Find the new coordinates using a starting point, a distance, and an angle

Question

Okay, say I have a point coordinate.

var coordinate = { x: 10, y: 20 };

Now I also have a distance and an angle.

var distance = 20;
var angle = 72;

The problem I am trying to solve is, if I want to travel 20 points in the direction of angle from the starting coordinate, how can I find what my new coordinates will be?

I know the answer involves things like sine/cosine, because I used to know how to do this, but I have since forgotten the formula. Can anyone help?

Answer 1

SOHCAHTOA

Sine = Opposite/Hypotenuse Cosine = Adjacent/Hypotenuse Tangent = Opposite/Adjacent

In your example:

Sine(72) = Y/20 -> Y = Sine(72) * 20
Cosine(72) = X/20 -> X = Cosine(72) *20

The problem is you have to be careful with what quadrant you are in. This works perfectly in the upper right quadrant, but not so nice in the other three quadrants.

Answer 2

Just to record a javascript adaptation from Movable Type Scripts

function createCoord(coord, bearing, distance){
    /** http://www.movable-type.co.uk/scripts/latlong.html
     φ is latitude, λ is longitude, 
     θ is the bearing (clockwise from north), 
     δ is the angular distance d/R; 
     d being the distance travelled, R the earth’s radius*
     **/

    var 
        radius = 6371e3, //meters
        δ = Number(distance) / radius, // angular distance in radians
        θ = Number(bearing).toRad();
        φ1 = coord[1].toRad(),
        λ1 = coord[0].toRad();

    var φ2 = Math.asin(Math.sin(φ1)*Math.cos(δ) + Math.cos(φ1)*Math.sin(δ)*Math.cos(θ));

    var λ2 = λ1 + Math.atan2(Math.sin(θ)*Math.sin(δ)*Math.cos(φ1), Math.cos(δ)-Math.sin(φ1)*Math.sin(φ2));

    λ2 = (λ2+3*Math.PI) % (2*Math.PI) - Math.PI; // normalise to -180..+180°

    return [λ2.toDeg(), φ2.toDeg()]; //[lon, lat]
}

Number.prototype.toDeg = function() { return this * 180 / Math.PI; }
Number.prototype.toRad = function() { return this * Math.PI / 180; }

Answer 3

i fount the equation here, i implemented it and it works perfectly.

this is my simple :

    const R = 6371e3; //rayon of the erth
    let lat1 = 36.7538 * Math.PI / 180; // latitude in rad
    let long1 = 3.0588 * Math.PI / 180; // longiture in rad
    let d = 5000; //distance between the two points
    let angle = (90-20)*Math.PI/180; // the angle between the 2 points in rad (20°)
    const sigma = d / R;
    const delLat = sigma * Math.cos(angle);
    let lat2 = (lat1 + delLat) * 180 / Math.PI;//latitude of the destination point in deg
    const del = Math.log(Math.tan(lat2 / 2 + Math.PI / 4) / Math.tan(lat1 / 2 + Math.PI / 4));
    // const q = Math.abs(del) > 10e-12 ? delLat / del : Math.cos(lat1); 
    const q = Math.cos(lat1)
    
    const delLong = sigma * Math.sin(angle) / q;
    let long2 = (long1 + delLong) * 180 / Math.PI; //longitude of the destination point in deg