Best way to approximate throw arcs?

[triplejam]

What's the best way to approximate the curved path an object may take due to gravity if thrown?

Also what if the intended destination is known and the throw is more underhanded rather then overhanded / shot? I assume the highest point will vary based on the force but how to make it still follow the intended arc and look natural?

I figure once the path / spline is known then it can be drawn simply by subdividing into line segments.

[jrouwe]

The simplest way of approximating a thrown object is by assuming the object is a point mass and that it has no damping. This means you can use the classical mechanics formula:

x(t) = x(0) + v(0) * t + 0.5 * g * t^2

x(t): 3 dimensional position at time t
v(t): 3 dimensional velocity at time t
g: 3 dimensional gravity vector
t: time

You simulate using fixed time steps (e.g. every 0.1 seconds) and approximate the arc as line segments.

I'm not really sure what you mean with the second question, but the formula is always the same, the only thing you can choose here is v(0), the initial velocity of the object. If you know your starting position x(0) and destination position x(t) then you can figure out what the initial velocity should be through:

x(t) = x(0) + vu * up + vf * forward + 0.5 * g * t^2

I've split the velocity in an up and forward components (vu and vf). forward is a unit horizontal vector pointing from x(0) to x(t), up is a unit vector pointing up (usually the y-axis).

The time it takes to travel the horizontal distance can be determined by:

(x(t) - x(0)) dot forward = vf * t

The velocity can be expressed as:

vu^2 + vf^2 = |v(0)|^2

This gives you 3 equations with 3 unknowns (t, vu and vf), so you can solve your initial velocity v(0).