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Advanced vector math

Planes

The dot product has another interesting property with unit vectors. Imagine that perpendicular to that vector (and through the origin) passes a plane. Planes divide the entire space into positive (over the plane) and negative (under the plane), and (contrary to popular belief) you can also use their math in 2D:

../../_images/tutovec10.png

Unit vectors that are perpendicular to a surface (so, they describe the orientation of the surface) are called unit normal vectors. Though, usually they are just abbreviated as normals. Normals appear in planes, 3D geometry (to determine where each face or vertex is siding), etc. A normal is a unit vector, but it's called normal because of its usage. (Just like we call (0,0) the Origin!).

The plane passes by the origin and the surface of it is perpendicular to the unit vector (or normal). The side towards the vector points to is the positive half-space, while the other side is the negative half-space. In 3D this is exactly the same, except that the plane is an infinite surface (imagine an infinite, flat sheet of paper that you can orient and is pinned to the origin) instead of a line.

Distance to plane

Now that it's clear what a plane is, let's go back to the dot product. The dot product between a unit vector and any point in space (yes, this time we do dot product between vector and position), returns the distance from the point to the plane:

var distance = normal.dot(point)

But not just the absolute distance, if the point is in the negative half space the distance will be negative, too:

../../_images/tutovec11.png

This allows us to tell which side of the plane a point is.

Away from the origin

I know what you are thinking! So far this is nice, but real planes are everywhere in space, not only passing through the origin. You want real plane action and you want it now.

Remember that planes not only split space in two, but they also have polarity. This means that it is possible to have perfectly overlapping planes, but their negative and positive half-spaces are swapped.

With this in mind, let's describe a full plane as a normal N and a distance from the origin scalar D. Thus, our plane is represented by N and D. For example:

../../_images/tutovec12.png

For 3D math, Godot provides a Plane built-in type that handles this.

Basically, N and D can represent any plane in space, be it for 2D or 3D (depending on the amount of dimensions of N) and the math is the same for both. It's the same as before, but D is the distance from the origin to the plane, travelling in N direction. As an example, imagine you want to reach a point in the plane, you will just do:

var point_in_plane = N*D

This will stretch (resize) the normal vector and make it touch the plane. This math might seem confusing, but it's actually much simpler than it seems. If we want to tell, again, the distance from the point to the plane, we do the same but adjusting for distance:

var distance = N.dot(point) - D

The same thing, using a built-in function:

var distance = plane.distance_to(point)

This will, again, return either a positive or negative distance.

Flipping the polarity of the plane can be done by negating both N and D. This will result in a plane in the same position, but with inverted negative and positive half spaces: