Paul wrote:"The face-lattice can be viewed as a poset (partially-ordered set). A node in the face-lattice a is less-than a node b if there is an always increasing path from a to b.
Every two elements a and b have a unique, minimal upper-bound called the join of a and b (the smallest-dimensional face which both of these faces belong to). Every two elements a and b have a unique, maximal lower-bound called the meet of a and b (the largest-dimensional face which both of these faces share). "
"convex combination"...? What precisely is a 'convex' combination? I don't understand...?
"in-hyperplane dot-product"... Is this something that has a general definition? It wouldn't be one of the various products that one might use in Clifford, Geometric, or Multi-linear algebra? It kinda sounds like it might be...
I don't have an advanced understanding of Geometric algebra, but if some of this can be described in terms of multi-vectors and products of Geometric algebra, I do have some significant familiarity with these... at least, basics of Geometric algebra.
I also just noticed... it appears you just added two images to your post. I was about to ask... I guess you use the html tag img within the square brackets, like when posting a URL to the board? Can the images be GIFs, JPEGs, or PNGs? What image formats can be posted? Are there any restrictions?
That's interesting... I put the regular html tags with the greater than and less than signs for the underlines here. Can you use the regular html tags for URLs and other html markup?
Paul wrote:I gather you're probably referring to something like the Lounesto (left) contractive inner product, or the Hestenes semi-symmetric inner product...
Paul wrote:what software did you use to generate the graphs in your last two posts?
Paul wrote:Would a 'linear' combination also be a 'convex' combination?
"If I read my tea-leaves correctly, I believe this is a 4-dimensional polytope with seven vertexes and thirty-five tetrahedronal facets."
If I'm following correctly, the information the dot product yielded tells us that those 4 points are in one hyperplane... in this case, one realm, one 3D space. It's one cell of the polytope.
However, although it's unlikely, is it possible, for instance, that all 4 points are coplanar?
Also... can we really be sure that all the other cells are going to be the same as this one? Even if each is composed of 4 points, couldn't some of them just be a 2D face,... or something?
So, one would be well-advised to determine the rank of the matrix formed by the vertexes before entering into all of this.
For example, using the wedge-product, one might be able to tell that the four vertexes on one face of a cube are coplanar, but would you be able to distinguish that face from one cut to include one edge and the diagonally opposed edge of the cube?
Paul wrote:My first thought is to perhaps consider those convex combinations...?
Paul (in email) wrote:Hence, I had thought that something like the Hodge dual would only work in 3D... perhaps 7D? However, I didn't think the Hodge dual would work in a general nD since there is no duality between vectors in bivectors in nD of the nature there is in 3D.
public Plane4D( Point4D aa, Point4D bb, Point4D cc, Point4D dd ) {
double ax = aa.getX();
double ay = aa.getY();
double az = aa.getZ();
double aw = aa.getW();
double e1 = ( bb.getX() - ax );
double e2 = ( bb.getY() - ay );
double e3 = ( bb.getZ() - az );
double e4 = ( bb.getW() - aw );
double cx = cc.getX() - ax;
double cy = cc.getY() - ay;
double cz = cc.getZ() - az;
double cw = cc.getW() - aw;
double e12 = e1 * cy - e2 * cx;
double e13 = e1 * cz - e3 * cx;
double e14 = e1 * cw - e4 * cx;
double e23 = e2 * cz - e3 * cy;
double e24 = e2 * cw - e4 * cy;
double e34 = e3 * cw - e4 * cz;
double dx = dd.getX() - ax;
double dy = dd.getY() - ay;
double dz = dd.getZ() - az;
double dw = dd.getW() - aw;
double e123 = e12 * dz - e13 * dy + e23 * dx;
double e124 = e12 * dw - e14 * dy + e24 * dx;
double e134 = e13 * dw - e14 * dz + e34 * dx;
double e234 = e23 * dw - e24 * dz + e34 * dy;
this.normal = new Point4D( e234, -e134, e124, -e123 );
this.offset = this.normal.dot( aa );
if ( this.offset < 0 ) {
this.normal = new Point4D( -e234, e134, -e124, e123 );
this.offset = this.normal.dot( aa );
}
};
if ( this.offset < 0 ) {
this.normal = new Point4D( -e234, e134, -e124, e123 );
this.offset = this.normal.dot( aa );
}
Paul wrote:That applet doesn't work in IE for me... it does work in Netscape, though... Do you need the Sun Java enabled?
Anyway... I'm not sure I understand. Is that a 3D cube and a 3D tetrahedron in the applet?
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if ( this.offset < 0 ) {
this.normal = new Point4D( -e234, e134, -e124, e123 );
this.offset = this.normal.dot( aa );
}
This part of your code looks like it might be taking the negative of some vector. Is it taking the negative of the normal if it turns out that we get the one pointing towards the centroid of the polytope?
Also... this hidden line stuff... Is the centroid of the polytope at the origin, at least of the main frame (is this usually called the World frame?) Then, you have position vectors (well, the points in the main frame) which locate the tail of the line of sight vector.
Do you use different frames? Like perhaps one for the line of sight vector, and perhaps one for the normal to the facet?
Then, you'd have to come up with the normal to each n-1 facet of the nD polytope. Presumably, you take the one pointing outward. Can you tell whether it's pointing inward or outward by... well, something perhaps to do with the parametric equations of the rays of one of the position vectors of a corner of the facet and the normal... They won't intersect, but can the range, or sign, of something... Seems to me I've seen this done sometime. Somehow I think you can tell whether the two vectors are pointing in the same direction...
Anyway... Once you've got the normal pointing outward (I'd think that would be the better...) then if you calculate the angle of incidence between the normal and a ray from the line of sight intersecting the normal... perhaps this angle of incidence would reveal if the facet is visible from the line of sight?
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