What is a 3D Vector and how does it differ from a

问题:

Does a 3D vector differ from a 3D point tuple (x,y,z) in the context of 3D game mathematics?

If they are different, then how do I calculate a vector given a 3d point?

回答1:

The difference is that a vector is an algebraic object that may or may not be given as the set of coordinates in some space. (thanks to bungalobill for correcting my sloppiness).

A point is just a point given by coordinates. Generally, one can conflate the two. If you are given a set of coordinates, and told that they constitute a 'point' with no further information (choice of basis, etc), then you can just hand that set of numbers back and legitimately claim to have produced a vector.

The largest difference between the two is that it makes no sense to do things to one that you can do to the other. For example,

1. You can add vectors: <1 2 3> + <3 2 1> = <4 4 4>

2. You can multiply (or scale) a vector by a number (generally called a scalar) 2 * <1 1 1> = <2 2 2>

3. You can ask how far apart two points are: d((1, 2, 3), (3, 2, 1) = sqrt((1 - 3)² + (2 - 2)² + (3 - 1)²) = sqrt(8) ~= 2.82

A good intuitive way to think about the association between a vector and a point is that a vector tells you how to get from the origin (that one point in space to which we assign the coordinates (0, 0, 0)) to its associated point.

If you translate your coordinate system, then you get a new vector for the same point. Although the coordinates that make up the point will undergo the same translation so it's a pretty easy conflation to make between the two.

Likewise if rotate the coordinate system or apply some other transformation (e.g. a shear), then the coordinates and vector associated to the point will also change.

It's also possible for a vector to be something else entirely, for example a bounded function on the interval [0, 1] is a vector because you can multiply it by a real number and add it to another function on the interval and it will satisfy certain requirements (namely the axioms of a vectorspace). In this case one thinks of having one coordinate for each real number, x, in [0, 1] where the value of that coordinate is just f(x). So that's the easiest example of an infinite dimensional vector space.

There are all sorts of vector spaces and the notion that a vector is a 'point and a direction' (or whatever it's supposed to be) is actually pretty vacuous.

回答2:

A vector represents a change from one state to another. To create one, you need two states (in this case, points), and then you subtract the initial state from the final state in order to get the resultant vector.

回答3:

Vectors are a more general idea that a point in 3D space.

Vectors can have 2, 3, or n dimensions. They represent many quantities in the physical world (e.g., velocity, force, acceleration) besides position.

A mathematician would say that a vector is a first order tensor that transforms according to this rule:

u(i) = A(i, j)v(j)

You need both point and vector because they are different. A point in 3D space denoting position is a vector, but every vector is not a point in 3D space.

Then there's the computer science notion of a vector as a container - it's an abstraction for an array of values or references. This is a different concept from a mathematician's idea of a vector, because every vector container need not obey the first order tensor transformation law (e.g. a Vector of OrderItems). That's yet another separate idea.

It's important to keep all these in mind when talking about vectors and points.

回答4:

Does a 3D vector differ from a 3D point tuple (x,y,z) in the context of 3D game mathematics?

Traditionaly vector means a direction and speed. A point could be considered a vector from the world orgin of one time step. (even though it may not be considered mathematically pure)

If they are different, then how do I calculate a vector given a 3d point?

target-tower is the common mnemonic.

Careful on your usage of this. The resulting vector is really normal*velocity. If you want to change it into something useful in a game application: you will need to normalize the vector first.

Example: Joe is at (10,0,0) and he wants to go to (10,10,0)
Target-Tower: (10,10,0)-(10,0,0)=(0,10,0)
Normalize the resulting vector: (0,1,0)
Apply "physics": (0,1,0) * speed*elapsed_time < speed = 3 and we'll say that the computer froze for a whole 2 seconds between the last step and this one for ease of computation > =(0,6,0)
Add the resulting vector to Joes current point in space to get his next point in space: ... =(10,6,0)

Normal = vector/(sqrt(x*x+y*y+z*z))

...I think I have everything here

回答5:

No, there is no difference. Points are vectors. If you like to think of a vector as direction + magnitude, then point is just a vector from the origin of coordinates.

回答6:

Vector is the change in the states. A point is the static point. Two vectors can be parallel or perpendicular. You can have product of two vectors which is a third vector. You can multiply a vector by a constant. You can add two vectors.
All these operations are not allowed on point. So program wise if you think both as a C++ class, there will be many such methods in the vector class but probably only Get and Set for point.

回答7:

A vector is a line, that is a sequence of points but that it can be represented by two points, the starting and the ending point.

If you take the origin as the starting point, then you can describe your vector giving only the ending point.