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Wizumwalt

what's the diff between point, vertex, tuple, etc...

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What's the difference between a point, node, grid, vertex, and 2d tuple. I'm trying to put together a uml class diagram of classes that I'd use in my package and not sure of the order of things like this. Thanks,

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This is not an authoratative answer, just some ideas.

First, you might add vector to that list. In math, at least when interpreted geometrically, a vector has a direction and magnitude (length). These are implicit in the representation, which is usually just a set of scalars (such as x,y,z for 3d).

A point is a location in space. It can also be considered as a vector from the origin. For this reason and others, points are often represented as vectors.

A vertex is, most generally, a point in space. We may speak of the 'vertices of a mesh', even if no information other than location is given. However, a vertex typically has additional information stored with it, such as color and texture coordinates.

A node can be various things, for example a junction or termination in a graph or tree.

At least according to one definition, a tuple is simply a set of n elements, where n is some fixed size.

A grid is, most generally, a regular subdivision of a surface.

All these terms probably have more rigorous or formal definitions than I've given here. Some research online will probably turn up plenty of info.

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Point - a geometric entity of no dimension, being specified by at least one coordinate.

Vertex - for an n-dimensional non-trivial geometric entity, it is a point which lies on such an entity such that it lies also on n sides of said geometric entity. For example: a vertex of a square is a point on the square such that the point happens to lie on two sides of the square simultaneously. For a cube, the point lies on three of it's edges; a 4-dimensional hypercube, four edges; a 5-dimensional hypercube, five edges; an n-dimensional hypercube, n edges.

Node - Several interpretations exist for this:

* the point at which a continuous curve crosses itself
* a "junction" point in a graph
* a major turning point in a path

Tuple - this is a very abstract concept. An n-tuple is essentially a collection of n mathematical "entities". One element can contain a mere scalar, while the next contains a list of 7-tensors. A tuple is in one sense a generalization of the concept of sequence: single, double, triple, quadruple, quintuple, ... n-tuple.

An example of a widely used tuple is the 4-tuple, or more colliquially, the quaternion.

EDIT: I'm not sure if a quaternion is formally defined as a 4-tuple. I do know that it is however often interpreted as such.

A grid is, like jyk said, a regular subdivision of a surface. Another definition would be an interconnected set of nodes.

HTH,
nilkn

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Quote:
Original post by nilkn
An example of a widely used tuple is the 3-tuple, or more colliquially, the quaternion.

Huh, how so? I would be more inclined to regard a quaternion as a 4-tuple (or a 2-tuple, if broken into scalar and vector parts).

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Quote:
Original post by Sneftel
Quote:
Original post by nilkn
An example of a widely used tuple is the 3-tuple, or more colliquially, the quaternion.

Huh, how so? I would be more inclined to regard a quaternion as a 4-tuple (or a 2-tuple, if broken into scalar and vector parts).


Heh, yeah sorry. I meant 4-tuple.

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Guest Anonymous Poster
Quote:
Original post by Wizumwalt
What's the difference between a point, node, grid, vertex, and 2d tuple.

I'm trying to put together a uml class diagram of classes that I'd use in my package and not sure of the order of things like this.

Thanks,


Mathematically:
A tuple is a finite sequence. The usual notation is (a1, a2,...,an).

A point is a 0-dimensional object. In n-dimensional space its coordinates are usually given by an n-tuple. For example, in 2-dimensional (2D) space, the point one unit in the positive y direction is given by the 2-tuple (0,1) in cartesian coordinates and (1,pi/2) in polar coordinates.

A vertex, in geometry and graph theory, is a point (defined above) where edges meet. In geometry and graph theory, node is usually synonymous with vertex.

A grid is formed by tessellating some figure, usually such that the lines are parallel (e.g. squares, hexagons, triangles all tessallate to form sets of parallel lines).

An n-vector (since you probably want vectors as well) is a direction and a magnitude. It is commonly represented by an n-tuple (e.g. (x, y, z) in 3D) or a linear combination of basis vectors (e.g. xi + yj + zk).

Quaternions can be thought of as one extension of the complex numbers (similar to how complex numbers are one extension of the real numbers). The three representations I've seen most often are:
w + xi + yj + zk where i2 = j2 = k2 = ijk = -1, ij = -ji = k, jk = -kj = i, ki = -ik = j.
(w, x, y, z) which is a 4-tuple of real numbers.
(w, v) where v = (x, y, z) which is a 2-tuple containing a real number in the first part and a 3-vector in the second part. This 3-vector is usually represented as a 3-tuple, as above.
There are other representations.

All that said, I'm not sure that all of these fit in one class tree.

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