# Rhombo

Over the past couple of weeks I wrote some code in C# to generate dissections of the rhombic triacontahedron into golden rhombohedrons. George Hart discusses these types of dissections here  and also talks about the problem of enumerating them in an appendix here — briefly, all this material by Hart and others is about how the fact that the rhombic triacontahedron and the rhombic enneacontahedron are zonohedra lead to both having interesting combinatoric properties which can be explored by coloring their dissections.

I was, however, more interested in how such dissections could be turned into an interlocking puzzle, akin to a traditional burr puzzle. amd as such needed code to generate 3D models of the dissections. My generation code is a dumb, constructive, brute force approach in which I just traverse the search space adding rhombohedrons to a candidate dissection in progress and backtracking when reaching a state in which it is impossible to add a rhombohedron without intersecting the one that was already added or the containing triacontahedron, keeping track of configurations that have already been explored.

Dissections of the rhombic triacontahedron into golden rhombohedrons (hereafter “blocks”) turns out to always need 10 and 10 of the two types of blocks that Hart refers to in the above as the “pointy” and “flat” varieties (and which I refer to as yellow and blue). Further it turns out that in all of these dissections there are four blocks that are completely internal, i.e. sharing no face with the triacontahedron; I also believe that the four internal blocks are always three blue and one yellow, but I’m not sure about that.

My strategy for finding an interlocking puzzle was the following:

• Generate a bunch of raw dissections into blocks
• For each dissection, search the adjacency graph for four pieces, the union of sets of five blocks, such that
• Each piece forms a simple path in the dissection; that is, each block in the piece
• is either an end block that is face adjacent to a next or previous block in the piece or is a non-end block that is face adjacent to a next block and a previous block.
• and does not share any edges with other blocks in the piece except for the edges of the face adjacencies.
• Each piece contains at least one fully internal block.
• Each piece is “single axis disentangle-able” from each other piece, where we mean by that that there exists some edge e in the complete construction such that if given piece p1 and piece p2, if you offset p1 in the direction of  e by a small amount p1 does not intersect p2.
• Each piece is not single axis disentangle-able from the union of the other three pieces.

I never managed to succeed in doing a complete enumeration, generating all of the dissections for reasons that I don’t feel like going into. (As I said above, I did not do anything fancy and it would be easier to just be smarter about how I do the generation than to make what I have more efficient; i.e. could have done the George Hart algorithm if I had known abouyt that or there are ways of transforming one dissection into another that I don’t do — I do an exhaustive search, period — but I never did the smarter stuff because I found what I was looking for, see below)

But from about 10 dissections I found one set of pieces that uniquely satisfies all of the above:

Here’s some video. (The individual blocks were 3D printed and super glued together)

I’m calling the above “rhombo”. Those pieces are rough because I only 3D printed the individual rhombohedrons and then superglued them together into the pieces, which is imprecise. I had to sand them heavily to get them to behave  nicely. I’ll eventually put full piece models up on Shapeways.

In the course of doing this work, it became apparent that there is no good computational geometry library for C# to use for something like this. There is one called Math.Net Numerics along with Math.Net Spatial that will get you vectors and matrices but not with all the convenience routines you’d expect to treat vectors like 3D points and so forth. What I ended up doing was extracting the vectors and matrices out of monogame and search-and-replacing “float” to “double” to get double precision. Here is that code on github. I also included in there 3D line segment/line segment intersection code and 3D triangle/triangle intersection code which I transliterated to C#. The line segment intersection code came from Paul Bourke’s web site. And the triangle intersection code came from running Tomas Moller’s C code through just a C preprocessor to resolve all the macros and then transliterating the result to C#.

# Basic signals & slots in C++

I put up some code on github that implements basic signals and slots functionality using standards compliant C++. It’s one header file “signals.hpp”. See here: link to github

This implementation of signals is a re-work of some code by a user _pi on the forums for the cross-platform application framework Juce — that thread is here — which was itself a re-work of some example code posted by Aardvajk in the GameDev.net forums (here). I just put it all together, cleaned things up, fixed some bugs, and added lambda support.

The basic idea is that given variadic templates being added to C++ it became possible for a more straight-forward implementation of signal and slot functionality beyond what was done for boost::signals et. al. — that is what Aardvajk’s original article was about. _pi changed Aardvajk’s code to make it such that slots are a thing you inherit from, which makes more sense to me. I changed _pi’s code so that it uses std::functions to store the handlers thus allowing lambdas with captures to be attached to a signal.

Usage is like the following:

```#include "signals.hpp"
#include <iostream>

// a handler is a kind of slot, that, as an implementation detail, requires usage of the
// "curiously recurring template pattern". That is, the intention is for instances of
// a class C that will react to a signal firing to have an is-a relationship with a slot
// parametrized on class C itself.

class CharacterHandler : public Slot<CharacterHandler>
{
public:
void HandleCharacter(char c)
{
std::cout << "The user entered '" << c << "'" << std::endl;
}
};

class DigitHandler : public Slot<DigitHandler>
{
public:
void HandleCharacter(char c)
{
if (c >= '0' && c <= '9') {
int n = static_cast<int>(c - '0');
std::cout << "  " << n << " * " << n << " = " << n*n << std::endl;
}
}
};

int main()
{
bool done = false;

char c;
Signal<char> signal;
CharacterHandler character_handler;
DigitHandler digit_handler;

// can attach a signal to a slot with matching arguments
signal.connect(character_handler, &CharacterHandler::HandleCharacter);

// can also attach a lambda, associated with a slot.
// (the lambda could capture the slot and use it like anything else it captures
// however technically all the associated slot is doing is allowing you to have
// a way of disconnecting the lambda e.g. in this case signal.disconnect(characterHandler)
signal.connect(character_handler,
[&](char c) -> void {
if (c == 'q')
done = true;
}
);

// can also attach a slot to a signal ... this means the same as the above.
digit_handler.connect(signal, &DigitHandler::HandleCharacter);

do
{
std::cin >> c;
signal.fire(c);

} while (! done);

// can disconnect like this
character_handler.disconnect(signal);

// or this
signal.disconnect(digit_handler);

//although disconnecting wasnt necessary here in that just gaving everything go out of scope
// wouldve done the right thing.

return 0;
}
```

# Mean Shift Segmentation in OpenCV

I’ve posted a new repository on GitHub for doing mean shift segmentation in C++ using OpenCV: see here.

OpenCV contains a mean shift filtering function and has a GPU, I think CUDA, implementation of mean shift segmentation. I didn’t evaluate the GPU implementation because I’m personally not interested in GPU for the project I am working on. I did take a look at turning cv::pyrMeanShiftFiltering(…) output into a segmentation but didn’t bother trying because pyrMeanShiftFiltering seems broken to me. This is my gut instinct — I can’t quantify it but basically I agree with this guy. The output just seems to not be as good as the output generated by codebases elsewhere online. I have no idea why … one interesting reason might be that OpenCV is doing mean shift on RGB rather than one of the color spaces that are supposed to be better at modeling human vision. Everybody always says to do things that involve treating colors as points in Euclidean space using L*a*b* or L*u*v* rather than RGB, but in practice, to be honest, it never seems to matter to me. Maybe this is an example of where it does. I don’t know but in any case cv::pyrMeanShiftFiltering in my opinion sucks.

The “elsewhere online” I mention above is the codebase of EDISON, “Edge Detection and Image SegmentatiON”, made freely available by Rutgers University’s “Robust Image Understanding Laboratory”. EDISON is a command line tool that parses a script specifying a sequence of computer vision operations that I wasn’t really interested in except for the part in which it does mean shift segmentation, as its mean shift output seems really good to me. What I have done is extracted the mean shift code, which was C, wrapped it thinly in C++, and ported it to use OpenCV types, e.g. cv::Mat, and OpenCV operations where possible. I also re-factored for concision and removed C-isms where possible, e.g. I replace naked memory allocations with std::vectors and so forth.

The most significant change coming out of this re-factoring work in terms of functionality and/or performance was replacing the EDISON codebase’s L*u*v*-to-RGB/RGB-to-L*u*v* conversion routines with OpenCV calls. This actually changes the output of this code relative to EDISON because OpenCV and EDISON give different L*u*v* values for the same image. Not sure who is right or the meaning of the difference but OpenCV is an industry standard so am erring on the side of OpenCV and further the segmentation this code outputs is in my opinion better that what results from EDISON’s L*u*v* routines while performance is unchanged.

Below is some output:

# Jesus Christ, internet…

Center window on primary screen — Win32, C/C++ — below, featuring workiness. Our long national nightmare is now over:

```void CenterWindowOnScreen(HWND hwnd)
{
RECT wnd_rect;
GetWindowRect(hwnd, &wnd_rect);

RECT screen_rect;
SystemParametersInfo(SPI_GETWORKAREA, 0, reinterpret_cast<PVOID>(&screen_rect), 0);

int scr_wd = screen_rect.right - screen_rect.left;
int scr_hgt = screen_rect.bottom - screen_rect.top;
int wnd_wd = wnd_rect.right - wnd_rect.left;
int wnd_hgt = wnd_rect.bottom - wnd_rect.top;

int x = (scr_wd - wnd_wd) / 2;
int y = (scr_hgt - wnd_hgt) / 2;

SetWindowPos(hwnd, 0, x, y, 0, 0, SWP_NOZORDER | SWP_NOSIZE);
}
```

I’ve been doing some Win32 programming lately and just wanted to flag this old Raymond Chen post in which he defines a simple function that fills a hole in the Win32 API and has a nice why-didnt-I-think-of-that quality to it.

```BOOL UnadjustWindowRectEx(
LPRECT prc,
DWORD dwStyle,
DWORD dwExStyle)
{
RECT rc;
SetRectEmpty(&rc);
if (fRc) {
prc->left -= rc.left;
prc->top -= rc.top;
prc->right -= rc.right;
prc->bottom -= rc.bottom;
}
return fRc;
}
```

Note, Chen says that AdjustWindowRect handles scroll bars but I think that is wrong. AdjustWindowRect works without a window handle so it is not possible for it to get scroll bars right. It can treat any window with the WS_VSCROLL and WS_HSCROLL styles as having two scroll bars (which I think it does) but a window can have those styles and have it’s scroll bars hidden (if all the content fits in the client area, say). You could write another function AdjustWindowRectNoReally that takes a window handle but this kind of defeats the purpose of AdjustWindowRect — you mostly use it before creating a window so don’t have a handle.

If you do have a handle, probably what you want is the following: (from here)

```void ClientResize(HWND hWnd, int nWidth, int nHeight)
{
RECT rcClient, rcWind;
POINT ptDiff;
GetClientRect(hWnd, &rcClient);
GetWindowRect(hWnd, &rcWind);
ptDiff.x = (rcWind.right - rcWind.left) - rcClient.right;
ptDiff.y = (rcWind.bottom - rcWind.top) - rcClient.bottom;
MoveWindow(hWnd, rcWind.left, rcWind.top, nWidth + ptDiff.x, nHeight + ptDiff.y, TRUE);
}
```

which sets a window’s size by its client rectangle rather than its window rectangle, and is more accurate than doing this via AdjustWindowRect anyway because it will do the right thing regarding menu wrapping space and scroll bar space.

# How to Convert a GDI+ Image to an OpenCV Matrix in C++…

You have to convert the gdi+ image to a gdi+ bitmap and then to an OpenCV matrix. There is no easier way to do the first conversion than creating a bitmap and painting the image into it, as far as I know.

To perform the second conversion (Gdiplus::Bitmap -> cv::Mat), note that Mat constructors order their parameters rows then columns so that is Bitmap height then width. The actual memory layout is row major, however, so we can just copy the data, but there is no need to do the actual copy yourself. You can use one of the Mat constructors that will wrap existing data without copying and then force it to copy by calling the clone member function.

The other trouble here however is that gdi+ supports, in theory at least, loads of exotic pixel formats so to handle them all would be a chore. The following handles the basic case:

```Gdiplus::Bitmap* GdiplusImageToBitmap(Gdiplus::Image* img, Gdiplus::Color bkgd = Gdiplus::Color::Transparent)
{
int wd = img->GetWidth();
int hgt = img->GetHeight();
auto format = img->GetPixelFormat();
Gdiplus::Bitmap* bmp = new Gdiplus::Bitmap(wd, hgt, format);

if (bmp == nullptr)
return nullptr; // this might happen if format is something exotic, not sure.

auto g = std::unique_ptr<Gdiplus::Graphics>(Gdiplus::Graphics::FromImage(bmp));
g->Clear(bkgd);
g->DrawImage(img, 0, 0, wd, hgt);

return bmp;
}

cv::Mat GdiPlusBitmapToOpenCvMat(Gdiplus::Bitmap* bmp)
{
auto format = bmp->GetPixelFormat();
if (format != PixelFormat24bppRGB)
return cv::Mat();

int wd = bmp->GetWidth();
int hgt = bmp->GetHeight();
Gdiplus::Rect rcLock(0, 0, wd, hgt);
Gdiplus::BitmapData bmpData;

if (!bmp->LockBits(&rcLock, Gdiplus::ImageLockModeRead, format, &bmpData) == Gdiplus::Ok)
return cv::Mat();

cv::Mat mat = cv::Mat(hgt, wd, CV_8UC3, static_cast<unsigned char*>(bmpData.Scan0), bmpData.Stride).clone();

bmp->UnlockBits(&bmpData);
return mat;
}

cv::Mat GdiplusImageToOpenCvMat(Gdiplus::Image* img)
{
auto bmp = std::unique_ptr<Gdiplus::Bitmap>(GdiplusImageToBitmap(img));
return (bmp != nullptr) ? GdiPlusBitmapToOpenCvMat(bmp.get()) : cv::Mat();
}
```

# Lifelike, or the Joy of Killing Time via Breeding Little Squiggles

A couple of weeks ago I got interested in this project but wanted full control of the code, wanted to know exactly what it is doing, wanted a bunch of features like the ability to import and export CA rules, and wanted to have the process not be seeded by cellular automata already featuring gliders (which the web app seems to be). To this end I have pushed a project to github that does something similar but, I hope, more transparently.

I’m calling it Lifelike Cellular Automata Breeder. It is a (C# WinForms) application in which given some settings a user can artificially select and breed cellular automata; i.e., it performs a genetic algorithm in which the user manually provides the fitness criteria interactively.

I decided I wanted to only allow a reproduction step in which I scramble together state tables in various ways, guessing that using “DNA” more complex than commensurate 2D tables of numbers wouldn’t work well for a genetic process in this case. I characterize CA rules as applying only relative to a given number of states and a given, what I call, “cell structure” and “neighborhood function”. Cell structure just means a lattice type and neighborhood –e.g. square, with four neighbors; square, with eight neighbors; hex, with six, etc. “Neighborhood function” is an arbitrary function that given the  states of the n cells in some cell’s neighborhood returns an integer from 0 to r where r is dependent on the neighborhood function and possibly number of states. For example, Conway Life uses the neighborhood function I refer to as “alive cell count”, and for an n-cell neighborhood, r equals n because the greatest number of alive cells that can surround a cell is just the size of the neighborhood. If the user has selected s total states, the state tables will be s by r.

Lifelike works as follows

1. The user selects a number of states, cellular structure, and neighborhood function and kicks off the genetic process.
2. Lifelike sets the current generation to nil, where by “generation” we just mean a set of cellular automata that have been tagged with fitness values.
3. While the user has not clicked the “go to the next generation” button,
• If the current generation is nil, Lifelike randomly generates a cellular automata, CA, from scratch by making an s by r state table filled with random numbers from 0 to s. (The random states are generated via a discrete distribution controllable by the user). If the generation is not nil, Lifelike selects a reproduction method requiring k parents, selects k parents from the current generation such that this selection is weighted on the fitness of the automata, generates CA using the reproduction method and parents, and then possibly selects a random mutation function and mutates CA, selecting the mutation function via a discrete distribution controllable by the user and applying it with a “temperature” controlled by the user.
• Lifelike presents CA in a window.
• The user either skips CA in which case it no longer plays a role in the algorithm or applies a fitness value to it and adds it to the next generation.
4. When the user decides to go to next generation, the selections the user just made become the new parent generation and processing continues.

Results, Musings, etc.

Briefly, Lifelike works.

You can produce interesting cellular automata with it and there is a weird feeling that is hard to describe when you first see a tiny glider wiggle across the screen; however, the way it works is somehow more mundane than I thought that it would be. I wonder if all genetic algorithms are like this. Most of what it produces is garbage. When you see something that isn’t garbage you can select for it. However Lifelike doesn’t do magic. It doesn’t magically find phenomena you want just because you have a fancy framework implemented to find such phenomena. For example, it is my belief at this point that there is no simple hex grid analog of Conway Life using alive cell count, a six cell neighborhood, and 2-states. I think you could probably prove the non-existence of gliders in this configuration but it would be a boring proof by exhaustion and running Lifelike on that configuration is boring as well. Just because you, the user, are a step in a “genetic algorithm” doesn’t somehow make it interesting.

The simple hex neighborhood negative result led me to ask the following question: What is the smallest change you can make to hex/6-cell/alive cell count/2 states to allow Conway Life-like behavior? If you google “Hex Life” you will find that it is well-known that interesting things can happen if you go to a 12-cell  Star of David shaped neighborhood, but this seems inelegant to me because the simple hex neighborhood is so nice. The question then is are gliders possible in the simple hex lattice and neighborhood if we add one state and modify “alive cell count” in a trivial way? The answer to this question turns out be yes. There are beautiful rules that live in the hex lattice with the natural neighborhood if we have 0 = death, 1 = Alive-A, 2 = Alive-B and instead of the simple alive cell count we use its natural analog when states can have values greater than one: sum of states.

Below is a such a rule set and is probably the best thing that has come out of my work with Lifelike as far as I am concerned (So I am naming it Joe Life, assuming that it is unknown in the literature).

The above has a nice quality that Conway Life also has that I call “burn”. This a qualitative thing that is really hard to define but it is what I look for now when I play with Lifelike: burn + gliders = a Life-like cellular automaton. “Burn” is the propensity of a cellular automata configuration to descend into segregated regions of chaos that churn for awhile before ultimately decaying into gliders, oscillators, and still lifes. Some CAs burn faster than others; the above has a nice slow burn. CAs that exhibit steady controlled burn turn out to be rare. Most CAs either die or devolve instantly into various flavors of unbounded chaos.

However there does turn out to be another quality that is not death or unbounded chaos that is sort of like the opposite of burn. See for example

(The above is hex 6-cell, four states, and using a neighborhood function I call “state-based binary”)  which I have been calling “Armada” and generally have been referring to these kind of CAs as being armada-like. Armada-like cellular automata quickly decay completely into only weakly interacting gliders. For example, one from the literature that I would characterize as armada-like is Brian’s Brain. Armada-like rules turn out to be more common than life-like rules. They’re impressive when you first start finding them but they are ultimately less interesting, to me at least. The best thing about armada-like rules is that they indicate that life-like rules are probably “nearby” in the space you are exploring in the genetic process.

Also they can breed weird hybrids that defy classification, such as the following which are all burn with large blob-like gliders and seem sometimes to live around the boundary between armada-like rules and life-like rules.

Magic Carpets (square, 4-cell/ 4 states / sum of states)

or Ink Blots (hex, 6-cell/ 3 states / “0-low-med-high”)

My other major result is that life-like rules exist in the square 4-cell neighborhood if we allow an extra state and use the simple sum of states as the neighborhood function, but they can be boring looking so instead here is an armada-like square 4-cell CA that is on the edge of being lifelike:

The above uses the neighborhood function I call “2-state count” which enumerates all possible combinations of c1, number of neighboring cells in state 1 and c2, number of neighboring cells in state 2 or above, in an n-cell neighborhood i.e. c1 + c2 ≤ n.

# Draak Update

Below is gameplay video of a prototype of the puzzle game “Draak” that I am working on. Draak will be an iPad-only iOS game; the prototype below is written to Windows in C# on top of MonoGame.

The initial idea for this game was about the art. I thought it would be cool to make a game that looks like an animated version of one of M. C. Escher’s tessellations, particularly one that involves a similarity symmetry. So I lifted mechanics from a 1990s era game that I always felt was a good game trapped in the typical square lattice for which it was particularly ill-suited. The old game’s Euclidean lay-out wasted a lot of space.

A lot of the work in the above may not be immediately apparent (which is good I think). Specifically, there are actually two shapes of butterflies in the above not just one. They look like this:

and are arrayed as in a checkerboard with a 90 degree rotation applied to cells of opposite parity — the geometry might be clearer here in which I create these butterfly tiles unadorned with my tool EscherDraw (before I had beziers working in EscherDraw) Thus any time a butterfly moves to a cell with opposite parity I need to not only handle the scale and rotation tweening imposed by the spiral lattice, I also have to simulatenously do the 90 degree rotation imposed by the butterfly tiles and simultaneously morph the actual sprites above from one to the other.

I created the above sprites as vector art using tile outlines exported from Escher draw as SVG and wrote a Python script that morphs SVG as the publically available software for morphing vector graphics is surprisingly non-existent. I am going to post about my vector morphing code here soon, I just need to clean it up for use by people other than me.

# My Code for Doing Two Things That Sooner or Later You Will Want to do with Bezier Curves

I just added full cubic bezier curve support to the vector tessellation creation tool I am developing (see here). I’ll include some video at a later date (Update: here) showing this feature in action but first I want to document and make publicly available some of the code involved: namely, a bezier class in C# that supports two functions that are not the easiest things in the world to implement. Both of these functions come up as soon as you try to do anything non-trivial with beziers and the literature online can be unhelpful.

The first function is splitting a cubic bezier into two curves at a given t parameter such that the concatenation of the two curves is congruent to the original. This turns out to be easy; however, I could find no actual code on offer, rather just descriptions of code. There is good coverage elsewhere on the theory so I won’t go into it deeply here, but, briefly, splitting a bezier is easy because how one does so follows naturally from the sort of recursive definition of the bezier that is implied by De Casteljau’s algorithm. Anyway here is my code:

```...
public Tuple<Bezier,Bezier> Split(double t) {
PointD E = Interpolate(t, CtrlPoint1, CtrlPoint2);
PointD F = Interpolate(t, CtrlPoint2, CtrlPoint3);
PointD G = Interpolate(t, CtrlPoint3, CtrlPoint4);
PointD H = Interpolate(t, E, F);
PointD J = Interpolate(t, F, G);
PointD K = Interpolate(t, H, J);

return new Tuple<Bezier, Bezier>(
new Bezier(CtrlPoint1, E, H, K),
new Bezier(K, J, G, CtrlPoint4)
);
}

static private PointD Interpolate(double t, PointD pt1, PointD pt2) {
double x = (1.0 - t) * pt1.X + t * pt2.X;
double y = (1.0 - t) * pt1.Y + t * pt2.Y;
return new PointD(x, y);
}
....
```

The other function is considerably harder: finding the nearest point on a bezier curve to a given point. If you scan the literature (i.e. perform the google search) you will find that what most people end up using is some code from an article in the book Graphics Gems I, “Solving the Nearest-Point-On-Curve Problem” by Philip J. Schneider. I, however, have a couple of problems with this code (1) I don’t really understand it, (2) it is long and I need to port it to C#, and (3) someone on the internet is claiming that it is incorrect but given (1) I can’t really evaluate the claim; see the comment that begins “There seem to be errors in the ControlPolygonFlatEnough function […]” in the C code here.

But more relevantly I just don’t get this code on a fundamental level. The nearest point on a bezier problem is difficult to solve mathematically but it is easy to formulate mathematically. I think that the Graphics Gems code obscures the straightforward aspect of how this problem can be formulated. Further, I don’t have a problem using code that I don’t understand; however, if I am going to port an opaque block of code from one language to another I’d like to keep that portion of code to a minimum and I can do this easily by structuring my code around the formulation of the problem that I understand.

The formulation I am talking about is as follows, if B(t) is a particular cubic bezier function, B’(t) is its derivative, and P is an arbitrary point then [B(t) – P] ⋅ B’(t) = 0 when t is the parameter of the closest point on B to P. [B(t) – P] is a vector that points from P towards some point on the curve and B’(t) is a vector that is tangent to the curve at this point, if the distance is going to be a minimum then these two vectors will be perpendicular and thus their dot product will be zero. Since B(t) is cubic, its derivative will be quadratic and thus when we take the scalar product of [B(t) – P] and B’(t) we will end up with a quintic equation of one variable. This leads naturally to a nearest point function that looks like the following on the top level:

``` public Tuple<PointD, double> FindNearestPoint(PointD pt)
{
var polyQuintic = GetNearestBezierPtQuintic(
_points[0].X, _points[0].Y,
_points[1].X, _points[1].Y,
_points[2].X, _points[2].Y,
_points[3].X, _points[3].Y,
pt.X, pt.Y
);
List<Complex> roots = FindAllRoots(polyQuintic);

// Filter out roots with nonzero imaginary parts and roots
// with real parts that are not between 0 and 1.
List<double> candidates = roots.FindAll(
root => root.Real > 0 && root.Real <= 1.0 && Math.Abs(root.Imaginary) < ROOT_EPS
).Select(
root => root.Real
).ToList();

// add t=0 and t=1 ... the edge cases.

// find the candidate that yields the closest point on the bezier to the given point.
double t = double.NaN;
PointD output = new PointD(double.NaN,double.NaN);
double minDistance = double.MaxValue;
foreach (double candidate in candidates)
{
var ptAtCandidate = GetPoint(candidate);
double distance = DistSqu(ptAtCandidate, pt);
if (distance < minDistance)
{
minDistance = distance;
t = candidate;
output = ptAtCandidate;
}
}

return new Tuple<PointD, double>(output, t);
}
```

which reduces the problem to implementing GetNearestBezierPtQuintic() and a numeric root finding function for polynomials, given that quintic equations cannot be solved via a closed-form formula like the quadratic equation.

GetNearestBezierPtQuintic() –which returns the coefficients of [B(t) – P] ⋅ B’(t) when fully expanded given P and the control points of — turns out to be a little two much algebra to work out comfortably by hand, so I ran the open source symbolic mathematics application Sage on this Sage script yielding the following when translated into C#: (For the first time on the internet! … as far as I can tell)

```static private List<Complex> GetNearestBezierPtQuintic(double x_0, double y_0, double x_1, double y_1,
double x_2, double y_2, double x_3, double y_3, double x, double y)
{
double t5 = 3 * x_0 * x_0 - 18 * x_0 * x_1 + 27 * x_1 * x_1 + 18 * x_0 * x_2 - 54 * x_1 * x_2 + 27 * x_2 * x_2 -
6 * x_0 * x_3 + 18 * x_1 * x_3 - 18 * x_2 * x_3 + 3 * x_3 * x_3 + 3 * y_0 * y_0 - 18 * y_0 * y_1 +
27 * y_1 * y_1 + 18 * y_0 * y_2 - 54 * y_1 * y_2 + 27 * y_2 * y_2 - 6 * y_0 * y_3 + 18 * y_1 * y_3 -
18 * y_2 * y_3 + 3 * y_3 * y_3;
double t4 = -15 * x_0 * x_0 + 75 * x_0 * x_1 - 90 * x_1 * x_1 - 60 * x_0 * x_2 + 135 * x_1 * x_2 -
45 * x_2 * x_2 + 15 * x_0 * x_3 - 30 * x_1 * x_3 + 15 * x_2 * x_3 - 15 * y_0 * y_0 + 75 * y_0 * y_1 -
90 * y_1 * y_1 - 60 * y_0 * y_2 + 135 * y_1 * y_2 - 45 * y_2 * y_2 + 15 * y_0 * y_3 - 30 * y_1 * y_3 +
15 * y_2 * y_3;
double t3 = 30 * x_0 * x_0 - 120 * x_0 * x_1 + 108 * x_1 * x_1 + 72 * x_0 * x_2 - 108 * x_1 * x_2 +
18 * x_2 * x_2 - 12 * x_0 * x_3 + 12 * x_1 * x_3 + 30 * y_0 * y_0 - 120 * y_0 * y_1 +
108 * y_1 * y_1 + 72 * y_0 * y_2 - 108 * y_1 * y_2 + 18 * y_2 * y_2 - 12 * y_0 * y_3 + 12 * y_1 * y_3;
double t2 = 3 * x * x_0 - 30 * x_0 * x_0 - 9 * x * x_1 + 90 * x_0 * x_1 - 54 * x_1 * x_1 + 9 * x * x_2 -
36 * x_0 * x_2 + 27 * x_1 * x_2 - 3 * x * x_3 + 3 * x_0 * x_3 + 3 * y * y_0 - 30 * y_0 * y_0 - 9 * y * y_1 +
90 * y_0 * y_1 - 54 * y_1 * y_1 + 9 * y * y_2 - 36 * y_0 * y_2 + 27 * y_1 * y_2 - 3 * y * y_3 + 3 * y_0 * y_3;
double t1 = -6 * x * x_0 + 15 * x_0 * x_0 + 12 * x * x_1 - 30 * x_0 * x_1 + 9 * x_1 * x_1 - 6 * x * x_2 +
6 * x_0 * x_2 - 6 * y * y_0 + 15 * y_0 * y_0 + 12 * y * y_1 - 30 * y_0 * y_1 + 9 * y_1 * y_1 -
6 * y * y_2 + 6 * y_0 * y_2;
double t0 = 3 * x * x_0 - 3 * x_0 * x_0 - 3 * x * x_1 + 3 * x_0 * x_1 + 3 * y * y_0 - 3 * y_0 * y_0 -
3 * y * y_1 + 3 * y_0 * y_1;

return new List<Complex> { (Complex)t0/t5, (Complex)t1/t5, (Complex)t2/t5, (Complex)t3/t5, (Complex)t4/t5, (Complex)1.0 };
}
```

and decided to use Laguerre’s Method to solve the equation.

I chose Laguerre’s Method because it is optimized for polynomials, is less flaky than Newton-Raphson, is relatively concise, and is easy to port from C++ given .Net’s System.Numerics’s implementation of complex numbers. I ported the implementation of Laguerre’s Method found in Data Structures and Algorithms in C++ by Goodrich et. al. (here). This solver works well enough for my purposes; however, I think the optimal thing to do to get the optimal nearest-point-on-a-bezier algorithm would be to implement one of a class of newish algorithms that are specifically targeted at solving the quintic, e.g. “Solving the Quintic by Iteration” by Doyle and McMullen. No implementations of these algorithms seem to be publicly available, however, and it would be a research project that I don’t have time for right now to translate that paper from academese into code.

# An Application for Creating Novel Escher-like Tessellations of the Plane

Over the past several weeks I have been investigating desktop applications for creating novel Escher-like tilings of the plane. Basically I’ve determined that none of what is publicly available is useful to me. The game I have in development will involve an animated Escher-like tessellation of a double logarithmic spiral — see here. The “logarithmic spiral” part of it means the publicly available applications can’t help me: they are too limited in what they will do and I can’t extend them because they are proprietary, so I have created my own tool.

These applications are “too limited” in the sense that none encompass everything possible that one would regard as a 2D tessellation. You have some applications that simplistically shoehorn you into a few kinds of geometries ignoring the richness of the whole domain of tiled patterns. You have other applications that try to take the domain seriously and follow the mathematical literature on the subject. What this means, generally, is that they allow the user to construct a tessellation in terms of its “wallpaper group” which is the mathematical classification of a plane pattern based on the symmetries the pattern admits.

The latter seems like what one would want but in practice it isn’t, at least it is not what I want. It is too heavy of a constraint to work only in terms of wallpaper group because there are all kinds of patterns one might be interested in that fall outside of this formalism: Escher’s tessellations of the Poincare disk model of the hyperbolic plane, for example, or, say, any kind of aperiodic tiling or any tessellations that admit similarity as a symmetry, i.e. scaling, such as Escher’s square limit etchings, or my case: various spiral patterns.

It is instructive that Escher himself did not work in terms of symmetry groups. He created his own classification system which Doris Schattschneider refers to as Escher’s “layman’s theory” of tilings in her book Visions of Symmetry (the authoritative text on Escher’s tessellations). Escher’s classification system comes from a pragmatic perspective rather than a formal mathematical perspective. He starts with basic cells — a square, a rectangle, an isosceles right triangle, etc. — and then enumerates the ways in which one can surround the cell with itself that result in tilings of the plane. There is a good overview of his system here.

What I like about Escher’s approach is that it is geared towards construction. When you try to make one of these things you start with a tile or some a small set of simple tiles that tile the plane. You try to modify them until you get something you like. I decided to implement software that captures that approach. I didn’t want to have to be too concerned with “wallpaper group” or other abstractions which I don’t personally find to be intuitive anyway.

What I came up with is the following application that is written in C#. Below is video in which I use the application to construct something like Escher’s butterfly tessellation — system IX-D in his layman’s theory — but on a double logarithmic spiral (This is going to be the basis for the art of the first level of my game; the butterflies will be animated, flapping their wings, etc.)

The way the application works is that there are two kinds of tiles: normal tiles and tile references. Normal tiles are composed of “sides”. There are two kinds of sides: normal sides and side references. Tile references and side references point to a another tile or side and say basically, “I am just like that tile or side, except apply this transformation to me”. The transformations that tile references can apply to a tile are defined in terms of affine transformation matrices. The transformations that side references can apply are just two kinds of flipping relative to the sides’ end points, either flipping “horizontally” or flipping “vertically” or both. The application then allows the user to add and move vertices on any sides that are on tiles that are marked as editable and then resolves all the references in real-time.

Right now, all of the information about tiles and sides and which is a reference to which has to be hand coded as an XML file, which is a pain (for the above I wrote a separate Python program that generated the XML defining a double logarithmic spiral of tiles that interact with each other according to Escher’s IX-D system), but it is an extremely flexible formulation (it could, for example, handle hyperbolic tessellations in theory … if you added a kind of side that is a circular arc and moebius transformations to the set of transformations the applications knows how to apply to referenced tiles).

Eventually I’d like to release this application  as open source but I am not sure it will be useful to anyone but me in its current form. I need to incorporate the part that one has to hand code in XML into the actual GUI … fix bugs and bezier curves and so forth, but please feel free to contact me if you are interested in the application in its current state or otherwise.