[HN Gopher] The Mighty Simplex (2023)
___________________________________________________________________
The Mighty Simplex (2023)
Author : just_human
Score : 27 points
Date : 2025-11-15 16:49 UTC (6 hours ago)
(HTM) web link (galileo-unbound.blog)
(TXT) w3m dump (galileo-unbound.blog)
| ceh123 wrote:
| On the topic of simplices! I did my PhD in dynamical systems and
| the space of invariant measures [0] is (in the compact setting)
| always a simplex and the extreme points are the ergodic measures.
| It's because of this that you can kind of assume your system is
| ergodic do work there and frequently be able to generalize to the
| non-ergodic case (through ergodic decomposition).
|
| But the real thing I wanted to mention here was the Poulsen
| Simplex [1]. This is the unique Choquet simplex [2] for which the
| extreme points are dense. This means that it's like an
| uncountably infinite dimensional triangle where no matter where
| you are inside the triangle, you're arbitrarily close to a
| corner. It's my favorite shape and absolutely wild and impossible
| to conceptualize (even though I worked with it daily for years!)
|
| [0] https://en.wikipedia.org/wiki/Invariant_measure
|
| [1] https://eudml.org/doc/74350
|
| [2] https://en.wikipedia.org/wiki/Choquet_theory
| ikjasdlk2234 wrote:
| The simplex method in linear programming was one of the first
| times I ever had something relatively complex "click" for me.
|
| It's funny, I still think about that feeling weekly. My memory of
| it happening while working out a homework problem is actually
| sort of like in the first Harry Potter movie when Harry gets his
| wand: warm, bright lighting, widening eyes, and a feeling like my
| head was a helium balloon.
| 01HNNWZ0MV43FF wrote:
| So a simplex is a hypertriangle? The minimal shape that can be
| made with planes in any dimension?
|
| Understanding that 10 years ago might have made it easier for me
| to understand collision detection algorithms for game physics
| ceh123 wrote:
| Exactly! It's n+1 points in n dimensions (when finite). Another
| way to think about it (the way that I know because it extends
| into general Banach spaces and not just n dimensional spaces)
| is that each point inside is the unique weighted average of the
| extreme points (corners). So in 2d, if you have a square you
| can get that middle point by averaging all the corners, or
| averaging two opposing corners, so it's not a simplex.
| DonHopkins wrote:
| https://news.ycombinator.com/item?id=12572696
|
| DonHopkins on Sept 24, 2016 | prev | next [-]
|
| That is a neat idea! It reminds me of another cool constraint
| based animation system called "Embedded Constraint Graphics" that
| Tom Ngo developed at Interval Research Corporation, for which he
| filed a patent on Aug 6, 1996. So hey, didn't that patent just
| expire a month ago?
|
| Golan Levin used the ECG graphical editor to create the vector
| based face cartoons for his "Mouther" project, by simply dragging
| the eyes and mouth and other features around like you'd naturally
| want to be able to do. [2]
|
| It's a really brilliant way to automatically create directly
| manipulatable interactive graphics from target examples, which
| you can interpolate between along multiple dimensions at once
| (zones of a simplicial complex [3]), by dragging different parts
| of the graphics appropriately. It would figure out how to map the
| direction and amount you're dragging at a particular location, to
| appropriate movement in the n-dimensional target interpolation
| space. It was great for making cartoony direct manipulation user
| interface widgets!
|
| "A constraint-based graphics system employs different examples of
| an image to define the constraints of the system. The examples
| are grouped into subsets which can be interpolated with one
| another, according to a user-specified input that determines the
| relative proportion of each example image. An animation can be
| created by defining a sequence of such interpolated images.
| Alternatively, a user can directly manipulate observable
| components of an image to define a particular state for the
| image. Automatic transformations are defined to provide
| registration of an image within an overall scene by shifting the
| frame of reference for an image so that unnatural movements do
| not occur as the animation proceeds through a sequence of states.
| The structure of the system permits an animation to be divided
| into distinct components that can be combined with complementary
| components of other animations, to provide new results. These
| components include a clip motion which defines a sequence of
| states, a clip character which applies the clip motion to a
| particular image configuration, and clip art."
|
| "The various image examples can be associated with one another in
| a manner that defines a topological data structure which
| identifies their relationships. An example of such a data
| structure for the images of FIGS. 2A-2D is shown in FIG. 3.
| Referring thereto, each of the four image examples is associated
| with a vertex of a geometric shape. Specifically, the three image
| examples of FIGS. 2A, 2B, and 2C, which form one subset, define a
| triangular shape. The second subset, comprising the examples of
| FIGS. 2A, 2B, and 2D, defines a second triangle. Since the
| examples of FIGS. 2A and 2B are shared between the two subsets,
| the two triangles are joined along a common interface. Each
| subset of examples constitutes a simplex, or zone, and all of the
| zones together form a combinatorial structure, or state space,
| known as a simplicial complex. In the case illustrated in FIG. 5,
| the state space is composed of two triangular zones, ABC and ABD.
| While both zones in this case are two-dimensional structures, it
| is also possible for a state space to include one-dimensional
| zones, i.e. a line whose end points are defined by two examples,
| as well as multi-dimensional zones such as a tetrahedron defined
| by four examples."
|
| "The combinatorial structure defines a state space for the
| graphics system. Any given point within this space defines a
| particular image, and movement within the state space causes the
| appearance of the image to change. More particularly, each of the
| vertices corresponds to one of the image examples shown in FIGS.
| 2A-2D. A point located between two vertices results in an
| interpolated image comprised of a representative portion of each
| of the two images associated with the respective vertices. Thus,
| as one moves from the lowest vertex A in the structure of FIG. 3
| up the vertical line 10 to the highest vertex B, the figure's
| arms smoothly move from the position shown in FIG. 2A to that
| shown in FIG. 2B. Movement from the lowest vertex A to the left
| vertex C causes a transition in the image from that of FIG. 2A to
| that of FIG. 2C. A point located somewhere within the triangle
| defined by the three vertices A, B and C corresponds to an image
| in which the arms are partially raised and the right leg is
| partially lifted. For example, the point 12 represents a position
| in which the image is a weighted composite consisting of 60% of
| FIG. 2B, 30% of FIG. 2A and 10% of FIG. 2C. The percentages sum
| up to unity, and the weight values which correspond to these
| percentages, i.e. 0.6, 0.3 and 0.1, constitute a vector in
| barycentric coordinates. In a relatively simple embodiment of the
| invention, the examples are interpolated linearly; but in a
| different embodiment, the interpolation could employ non-linear
| functions, such as cubic polynomials. Any state s within the
| state space can be specified by a zone, in this case the zone
| containing the examples A, B and C, together with the vector,
| i.e. 0.3, 0.6, 0.1!."
|
| [1] https://www.google.com/patents/US5933150
|
| [2] https://www.flong.com/archive/projects/mouther/
|
| [3] https://en.wikipedia.org/wiki/Simplicial_complex
___________________________________________________________________
(page generated 2025-11-15 23:01 UTC)