = w/2+h/2-b; b_ = w/2+h/2-a;\n c_ = w/2+h/2-d; d_ = w/2+h/2-c;\n\n // Reassign for 6\n a_ = w/2+h/2-b; b_ = h/2-w/2+a;\n c_ = w/2+h/2-d; d_ = h/2-w/2+c;\nTheir x coordinates are identical and their y coordinates are reversed to one another.\nThis one will be out final sketch. The circled point G has the coordinates (-2,3) and the circled point H has the coordinates (-2,-3).\nA sketch illustrating points G and H.\nThis is the code:\n // Reassign for 7\n a_ = w/2-h/2+b; b_ = w/2+h/2-a;\n c_ = w/2-h/2+d; d_ = w/2+h/2-c;\n\n // Reassign for 8\n a_ = w/2-h/2+b; b_ = h/2-w/2+a;\n c_ = w/2-h/2+d; d_ = h/2-w/2+c;\n //...code... \n}\nOnce again, the x coordinates of these two points are the same, while the y coordinates are different. And once again I won\u2019t go into the full details, since this has been a long enough journey as it is, and I think we\u2019ve covered all the important principles. But feel free to play around with the code and change it. I really recommend commenting out the code for some of the points to see what your drawing looks like without them.\nI hope you had fun learning! This is our final app:\nSee the Pen Mandala Maker Tutorial: Part 4 by Hagar Shilo (@hagarsh) on CodePen.", "year": "2018", "author": "Hagar Shilo", "author_slug": "hagarshilo", "published": "2018-12-02T00:00:00+00:00", "url": "https://24ways.org/2018/the-art-of-mathematics/", "topic": "code"}