You are given the center $$$(C_x, C_y)$$$ and the radii $$$(R_x, R_y)$$$ of an ellipse. You are also given a list $$$L$$$ of points, $$$(x_i, y_i)$$$ where $$$1\leq i\leq |L|$$$. You need to output the score of the list $$$L$$$ as described above.

I determine the error instead of the score since it's easier, and do:

$$$score = 1 - error$$$

The error of a point is scored as a combination of two different errors.

1. The distance error: How far a point is from the circumference.

2. The space error: How far a point is from its neighbor from the expected distance.

The root-mean square values of these errors are calculated and passed into a function, which returns a value $$$E$$$. There is a maximum tolerance value for the error $$$M$$$.

The score is $$$1 - E/M$$$.

function get_projection_on_circumference(point) {
	var angle = point_direction(center_x, center_y, point.x, point.y);
	return new Point(
		center_x + radius * cos(angle),
		center_y + radius * sin(angle),
	);
}

function get_score(points) {
	static max_error = 100;
	var n_points = array_length(points);
	
	// error: distance from circumference
	var dis_error = 0;
	for (var i = 0; i < n_points; ++i) {
		var point = points[i];
		var dis = distance(center_x, center_y, point.x, point.y);
		dis_error += sqr(radius - dis);
	}
	dis_error /= n_points;
	dis_error = sqrt(dis_error);
	
	// project points on the circumference
	for (var i = 0; i < n_points; ++i) {
		points[i] = get_projection_on_circumference(points[i]);
	}
	
	// sort "points" in anti-clockwise order
	array_sort(points, function(p1, p2) {
		var dir1 = point_direction(center_x, center_y, p1.x, p1.y);
		var dir2 = point_direction(center_x, center_y, p2.x, p2.y);
		return ((dir1 < dir2) ? -1 : 1);
	});
	
	// error: evenly spaced
	var space_error = 0;
	var expected_angle = 360 / n_points;
	var j = n_points-1;
	for (var i = 0; i < n_points; ++i) {
		var p1 = points[j];
		var p2 = points[i];
		var dis = distance(p1.x, p1.y, p2.x, p2.y);
		var _val = clamp(1 - sqr(dis)/(2 * sqr(radius)), -1, 1);
		var _angle = darccos(_val);
		space_error += sqr(_angle - expected_angle);
		j = i;
	}
	space_error /= n_points;
	space_error = sqrt(space_error);
	
	// normalize error
	var error = F(dis_error, space_error);
	var error_normalized = clamp(error/max_error, 0, 1);
	
	// get score
	return (1 - error_normalized);
}