We Keep Coding

Set player velocity to launch them from point A to point B in an arc with bukkit

I'm trying to code a plugin for a server and I want to launch a player from point A to point B.
I've tried this but a lot of the time it is inaccurate and misses the block by a few blocks.
private Vector calculateVelocity(Location fromLoc, Location toLoc, int heightGain) {
Vector from = fromLoc.toVector();
Vector to = toLoc.toVector();
//Block locations
int endGain = to.getBlockY() - from.getBlockY();
double horizDist = fromLoc.distance(toLoc);
//Height gain
int gain = heightGain;
double maxGain = (gain > (endGain + gain) ? gain : (endGain + gain));
//Solve quadratic equation for velocity
double a = -horizDist * horizDist / (4 * maxGain);
double b = horizDist;
double c = -endGain;
double slope = -b / (2 * a) - Math.sqrt(b * b - 4 * a * c) / (2 * a);
//Vertical velocity
double veloY = Math.sqrt(maxGain * 0.115);
//Horizontal velocity
double vH = veloY / slope;
//Calculate horizontal direction
int distX = to.getBlockX() - from.getBlockX();
int distZ = to.getBlockZ() - from.getBlockZ();
double mag = Math.sqrt(distX * distX + distZ * distZ);
double dirX = distX / mag;
double dirZ = distZ / mag;
//Horizontal velocity components
double veloX = vH * dirX;
double veloZ = vH * dirZ;
//Actual velocity
Vector velocity = new Vector(veloX, veloY, veloZ);
return velocity;
}
Any suggetions on what I should do to get it accurate 100% of the time?

Tile Projection Google Maps: Convert Distance to Screen Dimensions

I am using the CanvasTileProvider in Google Maps Android v2.
I can convert lat long points to screen pixels.
However I would like to create a method to convert a distance to screen pixels. This will allow me to draw a circle of x radius. Can anyone help with this?
The code below I have butchered and modified from somewhere else so credit to the original author.
/**
* Converts between LatLng coordinates and the pixels inside a tile.
*/
public class TileProjection {
public int x;
public int y;
private int zoom;
private int TILE_SIZE;
private DoublePoint pixelOrigin_;
private double pixelsPerLonDegree_;
private double pixelsPerLonRadian_;
TileProjection(int tileSize, int x, int y, int zoom) {
this.TILE_SIZE = tileSize;
this.x = x;
this.y = y;
this.zoom = zoom;
pixelOrigin_ = new DoublePoint(TILE_SIZE / 2, TILE_SIZE / 2);
pixelsPerLonDegree_ = TILE_SIZE / 360d;
pixelsPerLonRadian_ = TILE_SIZE / (2 * Math.PI);
}
/**
* Get the dimensions of the Tile in LatLng coordinates
*/
public LatLngBounds getTileBounds() {
DoublePoint tileSW = new DoublePoint(x * TILE_SIZE, (y + 1) * TILE_SIZE);
DoublePoint worldSW = pixelToWorldCoordinates(tileSW);
LatLng SW = worldCoordToLatLng(worldSW);
DoublePoint tileNE = new DoublePoint((x + 1) * TILE_SIZE, y * TILE_SIZE);
DoublePoint worldNE = pixelToWorldCoordinates(tileNE);
LatLng NE = worldCoordToLatLng(worldNE);
return new LatLngBounds(SW, NE);
}
/**
* Calculate the pixel coordinates inside a tile, relative to the left upper
* corner (origin) of the tile.
*/
public PointF latLngToPoint(LatLng latLng) {
DoublePoint result = new DoublePoint(1, 1);
// Log.d("Aero","x " + String.valueOf(x));
// Log.d("Aero","y " + String.valueOf(y));
latLngToWorldCoordinates(latLng, result);
worldToPixelCoordinates(result, result);
result.x -= x * TILE_SIZE;
int numTiles = 1 << zoom;
if (latLng.longitude < 0) {
result.x = result.x + (numTiles * TILE_SIZE);
}
result.y -= y * TILE_SIZE;
return new PointF((float) result.x, (float) result.y);
}
private DoublePoint pixelToWorldCoordinates(DoublePoint pixelCoord) {
int numTiles = 1 << zoom;
DoublePoint worldCoordinate = new DoublePoint(pixelCoord.x / numTiles,
pixelCoord.y / numTiles);
return worldCoordinate;
}
/**
* Transform the world coordinates into pixel-coordinates relative to the
* whole tile-area. (i.e. the coordinate system that spans all tiles.)
* <p/>
* <p/>
* Takes the resulting point as parameter, to avoid creation of new objects.
*/
private void worldToPixelCoordinates(DoublePoint worldCoord, DoublePoint result) {
int numTiles = 1 << zoom;
result.x = worldCoord.x * numTiles;
result.y = worldCoord.y * numTiles;
}
private LatLng worldCoordToLatLng(DoublePoint worldCoordinate) {
DoublePoint origin = pixelOrigin_;
double lng = (worldCoordinate.x - origin.x) / pixelsPerLonDegree_;
double latRadians = (worldCoordinate.y - origin.y)
/ -pixelsPerLonRadian_;
double lat = Math.toDegrees(2 * Math.atan(Math.exp(latRadians))
- Math.PI / 2);
return new LatLng(lat, lng);
}
/**
* Get the coordinates in a system describing the whole globe in a
* coordinate range from 0 to TILE_SIZE (type double).
* <p/>
* Takes the resulting point as parameter, to avoid creation of new objects.
*/
private void latLngToWorldCoordinates(LatLng latLng, DoublePoint result) {
DoublePoint origin = pixelOrigin_;
result.x = origin.x + latLng.longitude * pixelsPerLonDegree_;
// Truncating to 0.9999 effectively limits latitude to 89.189. This is
// about a third of a tile past the edge of the world tile.
double siny = bound(Math.sin(Math.toRadians(latLng.latitude)), -0.9999,
0.9999);
result.y = origin.y + 0.5 * Math.log((1 + siny) / (1 - siny))
* -pixelsPerLonRadian_;
}
;
/**
* Return value reduced to min and max if outside one of these bounds.
*/
private double bound(double value, double min, double max) {
value = Math.max(value, min);
value = Math.min(value, max);
return value;
}
/**
* A Point in an x/y coordinate system with coordinates of type double
*/
public static class DoublePoint {
double x;
double y;
public DoublePoint(double x, double y) {
this.x = x;
this.y = y;
}
}
}
This is what I am proposing to use:
public Double MetersToPixels(LatLng latLng, Double distance){
double tileScale = TILE_SIZE / 256;
double pixelsPerMeter =1 / (156543.03392 * Math.cos(latLng.latitude * Math.PI / 180) / Math.pow(2, zoom)) * tileScale;
return pixelsPerMeter * distance;
}

At first you should be aware of the fact, that a circle on the surface of the earth is not exactly a circle on the map. But if you ignore this inaccuracy, you just need to create a LatLng point in 25nm distance, and then use latLngToPoint method to get the pixels. Comparing them with the pixels of the center, gives you the radius. For creating a LatLng in a given distance see the answer to this SO question (method move)

Randomly generating a latlng within a radius yields a point out of bounds

I'm trying to generate a point within a radius and I'm getting incorrect values. Someone mind taking a look and telling me what I'm doing wrong for the longitude? This was a formulaic approach posted on a different question...
public static Location generateLocationWithinRadius(Location myCurrentLocation) {
return getLocationInLatLngRad(1000, myCurrentLocation);
}
protected static Location getLocationInLatLngRad(double radiusInMeters, Location currentLocation) {
double x0 = currentLocation.getLatitude();
double y0 = currentLocation.getLongitude();
Random random = new Random();
// Convert radius from meters to degrees
double radiusInDegrees = radiusInMeters / 111000f;
double u = random.nextDouble();
double v = random.nextDouble();
double w = radiusInDegrees * Math.sqrt(u);
double t = 2 * Math.PI * v;
double x = w * Math.cos(t);
double y = w * Math.sin(t);
double new_x = x / Math.cos(y0);
double new_y = y / Math.cos(x0);
double foundLatitude;
double foundLongitude;
boolean shouldAddOrSubtractLat = random.nextBoolean();
boolean shouldAddOrSubtractLon = random.nextBoolean();
if (shouldAddOrSubtractLat) {
foundLatitude = new_x + x0;
} else {
foundLatitude = x0 - new_x;
}
if (shouldAddOrSubtractLon) {
foundLongitude = new_y + y0;
} else {
foundLongitude = y0 - new_y;
}
Location copy = new Location(currentLocation);
copy.setLatitude(foundLatitude);
copy.setLongitude(foundLongitude);
return copy;
}
I should also say that for some reason the valid points yield a uniform line of coordinates when looking at them.
I think the latitude is processing correctly whereas the longitude is not.

Your code seems to be more or less based on an idea
which is presented at gis.stackexchange.com
and discussed some more there in this discussion
and in this discussion.
If we take a closer look at it based on those discussions then maybe it makes more sense.
To easily limit the values to a circle it uses the approach of randomizing a direction and a distance. First we get two random double values between 0.0 ... 1.0:
double u = random.nextDouble();
double v = random.nextDouble();
As the radius is given in meters and the calculations require degrees, it's converted:
double radiusInDegrees = radiusInMeters / 111000f;
The degrees vs. meters ratio of the equator is used here. (Wikipedia suggests 111320 m.)
To have a more uniform distribution of the random points the distance is compensated with a square root:
w = r * sqrt(u)
Otherwise there would be a statistical bias in the amount of points near the center vs. far from the center. The square root of 1 is 1 and 0 of course 0, so
multiplying the root of the random double by the intended max. radius always gives a value between 0 and the radius.
Then the other random double is multiplied by 2 * pi because there are 2 * pi radians in a full circle:
t = 2 * Pi * v
We now have an angle somewhere between 0 ... 2 * pi i.e. 0 ... 360 degrees.
Then the random x and y coordinate deltas are calculated with basic trigonometry using the random distance and random angle:
x = w * cos(t)
y = w * sin(t)
The [x,y] then points some random distance w away from the original coordinates towards the direction t.
Then the varying distance between longitude lines is compensated with trigonometry (y0 being the center's y coordinate):
x' = x / cos(y0)
Above y0 needs to be converted to radians if the cos() expects the angle as radians. In Java it does.
It's then suggested that these delta values are added to the original coordinates. The cos and sin are negative for half of the full circle's angles so just adding is fine. Some of the random points will be to the west from Greenwich and and south from the equator. There's no need to randomize
should an addition or subtraction be done.
So the random point would be at (x'+x0, y+y0).
I don't know why your code has:
double new_y = y / Math.cos(x0);
And like said we can ignore shouldAddOrSubtractLat and shouldAddOrSubtractLon.
In my mind x refers to something going from left to right or from west to east. That's how the longitude values grow even though the longitude lines go from south to north. So let's use x as longitude and y as latitude.
So what's left then? Something like:
protected static Location getLocationInLatLngRad(double radiusInMeters, Location currentLocation) {
double x0 = currentLocation.getLongitude();
double y0 = currentLocation.getLatitude();
Random random = new Random();
// Convert radius from meters to degrees.
double radiusInDegrees = radiusInMeters / 111320f;
// Get a random distance and a random angle.
double u = random.nextDouble();
double v = random.nextDouble();
double w = radiusInDegrees * Math.sqrt(u);
double t = 2 * Math.PI * v;
// Get the x and y delta values.
double x = w * Math.cos(t);
double y = w * Math.sin(t);
// Compensate the x value.
double new_x = x / Math.cos(Math.toRadians(y0));
double foundLatitude;
double foundLongitude;
foundLatitude = y0 + y;
foundLongitude = x0 + new_x;
Location copy = new Location(currentLocation);
copy.setLatitude(foundLatitude);
copy.setLongitude(foundLongitude);
return copy;
}

It is hard for me to provide you with a pure Android solution as I never used those API. However I am sure you could easily adapt this solution to generate a random point within a given radius from an existing point.
The problem is solved in a two dimensions space however it is easy to extend to support altitude as well.
Please have a look at the code below. It provides you with a LocationGeneratoras well as my own Location implementation and an unit test proving that it works.
My solution is based on solving the circle equation (x-a)^2 + (y-b)^2 = r^2
package my.test.pkg;
import org.junit.Test;
import java.util.Random;
import static org.junit.Assert.assertTrue;
public class LocationGeneratorTest {
private class Location {
double longitude;
double latitude;
public Location(double longitude, double latitude) {
this.longitude = longitude;
this.latitude = latitude;
}
}
private class LocationGenerator {
private final Random random = new Random();
Location generateLocationWithinRadius(Location currentLocation, double radius) {
double a = currentLocation.longitude;
double b = currentLocation.latitude;
double r = radius;
// x must be in (a-r, a + r) range
double xMin = a - r;
double xMax = a + r;
double xRange = xMax - xMin;
// get a random x within the range
double x = xMin + random.nextDouble() * xRange;
// circle equation is (y-b)^2 + (x-a)^2 = r^2
// based on the above work out the range for y
double yDelta = Math.sqrt(Math.pow(r, 2) - Math.pow((x - a), 2));
double yMax = b + yDelta;
double yMin = b - yDelta;
double yRange = yMax - yMin;
// Get a random y within its range
double y = yMin + random.nextDouble() * yRange;
// And finally return the location
return new Location(x, y);
}
}
#Test
public void shoulRandomlyGeneratePointWithinRadius () throws Exception {
LocationGenerator locationGenerator = new LocationGenerator();
Location currentLocation = new Location(20., 10.);
double radius = 5.;
for (int i=0; i < 1000000; i++) {
Location randomLocation = locationGenerator.generateLocationWithinRadius(currentLocation, radius);
try {
assertTrue(Math.pow(randomLocation.latitude - currentLocation.latitude, 2) + Math.pow(randomLocation.longitude - currentLocation.longitude, 2) < Math.pow(radius, 2));
} catch (Throwable e) {
System.out.println("i= " + i + ", x=" + randomLocation.longitude + ", y=" + randomLocation.latitude);
throw new Exception(e);
}
}
}
}
NOTE:
This is just a generic solution to obtain a random point inside a circle with the center in (a, b) and a radius of r that can be used to solve your problem and not a straight solution that you can use as such. You most likely will need to adapt it to your use case.
I believe this is a natural solution.
Regards

Kotlin version of Markus Kauppinen answer
fun Location.getRandomLocation(radius: Double): Location {
val x0: Double = longitude
val y0: Double = latitude
// Convert radius from meters to degrees.
// Convert radius from meters to degrees.
val radiusInDegrees: Double = radius / 111320f
// Get a random distance and a random angle.
// Get a random distance and a random angle.
val u = Random.nextDouble()
val v = Random.nextDouble()
val w = radiusInDegrees * sqrt(u)
val t = 2 * Math.PI * v
// Get the x and y delta values.
// Get the x and y delta values.
val x = w * cos(t)
val y = w * sin(t)
// Compensate the x value.
// Compensate the x value.
val newX = x / cos(Math.toRadians(y0))
val foundLatitude: Double
val foundLongitude: Double
foundLatitude = y0 + y
foundLongitude = x0 + newX
val copy = Location(this)
copy.latitude = foundLatitude
copy.longitude = foundLongitude
return copy
}

Longitude and Latitude uses ellipsoidal coordinates so for big radius (hundred meters) the error using this method would become sinificant. A possible trick is to convert to Cartesian coordinates, do the radius randomization and then transform back again to ellipsoidal coordinates for the long-lat. I have tested this up to a couple of kilometers with great success using this java library from ibm. Longer than that might work, but eventually the radius would fall off as the earth shows its spherical nature.

Convert latitude/longitude point to a pixels (x,y) on mercator projection

I'm trying to convert a lat/long point into a 2d point so that I can display it on an image of the world-which is a mercator projection.
I've seen various ways of doing this and a few questions on stack overflow-I've tried out the different code snippets and although I get the correct longitude to pixel, the latitude is always off-seems to be getting more reasonable though.
I need the formula to take into account the image size, width etc.
I've tried this piece of code:
double minLat = -85.05112878;
double minLong = -180;
double maxLat = 85.05112878;
double maxLong = 180;
// Map image size (in points)
double mapHeight = 768.0;
double mapWidth = 991.0;
// Determine the map scale (points per degree)
double xScale = mapWidth/ (maxLong - minLong);
double yScale = mapHeight / (maxLat - minLat);
// position of map image for point
double x = (lon - minLong) * xScale;
double y = - (lat + minLat) * yScale;
System.out.println("final coords: " + x + " " + y);
The latitude seems to be off by about 30px in the example I'm trying. Any help or advice?
Update
Based on this question:Lat/lon to xy
I've tried to use the code provided but I'm still having some problems with latitude conversion, longitude is fine.
int mapWidth = 991;
int mapHeight = 768;
double mapLonLeft = -180;
double mapLonRight = 180;
double mapLonDelta = mapLonRight - mapLonLeft;
double mapLatBottom = -85.05112878;
double mapLatBottomDegree = mapLatBottom * Math.PI / 180;
double worldMapWidth = ((mapWidth / mapLonDelta) * 360) / (2 * Math.PI);
double mapOffsetY = (worldMapWidth / 2 * Math.log((1 + Math.sin(mapLatBottomDegree)) / (1 - Math.sin(mapLatBottomDegree))));
double x = (lon - mapLonLeft) * (mapWidth / mapLonDelta);
double y = 0.1;
if (lat < 0) {
lat = lat * Math.PI / 180;
y = mapHeight - ((worldMapWidth / 2 * Math.log((1 + Math.sin(lat)) / (1 - Math.sin(lat)))) - mapOffsetY);
} else if (lat > 0) {
lat = lat * Math.PI / 180;
lat = lat * -1;
y = mapHeight - ((worldMapWidth / 2 * Math.log((1 + Math.sin(lat)) / (1 - Math.sin(lat)))) - mapOffsetY);
System.out.println("y before minus: " + y);
y = mapHeight - y;
} else {
y = mapHeight / 2;
}
System.out.println(x);
System.out.println(y);
When using the original code if the latitude value is positive it returned a negative point, so I modified it slightly and tested with the extreme latitudes-which should be point 0 and point 766, it works fine. However when I try a different latitude value ex: 58.07 (just north of the UK) it displays as north of Spain.

The Mercator map projection is a special limiting case of the Lambert Conic Conformal map projection with
the equator as the single standard parallel. All other parallels of latitude are straight lines and the meridians
are also straight lines at right angles to the equator, equally spaced. It is the basis for the transverse and
oblique forms of the projection. It is little used for land mapping purposes but is in almost universal use for
navigation charts. As well as being conformal, it has the particular property that straight lines drawn on it are
lines of constant bearing. Thus navigators may derive their course from the angle the straight course line
makes with the meridians. [1.]
The formulas to derive projected Easting and Northing coordinates from spherical latitude φ and longitude λ
are:
E = FE + R (λ – λₒ)
N = FN + R ln[tan(π/4 + φ/2)]
where λO is the longitude of natural origin and FE and FN are false easting and false northing.
In spherical Mercator those values are actually not used, so you can simplify the formula to
Pseudo code example, so this can be adapted to every programming language.
latitude = 41.145556; // (φ)
longitude = -73.995; // (λ)
mapWidth = 200;
mapHeight = 100;
// get x value
x = (longitude+180)*(mapWidth/360)
// convert from degrees to radians
latRad = latitude*PI/180;
// get y value
mercN = ln(tan((PI/4)+(latRad/2)));
y = (mapHeight/2)-(mapWidth*mercN/(2*PI));
Sources:
OGP Geomatics Committee, Guidance Note Number 7, part 2: Coordinate Conversions and Transformation
Derivation of the Mercator projection
National Atlas: Map Projections
Mercator Map projection
EDIT
Created a working example in PHP (because I suck at Java)
https://github.com/mfeldheim/mapStuff.git
EDIT2
Nice animation of the Mercator projection
https://amp-reddit-com.cdn.ampproject.org/v/s/amp.reddit.com/r/educationalgifs/comments/5lhk8y/how_the_mercator_projection_distorts_the_poles/?usqp=mq331AQJCAEoAVgBgAEB&amp_js_v=0.1

You cannot merely transpose from longitude/latitude to x/y like that because the world isn't flat. Have you look at this post? Converting longitude/latitude to X/Y coordinate
UPDATE - 1/18/13
I decided to give this a stab, and here's how I do it:-
public class MapService {
// CHANGE THIS: the output path of the image to be created
private static final String IMAGE_FILE_PATH = "/some/user/path/map.png";
// CHANGE THIS: image width in pixel
private static final int IMAGE_WIDTH_IN_PX = 300;
// CHANGE THIS: image height in pixel
private static final int IMAGE_HEIGHT_IN_PX = 500;
// CHANGE THIS: minimum padding in pixel
private static final int MINIMUM_IMAGE_PADDING_IN_PX = 50;
// formula for quarter PI
private final static double QUARTERPI = Math.PI / 4.0;
// some service that provides the county boundaries data in longitude and latitude
private CountyService countyService;
public void run() throws Exception {
// configuring the buffered image and graphics to draw the map
BufferedImage bufferedImage = new BufferedImage(IMAGE_WIDTH_IN_PX,
IMAGE_HEIGHT_IN_PX,
BufferedImage.TYPE_INT_RGB);
Graphics2D g = bufferedImage.createGraphics();
Map<RenderingHints.Key, Object> map = new HashMap<RenderingHints.Key, Object>();
map.put(RenderingHints.KEY_INTERPOLATION, RenderingHints.VALUE_INTERPOLATION_BICUBIC);
map.put(RenderingHints.KEY_RENDERING, RenderingHints.VALUE_RENDER_QUALITY);
map.put(RenderingHints.KEY_ANTIALIASING, RenderingHints.VALUE_ANTIALIAS_ON);
RenderingHints renderHints = new RenderingHints(map);
g.setRenderingHints(renderHints);
// min and max coordinates, used in the computation below
Point2D.Double minXY = new Point2D.Double(-1, -1);
Point2D.Double maxXY = new Point2D.Double(-1, -1);
// a list of counties where each county contains a list of coordinates that form the county boundary
Collection<Collection<Point2D.Double>> countyBoundaries = new ArrayList<Collection<Point2D.Double>>();
// for every county, convert the longitude/latitude to X/Y using Mercator projection formula
for (County county : countyService.getAllCounties()) {
Collection<Point2D.Double> lonLat = new ArrayList<Point2D.Double>();
for (CountyBoundary countyBoundary : county.getCountyBoundaries()) {
// convert to radian
double longitude = countyBoundary.getLongitude() * Math.PI / 180;
double latitude = countyBoundary.getLatitude() * Math.PI / 180;
Point2D.Double xy = new Point2D.Double();
xy.x = longitude;
xy.y = Math.log(Math.tan(QUARTERPI + 0.5 * latitude));
// The reason we need to determine the min X and Y values is because in order to draw the map,
// we need to offset the position so that there will be no negative X and Y values
minXY.x = (minXY.x == -1) ? xy.x : Math.min(minXY.x, xy.x);
minXY.y = (minXY.y == -1) ? xy.y : Math.min(minXY.y, xy.y);
lonLat.add(xy);
}
countyBoundaries.add(lonLat);
}
// readjust coordinate to ensure there are no negative values
for (Collection<Point2D.Double> points : countyBoundaries) {
for (Point2D.Double point : points) {
point.x = point.x - minXY.x;
point.y = point.y - minXY.y;
// now, we need to keep track the max X and Y values
maxXY.x = (maxXY.x == -1) ? point.x : Math.max(maxXY.x, point.x);
maxXY.y = (maxXY.y == -1) ? point.y : Math.max(maxXY.y, point.y);
}
}
int paddingBothSides = MINIMUM_IMAGE_PADDING_IN_PX * 2;
// the actual drawing space for the map on the image
int mapWidth = IMAGE_WIDTH_IN_PX - paddingBothSides;
int mapHeight = IMAGE_HEIGHT_IN_PX - paddingBothSides;
// determine the width and height ratio because we need to magnify the map to fit into the given image dimension
double mapWidthRatio = mapWidth / maxXY.x;
double mapHeightRatio = mapHeight / maxXY.y;
// using different ratios for width and height will cause the map to be stretched. So, we have to determine
// the global ratio that will perfectly fit into the given image dimension
double globalRatio = Math.min(mapWidthRatio, mapHeightRatio);
// now we need to readjust the padding to ensure the map is always drawn on the center of the given image dimension
double heightPadding = (IMAGE_HEIGHT_IN_PX - (globalRatio * maxXY.y)) / 2;
double widthPadding = (IMAGE_WIDTH_IN_PX - (globalRatio * maxXY.x)) / 2;
// for each country, draw the boundary using polygon
for (Collection<Point2D.Double> points : countyBoundaries) {
Polygon polygon = new Polygon();
for (Point2D.Double point : points) {
int adjustedX = (int) (widthPadding + (point.getX() * globalRatio));
// need to invert the Y since 0,0 starts at top left
int adjustedY = (int) (IMAGE_HEIGHT_IN_PX - heightPadding - (point.getY() * globalRatio));
polygon.addPoint(adjustedX, adjustedY);
}
g.drawPolygon(polygon);
}
// create the image file
ImageIO.write(bufferedImage, "PNG", new File(IMAGE_FILE_PATH));
}
}
RESULT: Image width = 600px, Image height = 600px, Image padding = 50px
RESULT: Image width = 300px, Image height = 500px, Image padding = 50px

Java version of original Google Maps JavaScript API v3 java script code is as following, it works with no problem
public final class GoogleMapsProjection2
{
private final int TILE_SIZE = 256;
private PointF _pixelOrigin;
private double _pixelsPerLonDegree;
private double _pixelsPerLonRadian;
public GoogleMapsProjection2()
{
this._pixelOrigin = new PointF(TILE_SIZE / 2.0,TILE_SIZE / 2.0);
this._pixelsPerLonDegree = TILE_SIZE / 360.0;
this._pixelsPerLonRadian = TILE_SIZE / (2 * Math.PI);
}
double bound(double val, double valMin, double valMax)
{
double res;
res = Math.max(val, valMin);
res = Math.min(res, valMax);
return res;
}
double degreesToRadians(double deg)
{
return deg * (Math.PI / 180);
}
double radiansToDegrees(double rad)
{
return rad / (Math.PI / 180);
}
PointF fromLatLngToPoint(double lat, double lng, int zoom)
{
PointF point = new PointF(0, 0);
point.x = _pixelOrigin.x + lng * _pixelsPerLonDegree;
// Truncating to 0.9999 effectively limits latitude to 89.189. This is
// about a third of a tile past the edge of the world tile.
double siny = bound(Math.sin(degreesToRadians(lat)), -0.9999,0.9999);
point.y = _pixelOrigin.y + 0.5 * Math.log((1 + siny) / (1 - siny)) *- _pixelsPerLonRadian;
int numTiles = 1 << zoom;
point.x = point.x * numTiles;
point.y = point.y * numTiles;
return point;
}
PointF fromPointToLatLng(PointF point, int zoom)
{
int numTiles = 1 << zoom;
point.x = point.x / numTiles;
point.y = point.y / numTiles;
double lng = (point.x - _pixelOrigin.x) / _pixelsPerLonDegree;
double latRadians = (point.y - _pixelOrigin.y) / - _pixelsPerLonRadian;
double lat = radiansToDegrees(2 * Math.atan(Math.exp(latRadians)) - Math.PI / 2);
return new PointF(lat, lng);
}
public static void main(String []args)
{
GoogleMapsProjection2 gmap2 = new GoogleMapsProjection2();
PointF point1 = gmap2.fromLatLngToPoint(41.850033, -87.6500523, 15);
System.out.println(point1.x+" "+point1.y);
PointF point2 = gmap2.fromPointToLatLng(point1,15);
System.out.println(point2.x+" "+point2.y);
}
}
public final class PointF
{
public double x;
public double y;
public PointF(double x, double y)
{
this.x = x;
this.y = y;
}
}

JAVA only?
Python code here! Refer to Convert latitude/longitude point to a pixels (x,y) on mercator projection
import math
from numpy import log as ln
# Define the size of map
mapWidth = 200
mapHeight = 100
def convert(latitude, longitude):
# get x value
x = (longitude + 180) * (mapWidth / 360)
# convert from degrees to radians
latRad = (latitude * math.pi) / 180
# get y value
mercN = ln(math.tan((math.pi / 4) + (latRad / 2)))
y = (mapHeight / 2) - (mapWidth * mercN / (2 * math.pi))
return x, y
print(convert(41.145556, 121.2322))
Answer:
(167.35122222222225, 24.877939817552335)

public static String getTileNumber(final double lat, final double lon, final int zoom) {
int xtile = (int)Math.floor( (lon + 180) / 360 * (1<<zoom) ) ;
int ytile = (int)Math.floor( (1 - Math.log(Math.tan(Math.toRadians(lat)) + 1 / Math.cos(Math.toRadians(lat))) / Math.PI) / 2 * (1<<zoom) ) ;
if (xtile < 0)
xtile=0;
if (xtile >= (1<<zoom))
xtile=((1<<zoom)-1);
if (ytile < 0)
ytile=0;
if (ytile >= (1<<zoom))
ytile=((1<<zoom)-1);
return("" + zoom + "/" + xtile + "/" + ytile);
}
}

I'm new here, just to write, as I've been following the community for some years. I'm happy to be able to contribute.
Well, it took me practically a day in search of that and your question encouraged me to continue the search.
I arrived at the following function, which works! Credits for this article: https://towardsdatascience.com/geotiff-coordinate-querying-with-javascript-5e6caaaf88cf
var bbox = [minLong, minLat, maxLong, maxLat];
var pixelWidth = mapWidth;
var pixelHeight = mapHeight;
var bboxWidth = bbox[2] - bbox[0];
var bboxHeight = bbox[3] - bbox[1];
var convertToXY = function(latitude, longitude) {
var widthPct = ( longitude - bbox[0] ) / bboxWidth;
var heightPct = ( latitude - bbox[1] ) / bboxHeight;
var x = Math.floor( pixelWidth * widthPct );
var y = Math.floor( pixelHeight * ( 1 - heightPct ) );
return { x, y };
}

Calculating the Moment Of Inertia for a concave 2D polygon relative to its orgin

I want to compute the moment of inertia of a (2D) concave polygon. I found this on the internet. But I'm not very sure how to interpret the formula...
Formula http://img101.imageshack.us/img101/8141/92175941c14cadeeb956d8f.gif
1) Is this formula correct?
2) If so, is my convertion to C++ correct?
float sum (0);
for (int i = 0; i < N; i++) // N = number of vertices
{
int j = (i + 1) % N;
sum += (p[j].y - p[i].y) * (p[j].x + p[i].x) * (pow(p[j].x, 2) + pow(p[i].x, 2)) - (p[j].x - p[i].x) * (p[j].y + p[i].y) * (pow(p[j].y, 2) + pow(p[i].y, 2));
}
float inertia = (1.f / 12.f * sum) * density;
Martijn

#include <math.h> //for abs
float dot (vec a, vec b) {
return (a.x*b.x + a.y*b.y);
}
float lengthcross (vec a, vec b) {
return (abs(a.x*b.y - a.y*b.x));
}
...
do stuff
...
float sum1=0;
float sum2=0;
for (int n=0;n<N;++n) { //equivalent of the Σ
sum1 += lengthcross(P[n+1],P[n])*
(dot(P[n+1],P[n+1]) + dot(P[n+1],P[n]) + dot(P[n],P[n]));
sum2 += lengthcross(P[n+1],P[n]);
}
return (m/6*sum1/sum2);
Edit: Lots of small math changes

I think you have more work to do that merely translating formulas into code. You need to understand exactly what this formula means.
When you have a 2D polygon, you have three moments of inertia you can calculate relative to a given coordinate system: moment about x, moment about y, and polar moment of inertia. There's a parallel axis theorem that allows you to translate from one coordinate system to another.
Do you know precisely which moment and coordinate system this formula applies to?
Here's some code that might help you, along with a JUnit test to prove that it works:
import java.awt.geom.Point2D;
/**
* PolygonInertiaCalculator
* User: Michael
* Date: Jul 25, 2010
* Time: 9:51:47 AM
*/
public class PolygonInertiaCalculator
{
private static final int MIN_POINTS = 2;
public static double dot(Point2D u, Point2D v)
{
return u.getX()*v.getX() + u.getY()*v.getY();
}
public static double cross(Point2D u, Point2D v)
{
return u.getX()*v.getY() - u.getY()*v.getX();
}
/**
* Calculate moment of inertia about x-axis
* #param poly of 2D points defining a closed polygon
* #return moment of inertia about x-axis
*/
public static double ix(Point2D [] poly)
{
double ix = 0.0;
if ((poly != null) && (poly.length > MIN_POINTS))
{
double sum = 0.0;
for (int n = 0; n < (poly.length-1); ++n)
{
double twiceArea = poly[n].getX()*poly[n+1].getY() - poly[n+1].getX()*poly[n].getY();
sum += (poly[n].getY()*poly[n].getY() + poly[n].getY()*poly[n+1].getY() + poly[n+1].getY()*poly[n+1].getY())*twiceArea;
}
ix = sum/12.0;
}
return ix;
}
/**
* Calculate moment of inertia about y-axis
* #param poly of 2D points defining a closed polygon
* #return moment of inertia about y-axis
* #link http://en.wikipedia.org/wiki/Second_moment_of_area
*/
public static double iy(Point2D [] poly)
{
double iy = 0.0;
if ((poly != null) && (poly.length > MIN_POINTS))
{
double sum = 0.0;
for (int n = 0; n < (poly.length-1); ++n)
{
double twiceArea = poly[n].getX()*poly[n+1].getY() - poly[n+1].getX()*poly[n].getY();
sum += (poly[n].getX()*poly[n].getX() + poly[n].getX()*poly[n+1].getX() + poly[n+1].getX()*poly[n+1].getX())*twiceArea;
}
iy = sum/12.0;
}
return iy;
}
/**
* Calculate polar moment of inertia xy
* #param poly of 2D points defining a closed polygon
* #return polar moment of inertia xy
* #link http://en.wikipedia.org/wiki/Second_moment_of_area
*/
public static double ixy(Point2D [] poly)
{
double ixy = 0.0;
if ((poly != null) && (poly.length > MIN_POINTS))
{
double sum = 0.0;
for (int n = 0; n < (poly.length-1); ++n)
{
double twiceArea = poly[n].getX()*poly[n+1].getY() - poly[n+1].getX()*poly[n].getY();
sum += (poly[n].getX()*poly[n+1].getY() + 2.0*poly[n].getX()*poly[n].getY() + 2.0*poly[n+1].getX()*poly[n+1].getY() + poly[n+1].getX()*poly[n].getY())*twiceArea;
}
ixy = sum/24.0;
}
return ixy;
}
/**
* Calculate the moment of inertia of a 2D concave polygon
* #param poly array of 2D points defining the perimeter of the polygon
* #return moment of inertia
* #link http://www.physicsforums.com/showthread.php?t=43071
* #link http://www.physicsforums.com/showthread.php?t=25293
* #link http://stackoverflow.com/questions/3329383/help-me-with-converting-latex-formula-to-code
*/
public static double inertia(Point2D[] poly)
{
double inertia = 0.0;
if ((poly != null) && (poly.length > MIN_POINTS))
{
double numer = 0.0;
double denom = 0.0;
double scale;
double mag;
for (int n = 0; n < (poly.length-1); ++n)
{
scale = dot(poly[n + 1], poly[n + 1]) + dot(poly[n + 1], poly[n]) + dot(poly[n], poly[n]);
mag = Math.sqrt(cross(poly[n], poly[n+1]));
numer += mag * scale;
denom += mag;
}
inertia = numer / denom / 6.0;
}
return inertia;
}
}
Here's the JUnit test to accompany it:
import org.junit.Test;
import java.awt.geom.Point2D;
import static org.junit.Assert.assertEquals;
/**
* PolygonInertiaCalculatorTest
* User: Michael
* Date: Jul 25, 2010
* Time: 10:16:04 AM
*/
public class PolygonInertiaCalculatorTest
{
#Test
public void testTriangle()
{
Point2D[] poly =
{
new Point2D.Double(0.0, 0.0),
new Point2D.Double(1.0, 0.0),
new Point2D.Double(0.0, 1.0)
};
// Moment of inertia about the y1 axis
// http://www.efunda.com/math/areas/triangle.cfm
double expected = 1.0/3.0;
double actual = PolygonInertiaCalculator.inertia(poly);
assertEquals(expected, actual, 1.0e-6);
}
#Test
public void testSquare()
{
Point2D[] poly =
{
new Point2D.Double(0.0, 0.0),
new Point2D.Double(1.0, 0.0),
new Point2D.Double(1.0, 1.0),
new Point2D.Double(0.0, 1.0)
};
// Polar moment of inertia about z axis
// http://www.efunda.com/math/areas/Rectangle.cfm
double expected = 2.0/3.0;
double actual = PolygonInertiaCalculator.inertia(poly);
assertEquals(expected, actual, 1.0e-6);
}
#Test
public void testRectangle()
{
// This gives the moment of inertia about the y axis for a coordinate system
// through the centroid of the rectangle
Point2D[] poly =
{
new Point2D.Double(0.0, 0.0),
new Point2D.Double(4.0, 0.0),
new Point2D.Double(4.0, 1.0),
new Point2D.Double(0.0, 1.0)
};
double expected = 5.0 + 2.0/3.0;
double actual = PolygonInertiaCalculator.inertia(poly);
assertEquals(expected, actual, 1.0e-6);
double ix = PolygonInertiaCalculator.ix(poly);
double iy = PolygonInertiaCalculator.iy(poly);
double ixy = PolygonInertiaCalculator.ixy(poly);
assertEquals(ix, (1.0 + 1.0/3.0), 1.0e-6);
assertEquals(iy, (21.0 + 1.0/3.0), 1.0e-6);
assertEquals(ixy, 4.0, 1.0e-6);
}
}

For reference, here's a mutable 2D org.gcs.kinetic.Vector implementation and a more versatile, immutable org.jscience.mathematics.vector implementation. This article on Calculating a 2D Vector’s Cross Product is helpful, too.

I did it with Tesselation. And take the MOI's all together.