Operators

Higher-dimensional geometry (previously "Polyshapes").

Operators

Postby moonlord » Sun Jun 11, 2006 9:31 am

It seems we need some strict definitions for the operators we use. I decided to create a distinct thread because it also belongs to "Naming and notations" and "cone-like objects". Maybe you can give some help, as my knowlegde in this area is limited. Just thought to ring a bell.

For example, E may be defined as:
Consider (...) given by function f(dimension variables)=0. (...)E will be given by equations
f(d.v.)=0 and |next dimension variable| <= length in new dimension.
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Postby Keiji » Sun Jun 11, 2006 11:00 am

That definition makes no sense whatsoever. Here's a better one.

In CSG notation, xE will duplicate the object x, translate the copies so that one is 1 unit and the other is -1 unit in the next dimension. It will then connect all existing hypercells from one copy to the corresponding hypercell of the other copy, including solidifying the object.
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Postby moonlord » Sun Jun 11, 2006 11:14 am

And how would you express that mathematically? The cartesian product has such a definition, as PWrong showed somewhere...
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Postby Keiji » Sun Jun 11, 2006 11:36 am

You asked for it.

Assuming object x is n-dimensional, all points in the (n+1)-dimensional object xE will have these coordinates:

(a<sub>1</sub>,a<sub>2</sub>,a<sub>3</sub>,...,a<sub>n</sub>,b)

where a<sub>n</sub> is the nth coordinate of a point in object x, and -1≤b≤1.

Satisfied?
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Postby moonlord » Sun Jun 11, 2006 11:45 am

Which sounds better but is exactly what I said earlier. Agreed. There are still other operators to be defined :).
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Postby Keiji » Sun Jun 11, 2006 11:48 am

Maybe so, but I find that my definition makes a lot more sense.
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Postby wendy » Tue Jun 13, 2006 8:57 am

This is the definition i use for the four products.

A base is a figure in a space of n dimensions.

A centre (o) of product is the point somewhere in the same space as the base. For the normal products, this is the same as the geometric centre. But for "off-centre" products, this can be anywhere.

For any other point (p), it is defined such that a ray from o through p, crosses the surface s at 1, ie the value of the radial function at p is r = (OP/OS).

The three dimension-summing products are:


1. prism: R = max(r1, r2, ...)
2. tegum: R = sum(r1, r2, ...)
3. sphere: R = rss(r1, r2, ...) [rss = root sum square]

When these are applied to an edge (u) from +1 to -1, with a centre at 0, this gives, respectively,

max(u1, u2, ...) -> line, square, cube, tesseract, ...
sum(u1, u2, ...) -> diagonal, rhombus, octahedron, 16choron, ...
rss(u1, u2, ...) -> diameter, circle, sphere, glome.

When the edge is set to unit, these can be used as a coherent measure scale. The dimension of the product is the sum of dimensions in the elements.

We have, for two lines X, Y,

max(X,Y) -> rectangle sides X, Y
sum(X,Y) -> rhombus diagonals X, Y
rss(X,Y) -> ellipse diagonals x, y

These can be continued upwards to any dimension.

The fourth product is the pyramid product. It is defined in terms of an altitude A, in the shape of a simplex:

A = point, line, triangle, tetrahedron, pentachoron, ....

We then a point in the simplex that fits the equation,

a1+a2+... = 1

For a given point a, it is a1 of the height to vertex 1, a2 of the height to vertex 2, etc. This varies from a1 = 1, at the vertex, to a1 = 0 at the opposite side. If one wants a simple example, consider the simplex as a face of a cross-product, where the points form the axies, A is the face, being a1+a2+... = 1.

The simplex product is then defined in terms of

1 = sum(a1,a2,....) over A
R = max(a1*r1, a2*r2, ... ) dim = Na+N1+N2+...

When r1 is a point, we have r1 = 1, and have, eg max(a1, a2*r1, ...)

For a pyramid p, we have A in the z-axis, running from r1=apex to r2=base.

a point x,y,z then gives a radial function a1+a2*r2, which gives a tapering of height of a2 going from full size when a2 = 1, to a point at a2 = 0.

When one has a tetrahedron, where x= line, y = line, z= height, one gets a tapering of height, implemented by rectangles a1.x * a2.y as z varies over a1 to a2.


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Postby moonlord » Wed Jun 14, 2006 11:16 am

Almost all I can gather from your post is that the tegum product gives the duals of the prism product... Sorry, can't follow you...
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Postby PWrong » Wed Jun 14, 2006 1:22 pm

First, a few short definitions.
codim(A) = codim A is the dimension of the object A
dim(A) = dim A is the dimension of the space of A
So if A is a kD shape in nD space, then codim A = k, and dim A = n

Here's my current definition for the torus product of A and B. This only works if B is a sphere/ball of some dimension. Also, we require that:
dim(B) => dim(A) - codim(A)
Otherwise B has a unique axis which could point in any direction, and there could be infinitely many of A#B.

A is a kD object in nD space. At any given point, it has a tangent space T (a k-hyperplane tangent to the surface) and a normal space N (a (n-k)D hyperplane perpendicular to the surface). Example: a 2D surface in 3D space has a tangent plane and a normal vector at every point. Don't worry about how we find the spaces themselves.

Now, the normal space is part of our new coordinate system for B. Let B be a sphere in mD. We add a new B at every point in A. If m > n-k, we have to add new dimensions.

Here's a step-by-step construction
A is a kD object in nD, and B is a sphere in mD
Construct an A.
Let u be the position vector of a point in A.
Let N = (n<sub>1</sub>, ... , n<sub>n-k</sub>) be the normal space of A at the point.
Construct a new B, with one axis pointing in the direction of n, and all other axes being perpendicular to the axes of A.
Let v be the position vector of a point in B.
The torus product of A and B is the set of all possible vectors u+v.

Finally, here's the short, but probably indecipherable mathematical definition:
A#B = {u + v| u ∈ A, v ∈ B(N, x<sub>n+1</sub> ... x<sub>k+m</sub>)}
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Postby wendy » Thu Jun 15, 2006 9:22 am

To understand my last message, one supposes that in a space (eg a plane), there is a function f(x,y) = r, such that

1. f(kx, ky) = kr

2. f(x,y) = 1 defines the surface of a figure.

What this gives, then is a radial function, which gives a value of 1 at the surface, and 2 at a copy twice as large. That is, all of the points f(x,y)=k gives the surface of a figure k times as large as f(x,y)=1.

A point (x,y,z) can then be comprised of two subspaces (x,y) and (z). One can then evaluate different figures f1(x,y), say a pentagon, and f2(z), say a line segment.

When one "multiplies" f1 and f2, one gets a product where the value of the function at x,y,z is fp(f1(x,y), f2(z)).

For example, when fp is max(), the maximum value applies over f1(x,y) and f2(z). When one of these is 1, and the other is smaller, then one is either on the walls or roof/floor of the prism. When one is inside the prism, the values of f1(x,y) and f2(z) are both less than one.

If one applies any of the other functions as above, one gets a definition for the radial function of the other products, given from their bases.

So, we have, eg for a cross-polytope, that

1. a line with the midpoint = 0, and the ends = 1

2. The surface of the cross polytope, defined as sum(abs(x), abs(y), ...) = 1.

But where instead of having absolute-value of x, we use a function that varies by angle, such that the raduis is measured as 1 in that direction.

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