How Can I add Latitude, Longitude dynamically in Javascript?

How can I represent motion vectors on the surface of a sphere and add them together?

• I'm making a planet simulator, which makes much use of a sphere. Currently, I use latitude/longitude pairs for coordinates on the sphere, which have been fine for things like calculating distances and projecting the sphere to a rectangle; however, I'm not sure how to represent motion on the sphere (i.e. vectors) and how to add these vectors to produce new coordinates.  I'm completely open to not using latitude/longitude if that makes the job easier. I'm using a sphere of radius 1 to simplify calculation.

• Answer:

  Motion of points on the surface of a sphere is analogous to angular motion in the 3D space around the origin. An angular motion can be represented by http://en.wikipedia.org/wiki/Angular_velocity, which is a 3D vector. If the direction of movement is perpendicular to the point's displacement from the origin (which is always the case if the point is moving along the surface of a sphere), angular velocity has the formula [math]\vec{\omega} = \frac{d\phi}{dt} \vec{u} = \frac{v}{r} \vec{u}[/math] where [math] \frac{d\phi}{dt} [/math] is the angular speed (the speed in radian the point is rotating), [math]\vec{u}[/math] is the axis of rotation (using right-hand rule, normalized to unit length). It can also be expressed in its speed of motion on the surface of the sphere [math]v[/math], and the radius of the sphere [math]r[/math]. When a point undergoes two angular motions [math]\vec{\omega}_1,\vec{\omega}_2[/math] (or the point is rotating at [math]\vec{\omega}_1[/math] relative to the sphere, and the sphere is itself rotating at [math]\vec{\omega}_2[/math]), the resultant angular velocity just adds up to  [math]\vec{\omega}_1+\vec{\omega}_2[/math]. To find the position of a point [math]\vec{x}[/math] after rotating by [math]\vec{\omega}[/math] for time [math]t[/math], you may first convert the angular velocity to axis-angle representation (the axis of rotation is [math]\vec{\omega}/\left\Vert \vec{\omega}\right\Vert [/math], and the angle of rotation is [math]t \left\Vert \vec{\omega}\right\Vert [/math]), then use the http://en.wikipedia.org/wiki/Rodrigues%27_rotation_formula to obtain the resultant position. Note that the lat/long has to be converted to Cartesian coordinates to apply these formulas. Using Cartesian coordinates usually gives you simpler formulas.