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Math.cos(angleStep / 2); const points = Array.from({ length: numberOfSidesCircumscribed }, (_, i) => { const angle = i * angleStep - Math.PI / 2; // rotate so the polygon points up const x = polygonCenterX + circumscribedRadius * Math.cos(angle); const y = polygonCenterY + circumscribedRadius * Math.sin(angle); return `${x},${y}`; }).join(" "); return svg``; }33:T75b,areaCompareSVG = { const padding = 80; const height = (circleRadius+padding) * 2; const width = (circleRadius+padding) * 2; const polygonCenterX = padding + circleRadius; const polygonCenterY = padding + circleRadius; const circleCenterX = padding + circleRadius; const circleCenterY = padding + circleRadius; const stroke = 2; const angleStep = (2 * Math.PI) / numberOfSidesCompareArea; const firstAngle = -Math.PI / 2; const vertexX = polygonCenterX + circumradiusAreaCompare * Math.cos(firstAngle); const vertexY = polygonCenterY + circumradiusAreaCompare * Math.sin(firstAngle); // Midpoint of the line const midX = (polygonCenterX + vertexX) / 2; const midY = (polygonCenterY + vertexY) / 2; // Offset coordinates (perpendicular to upward line = horizontal offset) const offset = 10; const labelX = midX + offset; const labelY = midY; const points = Array.from({ length: numberOfSidesCompareArea }, (_, i) => { const angle = i * angleStep - Math.PI / 2; // rotate so the polygon points up const x = polygonCenterX + circumradiusAreaCompare * Math.cos(angle); const y = polygonCenterY + circumradiusAreaCompare * Math.sin(angle); return `${x},${y}`; }).join(" "); return svg``; }34:T605,/** * Calculates the length of the circum radius of a regular polygon that best approximates a circle * with an equal area. The degree of the polygon is determined by steps. * * @param {number} circleRadiusInMeters The radius of the target circle in meters. * @param {number} steps The degree of the regular polygon. * @returns {number} The length of the circum radius. */ calculateRegularPolygonCircumRadiusInMetersToBestApproximateEqualAreaCircle = (circleRadiusInMeters, steps = 12) => { // METHOD: construct a polygon with the same area as the circle, this minimizes the overall distance from the circle to the perimeter of the polygon, and is a better estimate of the circle than say an inscribed polygon const circleArea = Math.PI * circleRadiusInMeters * circleRadiusInMeters; const phi = Math.PI / steps; // angle between apothem and radius // 1. calculate the polygonArea: // polygonArea = circumRadiusInMeters * sin(phi) * circumRadiusInMeters * cos(phi) * steps // polygonArea = circumRadiusInMeters * circumRadiusInMeters * 1/2 * sin(2 * phi) * steps // via double angle identity // 2. set circleArea equal to polygonArea: // circleArea = polygonArea // circleArea = circumRadiusInMeters * circumRadiusInMeters * 1/2 * sin(2 * phi) * steps // 3. solve for the circumRadiusInMeters // circleArea * 2.0 / (sin(2 * phi) * steps) = circumRadiusInMeters * circumRadiusInMeters const circumRadiusInMeters = Math.sqrt((circleArea * 2.0) / (Math.sin(2.0 * phi) * steps)); return circumRadiusInMeters; };35:T500,equalAreaStepsSVG = { const padding = 80; const circleRadius = 300; const height = (circleRadius+padding) * 2; const width = (circleRadius+padding) * 2; const polygonCenterX = padding + circleRadius; const polygonCenterY = padding + circleRadius; const circleCenterX = padding + circleRadius; const circleCenterY = padding + circleRadius; const stroke = 2; const angleStep = (2 * Math.PI) / numberOfSidesEqualAreaSteps; const circumradiusEqualAreaSteps = calculateRegularPolygonCircumRadiusInMetersToBestApproximateEqualAreaCircle(circleRadius, numberOfSidesEqualAreaSteps); const points = Array.from({ length: numberOfSidesEqualAreaSteps }, (_, i) => { const angle = i * angleStep - Math.PI / 2; // rotate so the polygon points up const x = polygonCenterX + circumradiusEqualAreaSteps * Math.cos(angle); const y = polygonCenterY + circumradiusEqualAreaSteps * Math.sin(angle); return `${x},${y}`; }).join(" "); return svg``; }36:T6c7,sectorSVG = { const padding = 20; const radius = 300; const numberOfSides = 8; const height = (radius + padding); const width = (radius + padding); const polygonCenterX = width/2; const polygonCenterY = padding + radius; const circleCenterX = width/2; const circleCenterY = padding + radius; const stroke = 2; const angleStep = (2 * Math.PI) / numberOfSides; // Rotate 90° CCW by subtracting π/2 const rotation = Math.PI / 2; const startAngle = -angleStep / 2 - rotation; const endAngle = angleStep / 2 - rotation; // Circle sector const x1 = circleCenterX + radius * Math.cos(startAngle); const y1 = circleCenterY + radius * Math.sin(startAngle); const x2 = circleCenterX + radius * Math.cos(endAngle); const y2 = circleCenterY + radius * Math.sin(endAngle); const largeArcFlag = 0; const circleSectorPath = ` M ${circleCenterX},${circleCenterY} L ${x1},${y1} A ${radius},${radius} 0 ${largeArcFlag} 1 ${x2},${y2} Z `; const circumradius = calculateRegularPolygonCircumRadiusInMetersToBestApproximateEqualAreaCircle(radius, numberOfSides); // Polygon sector const px1 = polygonCenterX + circumradius * Math.cos(startAngle); const py1 = polygonCenterY + circumradius * Math.sin(startAngle); const px2 = polygonCenterX + circumradius * Math.cos(endAngle); const py2 = polygonCenterY + circumradius * Math.sin(endAngle); return svg``; }37:T869,/** * Calculates the number of sides of a regular polygon that best approximates a circle * with an equal area, while not exceeding an allowable rim error. * * This function determines the optimal number of sides for a regular polygon that * closely matches the area of a given circle while not exceeding an acceptable error * tolerance in the polygon's circumradius. * * @param {number} circleRadiusInMeters The radius of the target circle in meters. * @param {number} [maxAllowableLinearPolygonRimErrorInMeters=4.0] The maximum allowable error in meters for the polygon's circumradius when approximating the circle. * @returns {number} The number of sides of the best-fit regular polygon, chosen from predefined values that ensure good meshing and clean angles. */ calculateNumberOfRegularPolygonSidesToBestApproximateEqualAreaCircle = (circleRadiusInMeters, maxAllowableLinearPolygonRimErrorInMeters = 4.0) => { // step 1: assume highest allowable error const circumRadiusInMeters = circleRadiusInMeters + maxAllowableLinearPolygonRimErrorInMeters; // step 2: equate the area of the circle to the area of the polygon and solve for the number of sides // Math.PI * circleRadiusInMeters * circleRadiusInMeters = circumRadiusInMeters * circumRadiusInMeters * 1/2 * sin(2 * Math.PI / numberOfSides) * numberOfSides // 2 * Math.PI * circleRadiusInMeters * circleRadiusInMeters / circumRadiusInMeters * circumRadiusInMeters = sin(2 * Math.PI / numberOfSides) * numberOfSides // let x = 2 * pi/numberOfSides // sin(x)* 2pi/x = 2 * pi * circleRadiusInMeters * circleRadiusInMeters / circumRadiusInMeters * circumRadiusInMeters // use taylor series to approximate sin(x) * x ≈ 1 - x*x/6 // 1 - x*x/6 ≈ (circleRadiusInMeters * circleRadiusInMeters) / (circumRadiusInMeters * circumRadiusInMeters) // re-sub and solve for numberOfSides const approximateNumberOfSides = 2 * Math.PI / Math.sqrt(6 * (1 - ((circleRadiusInMeters * circleRadiusInMeters) / (circumRadiusInMeters * circumRadiusInMeters)))); const numberOfSides = Math.max(4, Math.min(1440, Math.ceil(approximateNumberOfSides))); return numberOfSides; };38:T529,dynamicStepsSVG = { const padding = 80; const circleRadius = 300; const height = (circleRadius+padding) * 2; const width = (circleRadius+padding) * 2; const polygonCenterX = padding + circleRadius; const polygonCenterY = padding + circleRadius; const circleCenterX = padding + circleRadius; const circleCenterY = padding + circleRadius; const stroke = 2; const numberOfSides = calculateNumberOfRegularPolygonSidesToBestApproximateEqualAreaCircle(circleRadius, allowableError) const angleStep = (2 * Math.PI) / numberOfSides; const circumradius = calculateRegularPolygonCircumRadiusInMetersToBestApproximateEqualAreaCircle(circleRadius, numberOfSides); const points = Array.from({ length: numberOfSides }, (_, i) => { const angle = i * angleStep - Math.PI / 2; // rotate so the polygon points up const x = polygonCenterX + circumradius * Math.cos(angle); const y = polygonCenterY + circumradius * Math.sin(angle); return `${x},${y}`; }).join(" "); return svg``; }39:Tc6c1,geojsonData1 = ({ "type": "FeatureCollection", "features": [ { "type": "Feature", "properties": { "name": "True Arc", "estimation": false }, "geometry": { "type": "GeometryCollection", "geometries": [ { "type": 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circleRadiusInMeters - The radius of the circle. * @param {Object} [options={}] - Additional options for Turf circle (e.g., units, properties). * @returns {Feature} A GeoJSON polygon feature representing the circle. */ function betterCircle(center, circleRadiusInMeters, options = {}) { let steps = options?.steps || 64; if (options.maximumRimDeviationInMeters && options.maximumRimDeviationInMeters > 0) { steps = calculateNumberOfRegularPolygonSidesToBestApproximateEqualAreaCircle(circleRadiusInMeters, options.maximumRimDeviationInMeters); } const circumRadiusInMeters = calculateRegularPolygonCircumRadiusInMetersToBestApproximateEqualAreaCircle(circleRadiusInMeters, steps); return turf.circle(center, circumRadiusInMeters, { steps, ...options }); } ```3b:[] 2f:{"id":"83d4bacaa114f1c5","slug":"smarter-approximations-for-circles","title":"It’s Not You, It’s Your Area - Smarter Approximations for Circles and Arcs","thumbnail":"88b2aa95485c75f65600f8ffe4102ea4b41d128f562ec1ea19de91c5a4aa964e","access_level":"public","create_time":"$D2025-05-14T20:12:58.855Z","update_time":"$D2025-06-11T18:25:03.978Z","view_time":null,"first_public_time":"$D2025-05-14T20:15:29.326Z","fork":{"id":"e57692300453779b","slug":"interactive-svg","title":"Interactive SVG","owner":{"id":"f35c755083683fe5","login":"observablehq","name":"Observable"}},"stars":0,"starred":false,"owner":{"type":"team","id":"70fc645112965623","login":"lemmingapex","name":"Scott Wiedemann","avatar_url":"https://avatars-next.observableusercontent.com/156fb61f0950e54d15240309f62d1358eb6a4b638dd759bc3f3daa5e19ab77b4","tier":"starter_2024"},"version":712,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"332cc023e36212e5","login":"lemmingapex","name":"Scott Wiedemann","avatar_url":"https://avatars-next.observableusercontent.com/156fb61f0950e54d15240309f62d1358eb6a4b638dd759bc3f3daa5e19ab77b4","tier":null}],"members":[{"type":"individual","id":"332cc023e36212e5","login":"lemmingapex","name":"Scott Wiedemann","avatar_url":"https://avatars-next.observableusercontent.com/156fb61f0950e54d15240309f62d1358eb6a4b638dd759bc3f3daa5e19ab77b4","tier":null,"role":"editor"},{"type":"team","id":"70fc645112965623","login":"lemmingapex","name":"Scott Wiedemann","avatar_url":"https://avatars-next.observableusercontent.com/156fb61f0950e54d15240309f62d1358eb6a4b638dd759bc3f3daa5e19ab77b4","tier":"starter_2024","role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"subscribed":null,"forks":0,"collection":null,"type":"notebook","body":{"title":"It’s Not You, It’s Your Area - Smarter Approximations for Circles and Arcs","cells":[{"id":0,"value":"# It’s Not You, It’s Your Area - Smarter Approximations for Circles and Arcs","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":172,"value":"I noticed a problem with the accuracy of some of the geospatial estimations I had built for a customer. The large arcs being drawn on the map were pretty far off from where the true arcs were located. I'm here to share my solution: an accurate, fast, tunable dynamically-sized polygon construction for these approximations that completely solved our customer's needs, and which you may consider in your own applications.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":526,"value":"Take a look at the not-so-great approximation in blue on the map below. *The code below sets up the map, and you can rightly ignore it.*","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":483,"value":"// import leaflet\nleaflet = require(\"leaflet@1.7.1/dist/leaflet.js\");","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":485,"value":"// stylesheet as well\nhtml``","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":487,"value":"$31","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":489,"value":"mapContainer0 = html`

`\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":491,"value":"viewof map0 = {\n const map = L.map(mapContainer0).setView([39.562249, -104.845633], 14);\n\n L.tileLayer('https://{s}.tile.openstreetmap.org/{z}/{x}/{y}.png', {\n attribution: '© OpenStreetMap contributors'\n }).addTo(map);\n\n L.geoJSON(geojsonData0, {\n onEachFeature: (feature, layer) => {\n const name = feature.properties.name || \"Unnamed\";\n layer.bindPopup(name);\n },\n style: function(feature) {\n switch (feature.properties.estimation) {\n case true: return {color: \"#0000ff\"};\n case false: return {color: \"#ff0000\", opacity: 0.65};\n }\n },\n pointToLayer: (feature, latlng) => L.circleMarker(latlng)\n }).addTo(map);\n\n return mapContainer0;\n}\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":523,"value":"In the map above, the blue arc is a 30-point polygon approximation; the red arc shows the *true* curve. Our application was configured to use 30 points regardless of arc radius. Since our application renders hundreds of thousands of arcs and circles of varying sizes, both accuracy and performance matter. The constant 30 point estimation was good for average cases, but for larger arcs—as shown above—the approximation can deviate by over 100 meters. Conversely, using 30 points for very small arcs is excessive.\n\nBefore diving into how to improve this, let's first take a step back... Why are polygons used at all? Why not just draw the arc directly?","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":35,"value":"## History\n\nCircles, arcs, and curves are fundamental shapes that appear throughout nature, mathematics, engineering, and design. We encounter them everywhere in the digital world and they arise in computer graphics often. Computers use circles extensively, but ironically, they have difficulty rendering them accurately. Circles and arcs are continuous and curved, but our digital displays and computer hardware is built and optimized for the discrete world.\n\nTo avoid the challenges of continuous shapes, most graphics systems approximate circles and arcs using polygons—shapes made up of straight line segments. This approach is not only faster but also aligns better with how modern graphics hardware works. GPUs are highly optimized to render triangles, the simplest polygon, and nearly all 2D and 3D graphics are broken down into triangles before being displayed.\n\nThe more sides a polygon has, the closer it looks to a circle. ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":121,"value":"viewof numberOfSides = Inputs.range([3, 50], {label: \"Number of Sides\", step: 1, value: 6})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":67,"value":"polygonSVG = {\n const padding = 20;\n const radius = 200;\n const height = (radius+padding) * 2;\n const width = (radius+padding) * 2 * 2;\n const polygonCenterX = padding + radius;\n const polygonCenterY = padding + radius;\n const circleCenterX = width - (padding + radius);\n const circleCenterY = padding + radius;\n const stroke = 2;\n const angleStep = (2 * Math.PI) / numberOfSides;\n\n const points = Array.from({ length: numberOfSides }, (_, i) => {\n const angle = i * angleStep - Math.PI / 2; // rotate so the polygon points up\n const x = polygonCenterX + radius * Math.cos(angle);\n const y = polygonCenterY + radius * Math.sin(angle);\n return `${x},${y}`;\n }).join(\" \");\n\n return svg``;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":91,"value":"There is a trade off between rendering accuracy and performance. The more sides the polygon has, the better the estimation, but the more storage and computation is required. Our goal is to generate the most accurate approximation with the fewest number of points.\n\n\nThere are a number of ways to approximate a circle with a polygon. In many applications, the approximated polygon is one that could be inscribed in the circle - as shown here (and in the example above).","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":148,"value":"viewof numberOfSidesInscribed = Inputs.range([3, 50], {label: \"Number of Sides\", step: 1, value: 8})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":104,"value":"inscribedSVG = {\n const padding = 40;\n const radius = 800;\n const height = (400+padding);\n const width = (radius+padding);\n const polygonCenterX = 0;\n const polygonCenterY = padding + radius;\n const circleCenterX = 0;\n const circleCenterY = padding + radius;\n const stroke = 2;\n const angleStep = (2 * Math.PI) / numberOfSidesInscribed;\n\n const points = Array.from({ length: numberOfSidesInscribed }, (_, i) => {\n const angle = i * angleStep - Math.PI / 2; // rotate so the polygon points up\n const x = polygonCenterX + radius * Math.cos(angle);\n const y = polygonCenterY + radius * Math.sin(angle);\n return `${x},${y}`;\n }).join(\" \");\n\n return svg``;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":140,"value":"The inscribed polygon is also how [turf.js generates its circle](https://turfjs.org/docs/api/circle). turf.js was used in some parts of our application and it was one of many parts that were altered in the end.\n\nAnother option might be to construct a circumscribed polygon - where all its sides are tangent to a circle.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":157,"value":"viewof numberOfSidesCircumscribed = Inputs.range([3, 50], {label: \"Number of Sides\", step: 1, value: 8})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":155,"value":"$32","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":142,"value":"In both cases, as the number of sides approaches infinity, the polygon converges to a circle. But with a finite number of sides, we aim for the best possible approximation. Inscribed polygons tend to underestimate the circle, while circumscribed ones overestimate it. There’s a more optimal solution between these extremes. Here I present a construction that minimizes the overall distance between the edge of the polygon and the edge of circle. I'll also provide an informal proof (solid justification as least) that this approach is—in some sense—optimal.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":179,"value":"## Equate the Area\n\nConsider the case where the area of the polygon matches the area of the circle. Try to adjust the circumradius of the polygon, such that it results in an area equal to that of the area of the circle.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":212,"value":"viewof numberOfSidesCompareArea = Inputs.range([5, 20], {label: \"Number of Sides\", step: 1, value: 8})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":183,"value":"viewof circumradiusAreaCompare = Inputs.range([280, 350], {label: \"Circumradius\", step: 0.1, value: 290})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":210,"value":"polygonArea = (0.5 * numberOfSidesCompareArea * circumradiusAreaCompare * circumradiusAreaCompare * Math.sin(2.0*Math.PI / numberOfSidesCompareArea)).toFixed(3)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":199,"value":"circleRadius = 300","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":204,"value":"circleArea = (Math.PI * circleRadius ** 2).toFixed(3)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":541,"value":"areaDifference = (polygonArea - circleArea).toFixed(3);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":181,"value":"$33","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":233,"value":"You will find that when the area of the polygon is equal to that of the circle, the result looks pretty good: the size of the polygon is in between the inscribed and circumscribed cases shown above. This should make sense: part of the polygon sticks out of the circle and part of the circle is outside the polygon. The length of the circumradius that results in a polygon that has equal area to that of the circle is dependent on the number of sides of the polygon. Like we stated before, a polygon with infinitely many sides should have a circumradius equal to the radius of the circle.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":382,"value":"tex`\n\\begin{array}{rl}\nA_{c} & = \\pi r^2 \\\\[10pt]\nA_{p} & = R \\cdot \\sin(\\phi) \\cdot R \\cdot \\cos(\\phi) \\cdot n \\quad \\text{where } \\quad\\quad \\phi = \\frac{\\pi} n \\\\\nA_{p} & = \\frac{1}{2} R^2 \\cdot \\sin(2\\phi) \\cdot n \\quad\\quad \\text{via double angle identity} \\\\[10pt]\n\\text{equate areas: } \\\\\nA_{c} & = A_{p} \\\\\n\\pi r^2 & = \\frac{1}{2} R^2 \\cdot \\sin(2\\phi) \\cdot n \\\\\nR^2 & = \\frac{2 \\pi r^2}{\\sin(2\\phi) \\cdot \\text{n}} \\\\\nR & = \\sqrt{\\frac{2 \\pi r^2}{\\sin(2\\phi) \\cdot \\text{n}}}\n\\end{array}\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":282,"value":"$34","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":266,"value":"Let's try it out.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":272,"value":"viewof numberOfSidesEqualAreaSteps = Inputs.range([5, 20], {label: \"Number of Sides\", step: 1, value: 8})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":262,"value":"$35","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":287,"value":"The area of the polygon and circle are now always the same. This appears to be working well, but how can we be sure there's not a better construction?","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":604,"value":"## Informal Proof\n\nWe want to see if this construction minimizes the overall distance between the edge of the polygon and the edge of circle. Let's examine just one sector of the polygon and circle.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":291,"value":"$36","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":309,"value":"If we can prove that the distance between this single edge of the polygon and the edge of circular sector is minimized, then by symmetry, we have proven it for the entire polygon.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":365,"value":"The Intermediate Value Theorem for Integrals states:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":349,"value":"tex`\n\\text{If } f \\text{ is continuous on } [a,b], \\text{ then there exists } c \\in [a,b] \\text{ such that:} \\\\\n\\int_a^b f(x)\\,dx = f(c)(b - a) \\\\\n\\text{Interpretation: The average value of } f \\text{ over } [a,b] \\text{ is achieved at some } c \\in [a,b].\n`\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":368,"value":"Imagine that the top crust of the orange circle sector is made of sand. When we shake it back and forth, the sand will level-out and perfectly align with the green triangle. This is the average value of the integral over the interval. Remember that an *average* is the **single value** that's *closest* to all the values. So in this case, the green edge is the single value that's closest to the entire edge of the circular sector.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":371,"value":"## Dynamic Construction\n\nWhile we've made significant progress, one limitation remains: our function still requires the number of polygon sides as input. \n Ideally, we want to construct a polygon with the fewest number of sides while retaining high accuracy. In order to accomplish this, we could instead pass in an acceptable error (in say meters or pixels) that we are willing to allow. We can interpret this as saying: *the edge of the polygon will at most be distance `X` away from the edge of the circle*. This allows the function to adapt the number of sides dynamically—producing accurate, size-efficient polygons across a wide range of circle sizes.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":426,"value":"tex`\n\\begin{array}{rl}\n\\text{Assume greatest amount of error: } \\\\\nR = r + \\epsilon \\\\\n\\text{Equate areas: } \\\\\nA_{c} & = A_{p} \\\\\n\\pi r^2 & = \\frac{1}{2} R^2 \\cdot n \\cdot \\sin\\left(\\frac{2\\pi}{n}\\right) \\\\\n\\frac{2\\pi r^2}{R^2} & = n \\cdot \\sin\\left(\\frac{2\\pi}{n}\\right) \\\\[10pt]\n\\text{Let } x = \\frac{2\\pi}{n} \\implies n = \\frac{2\\pi}{x} \\\\[10pt]\n\\sin(x) \\cdot \\frac{2\\pi}{x} & = \\frac{2\\pi r^2}{R^2} \\\\\n\\frac{\\sin(x)}{x} & = \\frac{r^2}{R^2} \\\\\n\\text{Use Taylor Series: } \\frac{\\sin(x)}{x} \\approx 1 - \\frac{x^2}{6} \\\\[10pt]\n1 - \\frac{x^2}{6} & \\approx \\frac{r^2}{R^2} \\\\\nx & \\approx \\sqrt{6\\left(1 - \\frac{r^2}{R^2}\\right)} \\\\\nn & \\approx \\frac{2\\pi}{\\sqrt{6\\left(1 - \\frac{r^2}{R^2}\\right)}}\n\\end{array}\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":548,"value":"Please note that a second order taylor series is used to approximate `sin(x)/x`. This approximation is accurate when `x` is small. The actual error from ignoring higher-order terms is roughly:\n\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":689,"value":"tex`\nError \\approx \\frac{x^4}{120} \\\\\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":375,"value":"$37","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":457,"value":"We can now change our prior method signature to use a dynamic number of steps:\n```javascript\ncalculateRegularPolygonCircumRadiusInMetersToBestApproximateEqualAreaCircle = (circleRadiusInMeters, steps = calculateNumberOfRegularPolygonSidesToBestApproximateEqualAreaCircle(circleRadiusInMeters)) => {\n...\n};\n```","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":460,"value":"Finally, let's test it out!","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":466,"value":"viewof allowableError = Inputs.range([1, 100], {label: \"Allowable Error in Pixels\", step: 1, value: 20})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":462,"value":"$38","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":692,"value":"This construction now meets all of our goals.\n\n1. **Minimize error**: The edge of the polygon is *optimally close* (as defined above) to the edge of the circle.\n2. **Tunable precision**: The maximum deviation from the true circle is guaranteed to be no more than `X` meters.\n3. **Good Performance**: The calculations are fast and efficient. Polygon complexity scales with size—smaller circles use fewer points, larger ones use more—while always staying within the allowed error. In fact, many cases in our application now use fewer than 30 points, resulting in noticeable storage savings.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":599,"value":"## Conclusion\n\nThe map below now shows our new polygon approximation in blue with an error of no more than 15 meters. The true circle is still shown in red. (You will have to zoom in to see the difference.) While this approach uses more sides (~90 at 15 meters), a polygon with greater error and fewer sides would provide a much better approximation than our original construction as well. It's important to note that the maximum allowable error can be tuned for the needs of the application and users.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":595,"value":"$39","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":586,"value":"mapContainer1 = html`

`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":577,"value":"viewof map1 = {\n const map = L.map(mapContainer1).setView([39.562249, -104.845633], 14);\n \n L.tileLayer('https://{s}.tile.openstreetmap.org/{z}/{x}/{y}.png', {\n attribution: '© OpenStreetMap contributors'\n }).addTo(map);\n\n L.geoJSON(geojsonData1, {\n onEachFeature: (feature, layer) => {\n const name = feature.properties.name || \"Unnamed\";\n layer.bindPopup(name);\n },\n style: function(feature) {\n switch (feature.properties.estimation) {\n case true: return {color: \"#0000ff\"};\n case false: return {color: \"#ff0000\", opacity: 0.65};\n }\n },\n pointToLayer: (feature, latlng) => L.circleMarker(latlng)\n }).addTo(map);\n\n return mapContainer1;\n}\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":619,"value":"The methods above should work seamlessly with other libraries you might be using, such as turf.js.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":624,"value":"$3a","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":614,"value":"In addition to the changes mentioned above, we updated several calculations across the application to improve the accuracy of arc rendering. The most significant of these changes was replacing the Haversine formulas with Vincenty’s forward and inverse methods. Haversine assumes a spherical Earth and can introduce errors of up to ~0.5%, while Vincenty’s formulas account for Earth’s ellipsoidal shape (WGS84) and offer millimeter-level precision. Although Vincenty’s methods are slower due to their iterative nature and reliance on more complex math, the accuracy gained was essential to meet our customer’s requirements. If you're working on geospatial applications, I recommend evaluating both methods to determine which best suits your precision and performance needs.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":144,"value":"While circles and arcs are conceptually simple, they’re surprisingly difficult for computers to render precisely. Polygons offer a practical workaround—enabling smooth, efficient rendering that works well with pixel-based displays and modern graphics hardware. By exploring various circle approximation strategies, we developed a polygon construction method that outperforms many standard or off-the-shelf approaches—with minimal additional complexity. Geospatial libraries in the future could provide better approximations natively.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"}],"files":"$W3b","stdlib":"1","titling":"auto"}} 30:"$2f:body"