I've recently learned that we use Pythagorean theorem a lot in our physics calculations and I'm afraid I don't really get the point.
Here'
s an example from a book to make sure an object doesn't travel faster than a MAXIMUM_VELOCITY constant in the horizontal plane:
MAXIMUM_VELOCITY = <any number>;
SQUARED_MAXIMUM_VELOCITY = MAXIMUM_VELOCITY * MAXIMUM_VELOCITY;
function animate(){
var squared_horizontal_velocity = (x_velocity * x_velocity) + (z_velocity * z_velocity);
if( squared_horizontal_velocity <= SQUARED_MAXIMUM_VELOCITY ){
scalar = squared_horizontal_velocity / SQUARED_MAXIMUM_VELOCITY;
x_velocity = x_velocity / scalar;
z_velocity = x_velocity / scalar;
}
}
Let'
s try this with some numbers:
An object is attempting to move 5 units in x and 5 units in z. It should only be able to move 5 units horizontally in total!
MAXIMUM_VELOCITY = 5;
SQUARED_MAXIMUM_VELOCITY = 5 * 5;
SQUARED_MAXIMUM_VELOCITY = 25;
function animate(){
var x_velocity = 5;
var z_velocity = 5;
var squared_horizontal_velocity = (x_velocity * x_velocity) + (z_velocity * z_velocity);
var squared_horizontal_velocity = 5 * 5 + 5 * 5;
var squared_horizontal_velocity = 25 + 25;
var squared_horizontal_velocity = 50;
// if( squared_horizontal_velocity <= SQUARED_MAXIMUM_VELOCITY ){
if( 50 <= 25 ){
scalar = squared_horizontal_velocity / SQUARED_MAXIMUM_VELOCITY;
scalar = 50 / 25;
scalar = 2.0;
x_velocity = x_velocity / scalar;
x_velocity = 5 / 2.0;
x_velocity = 2.5;
z_velocity = z_velocity / scalar;
z_velocity = 5 / 2.0;
z_velocity = 2.5;
// new_horizontal_velocity = x_velocity + z_velocity
// new_horizontal_velocity = 2.5 + 2.5
// new_horizontal_velocity = 5
}
}
Now this works well, but we can do the same thing without Pythagoras:
MAXIMUM_VELOCITY = 5;
function animate(){
var x_velocity = 5;
var z_velocity = 5;
var horizontal_velocity = x_velocity + z_velocity;
var horizontal_velocity = 5 + 5;
var horizontal_velocity = 10;
// if( horizontal_velocity >= MAXIMUM_VELOCITY ){
if( 10 >= 5 ){
scalar = horizontal_velocity / MAXIMUM_VELOCITY;
scalar = 10 / 5;
scalar = 2.0;
x_velocity = x_velocity / scalar;
x_velocity = 5 / 2.0;
x_velocity = 2.5;
z_velocity = z_velocity / scalar;
z_velocity = 5 / 2.0;
z_velocity = 2.5;
// new_horizontal_velocity = x_velocity + z_velocity
// new_horizontal_velocity = 2.5 + 2.5
// new_horizontal_velocity = 5
}
}
Benefits of doing it without Pythagoras:
Less lines
Within those lines, it'
s easier to read what'
s going on
...and it takes less time to compute, as there are less multiplications
Seems to me like computers and humans get a better deal without Pythagorean theorem! However, I'm sure I'm wrong as I've seen Pythagoras' theorem in a number of reputable places, so I'd like someone to explain me the benefit of using Pythagorean theorem to a maths newbie.
Does this have anything to do with
unit vectors? To me a
unit vector is when we normalize a vector and turn it into a fraction. We do this by dividing the vector by a larger constant. I'm not sure what constant it is. The total size of the graph? Anyway, because it'
s a fraction, I take it, a
unit vector is basically a graph that can fit inside a 3D grid with the x-axis running from -1 to 1, z-axis running from -1 to 1, and the y-axis running from -1 to 1. That'
s literally everything I know about
unit vectors... not much :P And I fail to see their usefulness.
Also, we're not really creating a
unit vector in the above examples. Should I be determining the scalar like this:
// a mathematical work-around of my own invention. There may be a cleverer way to do this! I've also made up my own terms such as 'divisive_scalar' so don't bother googling
var divisive_scalar = (squared_horizontal_velocity / SQUARED_MAXIMUM_VELOCITY);
var divisive_scalar = ( 50 / 25 );
var divisive_scalar = 2;
var multiplicative_scalar = (divisive_scalar / (2*divisive_scalar));
var multiplicative_scalar = (2 / (2*2));
var multiplicative_scalar = (2 / 4);
var multiplicative_scalar = 0.5;
x_velocity = x_velocity * multiplicative_scalar
x_velocity = 5 * 0.5
x_velocity = 2.5
Again, I can't see why this is better, but it'
s more "unit-vector-y" because the multiplicative_scalar is a unit_vector? As you can see, I use words such as "unit-vector-y" so I'm really not a maths whiz! Also aware that
unit vectors might have nothing to do with Pythagorean theorem so ignore all of this if I'm barking up the wrong tree.
I'm a very visual person (3D modeller and concept artist by trade!) and I find diagrams and graphs really, really helpful so as many as humanely possible please!
