glen/dyna3
1
0
Fork 0
Code Issues 11 Pull Requests 1 Actions Projects Releases 2 Wiki Activity

doc: Add new notes directory with design notes

Browse Source
This commit is contained in:
Glen Whitney 2023-11-01 12:58:08 -07:00
parent fce8be5b56
commit 2cfcfacb5a
2 changed files with 92 additions and 0 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

53
notes/design.md Normal file
View File

@ -0,0 +1,53 @@
#### Design and Trajectory
The development goal is to get some sort of end-to-end system doing something as quickly as possible.
To that end, we should likely start with only points, planes, and incidence constraints. That will let us make some simple polyhedra; not sure if there are any challenging problems with just those elements/constraints. If not, it's probably good, because we could get away with just rational numbers as our underlying mathematical space and a toy handwritten solver. It will also be a good start on gluing different components together to make a working system.
Speaking of components, here are some different pieces we need to consider:
##### Language
Top candidates include C++, likely with a syntactic sugar preprocessor to make coding in it bearable, compiled into WASM via Emscripten, and Lobster, a significant-whitespace language with an interesting type system that appears on the Awesome WASM Languages list (link should go here).
Other candidates include Civet (as it's already very comfortable to code in, but it mires us in the JavaScript-family morass), and Python (also very comfortable for coding, speed might not be an issue if we're mostly using other optimized components, but not sure how mature WASM compilation is).
##### Display and Interaction
Possibilities include three.js, D3.js, writing `<x3d>` elements to the DOM and then using either x3DOM or X_ITE, or direct WebGL programming. See also possibly ganja.js, mentioned in the coordinate representation as well.
##### User Interface
The goal is to take the best parts of the UIs of GeoGebra, Geometry Expressions, LibreCAD, and FreeCAD. Some lessons therefrom:
* Multi-selection should be effortless, probably click to toggle whether something is in the current selection -- but if so, need a really easy gesture to unselect everything.
* Modality: Should tools reset themselves after one use, or persist until the tool is canceled? One option is have the idiom that "select operands, then select tool" is a one-shot use of the tool, whereas selecting a tool when there are no operands selected "activates" that tool until it is "deactivated" by selecting another tool or deselecting that tool.
* Parallelism: Every operation should be equally invocable either by a menu item, toolbar button (if it is configured to be on one), keystroke (ditto), or textual "command" (e.g., in some functional notation). The menu layout should be fixed and comprehensive; the textual commands should be internationalized and comprehensive; and the program should offer a default toolbar layout and keyboard shortcuts that need not be comprehensive and which should be easily configurable.
##### Solver
SolveSpace could well provide a general numeric solver that would work for us. There is a new Julia-based computer algebra system OSCAR that could be of interest. Sage may have pieces we can use. There may be generic Gröbner basis implementations out there we can use.
A closely related question is the representation of numbers that appear in coordinates. We can start with exact rationals, which should be implemented or easy to implement in whatever language we choose. But as soon as we have any quadratic constraints (like equal distances) we will need at least quadratic algebraic numbers if not arbitrary algebraic numbers, and a package to represent/manipulate those will be needed.
##### Coordinate Representation
Besides just the representation of numbers, we have to decide how to coordinatize various geometric entities. The default starting place is just triples of numbers for points in $\mathbb{R}^3$, but we may quickly decide to use more elaborate coordinates that allow more operations or entities to be easily coordinatized. For example, many systems use homogeneous coordinates with one extra dimension so that all rigid motions and scalings can be represented as (in our case) 4×4 matrices. Systems that have been suggested are inversive coordinates (Alex Kontorovich, see [inversive.md](inversive.md)) and various Geometric Algebras (like Clifford Algebras and many variants). Related to this last operation, see ganja.js, which could also bear on the the display/interaction item above as well.
###### Representing lines
In general it seems in 3D it's more comfortable to represent planes and points than lines. There are appear to be numerous options, not clear if any are really perfect or canonical:
* [Equivalence classes of] pairs of points
* [Equivalence classes of] pairs of planes
* A bag of arbitrarily many collinear points [of course this is still equivalence classes, but the practical computational aspect is that when there's another point of interest that turns out to be on the line, you just throw it in the bag.]
* Plücker coordinates
* A plane through the origin and a point on it. The line is perpendicular to that plane and goes through that point. This representation at least seems to be 1-1. Easy to tell if lines are parallel. Maybe not as easy to tell if they intersect, but not particularly worse than pairs of points, for example.
* The unit normal vector of the plane from the last option, but projected to the xy-plane, and that point of intersection, but projected to the xy-plane, so that there are just four numbers corresponding to the four-dimensionality of the space of lines. This representation has some discontinuities: very close lines might be represented by faraway coordinates, and (partly as a result) it might be tricky to compute with in general.

39
notes/inversive.md Normal file
View File

@ -0,0 +1,39 @@
#### Inversive Coordinates
(proposed by Alex Kontorovich as a practical system for doing 3D geometric calculations)
These coordinates are of form $I=(c, r, x, y, z)$ where we think of $c$ as the co-radius, $r$ as the radius, and $x, y, z$ as the "Euclidean" part, which we abbreviate $E_I$. There is an underlying basic quadratic form $Q(I_1,I_2) = (c_1r_2+c_2r_1)/2 - x_1x_2 -y_1y_2-z_1z_2$ which aids in calculation/verification of coordinates in this representation. We have:
| Entity or Relationship | Representation | Comments/questions |
| ---------------------------------------------------------- | -------------------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |
| Sphere s with radius r>0 centered on P = (x,y,z) | $I_s = (1/c, 1/r, x/r, y/r, z/r)$ satisfying $Q(I_s,I_s) = -1$, i.e., $c = r/(\|P\|^2 - r^2)$. | Can also write $I_s = (\|P\|^2/r - r, 1/r, x/r. y/r, z/r)$ -- so there is no trouble if $\|E_{I_s}\| = r$, just get first coordinate to be 0. |
| Plane p with unit normal (x,y,z), a distance s from origin | $I_p = (2s, 0, x, y, z)$ | Note $Q(I_p, I_p)$ is still -1. |
| Point P with Euclidean coordinates (x,y,z) | $I_P = (\|P\|^2, 1, x, y, z)$ | Note $Q(I_P,I_P) = 0$.  Why are we taking this rather than $(\|P\|, \|P\|, x, y, z)$ or $(1/\|P\|,\|P\|, x/\|P\|, y/\|P\|, z/\|P\|)$ ? I guess latter has trouble with the origin, for one. Perhaps this is the (only) choice that make some of the following true? |
| P lies on sphere or plane given by I | $Q(I_P, I) = 0$ | |
| Sphere/planes represented by I and J are tangent | $Q(I,J) = 1$ | |
| P is center of sphere represented by I | Well, $Q(I_P, I)$ comes out to be $(\|P\|^2/r - r + \|P\|^2/r)/2 - \|P\|^2/r$ or just $-r/2$ . | Is it if and only if ?   No this probably doesn't work because center is not conformal quantity. |
| Distance between P and R is d | $Q(I_P, I_R) = d^2/2$ | |
| Distance between P and sphere/plane rep by I | | |
| Distance between sphere/planes rep by I and J | | Q(I,J)=cosh^2 (d/2) maybe where d is distance in usual hyperbolic metric. Or maybe cosh d |
| Sphere centered on P through R | | Probably just calculate distance etc. |
| Dihedral angle between planes (or spheres?) rep by I and J | | cosh t = cos it s Q(I, J) = cos theta. |
| R, P, S are collinear | Maybe just cross product of two differences. | Not a conformal property. |
| Plane through noncollinear R, P, S | Should be, just solve Q(I, I_R) = 0 etc. | |
| circle | Maybe concentric sphere and the containing plane? | Defn: circle is intersection of two spheres. That does cover lines. But you lose the canonicalness |
| line | Maybe two containing planes? Maybe the perpendicular plane through origin and the point of line on it? Or maybe just as a bag of collinear points? | The first is the limiting case of the possible circle rep, but it is not canonical. The second appears to be canonical, but I don't see a circle rep that corresponds to it. |
The unification of spheres/planes is indeed attractive for a project like Dyna3. The relationship between this representation and Geometric Algebras is a bit murky; likely it somehow fits under the Geometric Algebra umbrella.
##### Additional more disorganized notes
Discussed coordinates with Alex Kontorovich. He was suggesting "inversive coordinates" -- for a sphere, that's 1/coradius, 1/radius, center/radius (where coradius is radius of sphere inverted in the unit sphere.) The advantage is tangent to and perpendicular to are linear in these coordinates (in the sense that if one is known, the condition of being tangent to or perpendicular to that one are linear). Planes have 1/radius = 0, and in fact, you can take the coordinates to be (2s, 0, x, y, z) where s is the distance to the origin and x,y,z are the normal direction. (Note the normal direction is only determined up to a scalar multiple. So could always scale so that the first non-zero coordinate is 1, or if you like only allow x, y to vary and let z be determined as sqrt(1-x^2^-y^2^). ) Points can be given by (r^2,1,x,y,z) where x,y,z are the coordinates and r is the distance to the origin. Quadratic form that tells you if something is a sphere/plane, or in the boundary, or up in the hyperbolic plane above. There are some details, but not quite explicit for modeling R^3, at http://sites.math.rutgers.edu/~alexk/files/LetterToDuke.pdf -- all this emphasize need to be agnostic with respect to geometric model so that we can experiment. Not really sure exactly how this relates or not to conformal geometric algebra, and whether it can be combined with geometric algebra. As formulated, there are clear-ish reps for planes/spheres and for points, but not as clear for lines. Have to see how to compute distance and/or specify a given distance. To combine inversive coordinates and geometric algebra, maybe think dually; there should be a lift from a normal vector and distance from origin to the five-vector; bivectors would rep circles/lines; trivectors would rep point pairs/points. What is the signature of this algebra, i.e. how many coordinates square to +1, -1, or 0? But it doesn't seem worth it for three dimensions, because there is a natural representation of points, as follows:
The signature of Q will be (1,4), and in fact Q(I1,I2) = 1/2(ab+ba) - E1\dot E2, where a is the "first" or "coradius" coordinate, "b" is the "second" or "radius" coordinate, and E is the Euclidean part (x,y,z). Then the inversive coordinates of a sphere with center (x,y,z) and radius r will be I = (1/\hat{r},1/r,x/r,y/r,z/r) where \hat{r} = r/(|E|^2 -r^2). These coordinates satisfy Q(I,I) = -1. For this to make sense, of course r > 0, but we get planes by letting the radius of a tangent sphere to the plane go to infinity, and we get I = (2s, 0, x0, y0, z0) where (x0,y0,z0) is the unit normal to the plane and s is the perpendicular distance from the plane to the origin. Still Q(I,I) = -1.
Since r>0, we can't represent individual points this way. Instead we will use some coordinates J for which Q(J,J) = 0. In particular, if you take for the Euclidean point E = (u,v,w) the coordinates J = (`|E|`^2,1,u,v,w) then Q(J,J) = 0 and moreover it comes out that Q(I,J) = 0
whenever E lies on the sphere or plane described by some I with Q(I,I) = -1.
The condition that two spheres I1 and I2 are tangent seems to be that Q(I1,I2) = 1. So given a fixed sphere, the condition that another sphere be tangent to it is linear in the coordinates of that other sphere.
This system does seem promising for encoding points, spheres, and planes, and doing basic computations with them. I guess I would just encode a circle as the intersection of the concentric sphere and the containing plane, and a line as either a pair of points or a pair of planes (modulo some equivalence relation, since I can't see any canonical choice of either two planes or two points). Or actually as described below, there is a more canonical choice.
I will have to work out formulas for the Euclidean distance between two entities, and the angle between them, and especially the intersection of two lines and the condition that three points are collinear.
In this vein, it seems as though if J1 and J2 are the reps of two points, then Q(J1,J2) = d^2/2. So then the sphere centered at J1 through J2 is (J1-(2Q(J1,J2),0,0,0,0))/sqrt(2Q(J1,J2)). Ugh has a sqrt in it. Similarly for sphere centered at J3 through J2, (J3-(2Q(J3,J2),0000))/sqrt(2Q(J3,J2)). J1,J2,J3 are collinear if these spheres are tangent, i.e. if those vectors have Q-inner-product 1, which is to say Q(J1,J3) - Q(J1,J2) - Q(J3,J2) = 2sqrt(Q(J1,J2)Q(J2,J3)). But maybe that's not the simplest way of putting it. After all, we can just say that the cross-product of the two differences is 0; that has no square roots in it.
One conceivable way to canonicalize lines is to use the *perpendicular* plane that goes through the origin, that's uniquely defined, and anyway just amounts to I = (0,0,d) where d is the ordinary direction vector of the line; and a point J in that plane that the line goes through, which just amounts to J=(r^2,1,E) with Q(I,J) = 0, i.e. E\dot d = 0. It's also the point on the line closest to the origin. The reason that we don't usually use that point as the companion to the direction vector is that the resulting set of six coordinates is not homogeneous. But here that's not an issue, since we have our standard point coordinates and plane coordinates; and for a plane through the origin, only two of the direction coordinates are really free, and then we have the one dot-product relation, so only two of the point coordinates are really free, giving us the correct dimensionality of 4 for the set of lines. So in some sense this says that we could take naively as coordinates for a line the projection of the unit direction vector to the xy plane and the projection of the line's closest point to the origin to the xy plane. That doesn't seem to have any weird gimbal locks or discontinuities or anything. And with these coordinates, you can test if the point E=x,y,z is on the line (dx,dy,cx,cy) by extending (dx,dy) to d via dz = sqrt(1-dx^2 - dy^2), extending (cx,cy) to c by determining cz via d\dot c = 0, and then checking if d\cross(E-c) = 0. And you can see if two lines are parallel just by checking if they have the same direction vector, and if not, you can see if they are coplanar by projecting both of their closest points perpendicularly onto the line in the direction of the cross product of their directions, and if the projections match they are coplanar.