Moving slowly an initial angle until it reach a final angle

Question

I will try to be very descriptive with this. I'm editing a game right now and the scenario is a 3D area.

I have an initial angle, writen as a direction vector, and another vector which haves different coordinates. As we know, the angle between 2 vectors is given by the formula: Theta = ACos( DotProduct( vec1, vec2 ) / ( VectorLength( vec1 ) * VectorLength( vec2 ) ) )

So let's describe the scenario: I'm currently programming some kind of stationary weapon, a sentry gun, this thing moves slowly his "head", shooting bullets to enemies. That angle rotation thing is my problem.

Let's imagine this: I have my sentry gun on a empty 3D area, and a "enemy" spawns over there. I can currently get the direction vector of my sentry's view angle, and the direction vector between my sentry and the player. Let's guess, using the formula described, his separation angle is 45 degrees. My sentry gun thinks (calls a function) at every 0.1 seconds, and I want to move his head 5 degrees at every thinking function until it reach the the player (ie, both vectors are nearly equal), and that means it will reach the player (if player keeps on its position...) in 0.9 seconds (5 degrees from 45)

How I can move sentry's view angle slowly until it reach a target? In 2D is easily but know I'm fighting with a 3D scenario, and I'm currently lost with this.

Any help would be appreciated, and about coding, I will be grateful with a pseudocode. Thanks! (and sorry for my english)

Answer 1

What you need is called SLERP - spherical linear interpolation

Your starting direction vector is p0 there, goal direction is p1, Omega is your Theta, and t parameter varies in range 0..1 with needed step

Delphi example for 2D case (it is easy to control)

var
  p0, p1: TPoint;
  i, xx, yy: Integer;
  omega, InvSinOmega, t, a0, a1: Double;
begin
  P0 := Point(0, 200);
  P1 := Point(200, 0);
  omega := -Pi / 2;
  InvSinOmega := 1.0 / Sin(omega);
  Canvas.Brush.Color := clRed;
  Canvas.Ellipse(120 + P0.X, 120 + P0.Y, 120 + P0.X + 7, 120 + P0.Y + 7);
  Canvas.Ellipse(120 + P1.X, 120 + P1.Y, 120 + P1.X + 7, 120 + P1.Y + 7);

  for i := 1 to 9 do begin
    t := i / 10;

    a0 := sin((1 - t) * omega) * InvSinOmega;
    a1 := sin(t * omega) * InvSinOmega;
    xx := Round(P0.X * a0 + P1.X * a1);
    yy := Round(P0.Y * a0 + P1.Y * a1);

    Canvas.Brush.Color := RGB(25 * i, 25 * i, 25 * i);
    Canvas.Ellipse(120 + xx, 120 + yy, 120 + xx + 9, 120 + yy + 9);
  end;