Hello Otis and welcome to PF. It is not clear to me exactly what you want to do. At first sight, it would seem to me that any point within the original circle would still be inside the rotated circle. Obviously, I have not understood what you want. However, if CG means
computer graphics, then...
I would love to know the answer to this question too. There is probably not much available for free. You might try the Gutenberg project
http://www.gutenberg.org/catalog/world
A search for "physics" returns some things by Einstein and something by Poincare.
Ah! That's the spirit! Does the wedge product ever fail?
' "No, never."
"What, never?"
"Well, hardly ever!" ' (G&S: HMS Pinafore)
To tickle your curiosity: In the 5-dimensional conformal model of Euclidean 3D space, the expression for a sphere in terms of 4 vectors (in...
Yes, that is fine, but here is
another, more direct, way to get the area by calculating the magnitude
of the bivector representing the parallelogram. In terms of a set of
orthonormal basis vectors \{ e_i }, i = 1,N \} in N dimensions. The vectors are
\vec{a} = \sum_i a_i e_i...
Hello and welcome to PF!
To rotate one vector into another one, you
need only a single rotation in the plane containing the two vectors.
Here is a general method of finding such a single rotation.
The angle of rotation is a
bivector whose direction specifies the plane of rotation and...
To get you started, here is a link to a PhD thesis devoted to this topic.
http://deposit.ddb.de/cgi-bin/dokserv?idn=969279493&dok_var=d1&dok_ext=pdf&filename=969279493.pdf
Yes, and there are some intervals that can sound both pleasant or
unpleasant depending on the context. The interval C to G#, an
augmented fifth, is dissonant, whereas C to A-flat, a minor sixth, is
consonant. This is so even on a piano, where these intervals are exactly
the same. To hear...
Hmmm. Does this really work? Consider a point on the sphere whose vector is perpendicular to one of the faces (i.e. the reference vector for the face, as defined in my post above). The projection of this vector onto its face would be zero, whereas the projection onto the other faces (excluding...
Welcome to PF! Never having done such a calculation, I am not sure of my
ground here, but since no one else has yet replied, here is a suggestion
that might work: Let a, b, and c be unit vectors (relative to an origin at
the centre of the sphere) representing the vertices of one of the...
You would probably always use a Euclidean metric because the outer product factorization should not be dependent on the metric. I vaguely recall seeing this somewhere else in the book (probably in dealing with the meet and the join). Glad you like the book---I'm sure it will be a valuable asset.
Algorithm for factoring a blade (from Dorst et al. GA4CS p. 535):
1. Input is a non-zero blade B of grade k.
2. Determine the norm s =||B||
3. Represent the blade in a basis and then find the basis blade E in this
representation with the largest coordinate; now you have a list of k basis...
As we did not own a car, my mother was fond of referring to her
wash machine as our "pseudo-automobile" because it was not a car
but had four wheels. My baby brother grew up speaking in this way
and became a Great Physicist, since he grasped the notion of
pseudovector at once: it is not a...
On the contrary, it means that k lies in the subspace F, and this implies that F can be factored as F=k\wedge a, where
a is some vector. i.e. F is not only a bivector but is also a blade. Then
k\wedge F = k\wedge k\wedge a = 0 . (Recall that, in 4 dimensions, a bivector does not in...