/**
* 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.
*/
calculatePolygonCircumRadiusInMetersToBestApproximateEqualAreaCircle = (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 cirle 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 fot 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;
};
/**
* Calculates the number of sides of a regular polygon that best approximates a circle
* with an equal area, while maintaining an allowable linear rim error.
*
* This function determines the optimal number of sides for a regular polygon that
* closely matches the area of a given circle while considering 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 = circumRadius * circumRadius * 1/2 * sin(2 * Math.PI / numberOfSides) * numberOfSides
// 2 * Math.PI * circleRadiusInMeters * circleRadiusInMeters / circumRadius * circumRadius = sin(2 * Math.PI / numberOfSides) * numberOfSides
// let x = 2 * pi/numberOfSides
// sin(x)* 2pi/x = 2 * pi * circleRadiusInMeters * circleRadiusInMeters / circumRadius * circumRadius
// use taylor series to approximate sin(x) * x ≈ 1 - x*x/6
// 1 - x*x/6 ≈ (circleRadiusInMeters * circleRadiusInMeters) / (circumRadius * circumRadius)
// re-sub and solve for numberOfSides
const approimateNumberOfSides = 2 * Math.PI / Math.sqrt(6 * (1 - ((circleRadiusInMeters * circleRadiusInMeters) / (circumRadiusInMeters * circumRadiusInMeters))));
const numberOfSides = Math.max(4, Math.min(1440, Math.ceil(approimateNumberOfSides)));
return numberOfSides;
};
Purpose-built for displays of data
Observable is your go-to platform for exploring data and creating expressive data visualizations. Use reactive JavaScript notebooks for prototyping and a collaborative canvas for visual data exploration and dashboard creation.
