To finish out this section, let’s try something a bit more complex and a great deal more useful. We’ll dynamically calculate an object’s acceleration according to a rule stated in Algorithm #3 — the object accelerates towards the mouse.
Diagram of mouse acceleration vector
Anytime we want to calculate a vector based on a rule or a formula, we need to compute two things: magnitude and direction. Let’s start with direction. We know the acceleration vector should point from the object’s position towards the mouse position. Let’s say the object is located at the point (x,y) and the mouse at (mouseX,mouseY).
Diagram of dx, dy
In that diagram, we see that we can get a vector (dx,dy) by subtracting the object’s position from the mouse’s position.:
  • dx = mouseX - x
  • dy = mouseY - y
Let’s rewrite the above using PVector syntax. Assuming we are in the Mover object definition and thus have access to the object’s PVector position, we then have:
var mouse = new PVector(mouseX, mouseY);
// Look! We’re using the static sub() because we want a completely new PVector
var dir = PVector.sub(mouse, position);
We now have a PVector that points from the mover’s position all the way to the mouse. If the object were to actually accelerate using that vector, it would appear instantaneously at the mouse position. This does not make for good animation, of course, and what we want to do now is decide how quickly that object should accelerate toward the mouse.
In order to set the magnitude (whatever it may be) of our acceleration PVector, we must first ___ that direction vector. That’s right, you said it. Normalize. If we can shrink the vector down to its unit vector (of length one) then we have a vector that tells us the direction and can easily be scaled to any value. One multiplied by anything equals anything.
var anything = ??;
To summarize, we take the following steps:
  1. Calculate a vector that points from the object to the target position (mouse)
  2. Normalize that vector (reducing its length to 1)
  3. Scale that vector to an appropriate value (by multiplying it by some value)
  4. Assign that vector to acceleration
Here's what the program looks like, with those steps fully implemented:
You may be wondering why the circle doesn’t stop when it reaches the target. It’s important to note that the object moving has no knowledge about trying to stop at a destination; it only knows where the destination is and tries to go there as quickly as possible. Going as quickly as possible means it will inevitably overshoot the position and have to turn around, again going as quickly as possible towards the destination, overshooting it again, and so on and so forth. Stay tuned; in later sections we’ll learn how to program an object to arrive at a position (slow down on approach).
This example is remarkably close to the concept of gravitational attraction (in which the object is attracted to the mouse position). Gravitational attraction will be covered in more detail in the next section. However, one thing missing here is that the strength of gravity (magnitude of acceleration) is inversely proportional to distance. This means that the closer the object is to the mouse, the faster it accelerates.
Let’s see what this example would look like with an array of movers (rather than just one).

This "Natural Simulations" course is a derivative of "The Nature of Code" by Daniel Shiffman, used under a Creative Commons Attribution-NonCommercial 3.0 Unported License.