Conversation started Jun 10, 2022 at 3:07.
D.C. the III
D.C. the III
Jun 10, 2022 03:07
been pondering for awhile @TedShifrin about finding equation of plane through point#(1,2,2)$ that cuts off smallest possible volume in first octant....
after some doodling.... I have this nebulous idea that I need to minimize the "slope" of my plane. THis idea comes from me picturing that I have my plane at the point and a bearing is attached to it so I am changing the direction of the plane all over to try and maximze the volume of my object.
Alexandru Ionut
Alexandru Ionut
@TedShifrin I have a little result you might find aesthetic that you told me you would like to see a year or so ago (if I ever get it)
CroCo
CroCo
Hi everyone, how can I compute this derivative where a and b are unit dual quaternion?
user image
K is symmetric positive definite matrix.
porridgemathematics
porridgemathematics
Jun 10, 2022 03:23
@monoidaltransform there is a natural extension of $g$ to the tensor bundle of $TM$ , and in the context of sobolev spaces, you can choose nice coordinates, say geodesic coordinates, to see that $|\nabla^{k} u|^p$ in these coordinates is basically a sum of the norms of the partial derivatives of $u$ of order $k$ raised to the power $p$
so working locally you get the usual euclidean sobolev norms
and you can show with some extra work that if you defined sobolev norms locally and patched them together with partitions of unity, they would all be equivalent to this one stated coordinate free using covariant derivatives
Ted Shifrin
Ted Shifrin
@D.C.theIII what are the xyz-intercepts of a plane?
@AlexandruIonut I don’t remember, but sure!
porridgemathematics
porridgemathematics
of course there would be some error terms you would need to make small to do what I am saying is possible
Alexandru Ionut
Alexandru Ionut
@TedShifrin Do you remember when we spoke about clothoids?
D.C. the III
D.C. the III
$(x,0,0), (0,y,0), (0,0,z)$
Ted Shifrin
Ted Shifrin
Vaguely, Alexandru.
porridgemathematics
porridgemathematics
Jun 10, 2022 03:27
@monoidaltransform the actual extension to higher form is as you would expect, you first define it on pure tensors by $g(a \otimes b, c \otimes d) = g(a,c) g(b,d)$ and then extend linearly
Ted Shifrin
Ted Shifrin
@D.C.theIII But you need an equation of the plane. Then tell me.
CroCo
CroCo
user image
correction.
any help.
D.C. the III
D.C. the III
So my method for "thinking" out things revovles around me writing out "Key questions", I am literally right now trying to answer the key question of "how canI characterize the plane?"...funny. ..gimme a minute
Alexandru Ionut
Alexandru Ionut
There is a 3d generalisation that was studied by a few folk in the 80s and 90s. Mehlum found expressions for parameter functions using some pretty exotic multivariable hypergeometric series and Ron Resch found a geometric construction for it. It is the spherical clothoid, the spherical curve with geodesic curvature a linear function of arc length (term coined by Ulo Lumiste in unrelated work). Well after 2 years of work, I found MUCH simpler expressions for the parameter functions :)
It's the best result of my life
arxiv.org/pdf/2203.07963.pdf
Ted Shifrin
Ted Shifrin
Congrats! This is stuff I don’t know at all, but I’ll read through it tomorrow.
Alexandru Ionut
Alexandru Ionut
Jun 10, 2022 03:32
I appreciate it greatly. I need to fix a sentence or two in my derivation (the way I worded some of the reasoning is wrong but the results all hold)
but you can still appreciate 99% of the paper I think
Ted Shifrin
Ted Shifrin
I know nothing about special functions …
Alexandru Ionut
Alexandru Ionut
I know it is quite niche.
D.C. the III
D.C. the III
let $v_1 = (x,0,0) - (0,y,0) = (x,-y,0)$ and $v_2 = (0, - y, z)$. I now have two direction vectors. Take the cross product, get a normal, call it $A$. Then eqn of plane will be $A \cdot \mathbf{(x - x_0)}$. where $x_0 = (1,2,3)$. Now this would be my constraint.
Ted Shifrin
Ted Shifrin
You’re destined to fail using xyz as two different things.
D.C. the III
D.C. the III
did I inadvertently use xyz as a set of points and as legths?
Ted Shifrin
Ted Shifrin
Jun 10, 2022 03:47
As intercepts and as coordinates of a generic point.
D.C. the III
D.C. the III
hmm.
Ted Shifrin
Ted Shifrin
It’s ok to use xyz for intercepts, but then you need XYZ for coordinates.
You want to maximize what?
D.C. the III
D.C. the III
so using a,b,c as the intercept pieces....I see the reasoning, but right now what is making me feel weird is that wouldn't this mean I'm fixing some arbitrary values for my vectors. In essence already giving the elevation of my plane?
I want to maximize volume of the octant
Ted Shifrin
Ted Shifrin
So that’s some constant times xyz.
D.C. the III
D.C. the III
Ok yeah, that makes sense thinking about it for a minute. Going to eat then write out my solution
 
Conversation ended Jun 10, 2022 at 3:59.