= Draw Me A Diagram = https://eprints.soton.ac.uk/257577/1/funcgeo2.pdf[Peter Henderson's _Functional Geometry_] composes a few disarmingly succinct functions to produce striking pictures. It is another satisfying example of simple building blocks unexpectedly leading to complex creations. See also https://dl.acm.org/doi/pdf/10.1145/800068.802148[the original 1982 version of the paper]. https://diagrams.github.io/[The `diagrams` Haskell package] builds on these ideas to provide a rich powerful language for vector graphics. I had used it to generate images for my webpages, but its many dependencies can be unpleasant when switching GHC versions. As I only need a tiny subset of its features, why not roll my own version?

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-- Placeholder for examples.

== Complex Numbers == We focus on 2D diagrams, representing points with complex numbers. We define the _dot product_ of two complex numbers to be: \[ (a + bi) \cdot (c + di) = ac + bd \] The resulting real is the product of their magnitudes and the cosine of the angle between them.

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infixl 8 |> (|>) = flip ($) infix 6 :+ data Com = Double :+ Double deriving (Eq, Show) dot (a :+ b) (c :+ d) = a*c + b*d dilate r (a :+ b) = r*a :+ r*b realPart (a :+ _) = a conjugate (a :+ b) = a :+ -b norm x = dot x x magnitude = sqrt . norm i = 0 :+ 1 instance Ring Com where (a :+ b) + (c :+ d) = (a + c) :+ (b + d) (a :+ b) - (c :+ d) = (a - c) :+ (b - d) (a :+ b) * (c :+ d) = (a*c - b*d) :+ (a*d + b*c) fromInteger a = fromInteger a :+ 0 instance Field Com where recip x = dilate (recip $ norm x) $ conjugate x

We also roll our own trigonometric functions for our homebrew Haskell compiler. We approximate \(\arctan(x)\) for \(x\in [0..\frac{1}{2}]\) with its Taylor series expansion: \[ \arctan(x) = x - \frac{x^3}{3} + \frac{x^5}{5} ... \] and hardcode \(\arctan(1) = \pi/4\). We ensure the smallest items are added first with `foldr` to fight floating-point rounding error.

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pi = 3.141592654 tau = 2*pi atanTaylor 1 = pi * 0.25 atanTaylor x = foldr (+) 0 $ reverse $ take 25 $ zipWith (/) (iterate (*(-x*x)) x) (iterate (+2) 1)

We build our `phase` function on top, which is also known as https://en.wikipedia.org/wiki/Atan2[`atan2`].

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phase (x :+ y) | x < 0 = if y < 0 then phase' -x -y - pi else pi - phase' -x y | y < 0 = -(phase' x -y) | otherwise = phase' x y phase' x y | y > x = 0.5*pi - phase'' x y | otherwise = phase'' y x phase'' y x | x == 0 = if y == 0 then 0 else pi * 0.5 | 2*y > x = atanTaylor 0.5 + atanTaylor ((r - 0.5) / (1 + r*0.5)) | otherwise = atanTaylor r where r = y / x

We use https://en.wikipedia.org/wiki/CORDIC[CORDIC] for computing sines and cosines of a given small angle \(\theta\). This algorithm reminds me of binary search. In brief, we start with: \[ \begin{aligned} \alpha &= 0 \\ \sin\alpha &= 0 \\ \cos\alpha\ &= 1 \end{aligned} \] then add to or subtract from \(\alpha\) successively the values: \[ \arctan(2^0), \arctan(2^{-1}), \arctan(2^{-2}), ... \] until \(\alpha\approx\theta\). All the while, we update \(\sin\alpha\) with corresponding additions or subtractions of \(2^{-k} \cos\alpha\), with a similar update for \(\cos\alpha\). There is a wrinkle. At each step, a scaling factor creeps in, which we could normalize away immediately. However, it's better to defer this so that we need only one final multiplication by a precomputed constant. More concretely, the algorithm follows the identities: \[ \begin{aligned} \sqrt{1+2^{-2k}} \sin(\alpha \pm \arctan(2^{-k})) &= \sin\alpha \pm 2^{-k}\cos\alpha \\ \sqrt{1+2^{-2k}} \cos(\alpha \pm \arctan(2^{-k})) &= \cos\alpha \mp 2^{-k}\sin\alpha \end{aligned} \] for \(k\in[0..n]\), where \(n\) is number of steps. The final normalization multiplies by: \[ \prod_{k=0}^n \frac{1}{\sqrt{1+2^{-2k}}} \] We work with \(\beta = \theta - \alpha\) instead of \(\alpha\) directly so we can compare against zero.

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cossinSmall theta = cordic lim tab theta 1 0 where lim = 25 cordic n ((p2, a):rest) beta x y | n == 0 = (kn * x, kn * y) | otherwise = cordic (n - 1) rest (beta - sig a) x' y' where sig = if beta < 0 then negate else id x' = x - sig p2 * y y' = sig p2 * x + y kn = kvalues!!lim kvalues = scanl1 (*) $ (\k -> 1/sqrt(1+0.5^(2*k))) <$> [0..] tab = zip pows $ atanTaylor <$> pows where pows = iterate (*0.5) 1 cossin theta | theta < 0 = second negate $ cossin (-theta) | theta <= pi/4 = cossinSmall theta | theta <= pi/2 = (\(a,b) -> (b,a)) $ cossin (pi/2 - theta) | theta <= pi = first negate $ cossin (pi - theta) | theta <= 2*pi = (\(a,b) -> (-a, -b)) $ cossin $ theta - pi | otherwise = cossin $ theta - 2*pi cos = fst . cossin sin = snd . cossin cis = uncurry (:+) . cossin

Outside of elementary arithmetic, the only mathematical operation our code depends on is taking the square root, which WebAssembly conveniently provides. But if I had to code it myself, I'd likely refer to http://www.numberworld.org/y-cruncher/algorithms.html[the algorithms used by y-cruncher]. == Shapes == Our core data type is `Shape`:

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data Shape = Shape { _envelope :: Com -> Double , _trace :: Com -> Com -> [Double] , _svg :: Double -> String -> String , _named :: [(String, (Com -> Com, Shape))] }

Each `Shape` is implicitly equipped with a point that we call its _local origin_. \( \newcommand\O{\textbf{O}} \newcommand\P{\textbf{P}} \newcommand\Q{\textbf{Q}} \newcommand\v{\textbf{v}} \newcommand\w{\textbf{w}} \) Let \(D\) be a `Shape` with local origin \(\O\). The _envelope_ of \(D\) is a function that takes a direction \(\v\) and returns the scalar \(s\) given by: \[ s = \sup \{ (\P - \O) \cdot \v | \P \in D \} \] In other words, the scalar \(s\) is the smallest value for which the plane through \(\O + s \v\) normal to \(\v\) partitions space so that one half contains the entirety of \(D\). Roughly speaking, if you were to walk in the direction \(v\) starting from \(O\), then \(s\) tells you how far you must travel so you no longer see \(D\), not even in your peripheral vision. *Example*: for the unit circle whose local origin is its center, the envelope is `recip . magnitude`. *Example*: for the 1D unit circle, that is, the points -1 and 1, with local origin 0, the envelope is the normalized projection along the real axis: ------------------------------------------------------------------------ \v@(x :+ _) -> abs x/norm v ------------------------------------------------------------------------ The _trace_ of \(D\) is a function that takes a point \(\P\) and a direction \(\v\) and returns the set: \[ \{ s | \P + s \v \in D \} \] (I haven't decided what to do about intervals within this set. Perhaps I could replace them with their endpoints, or remove them entirely.) In other words, the `trace` function identifies all the boundary points along a given ray. In fact, this function is so named because of ray-tracing, and has nothing to do with other trace functions in mathematics. While the envelope function always returns results with respect to the local origin \(\O\) of a diagram, the trace function must be given a starting point \(\P\). We represent the set of scalars with a sorted list. The examples on this page only use the largest element, that is, the outermost boundary point:

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maxTraceV p pt dir = case _trace p pt dir of [] -> Nothing ss -> let s = last ss in if s <= 0 then Nothing else Just s maxTraceP p pt dir = ($ dir) . dilate <$> maxTraceV p pt dir

The `_svg` function returns a snippet of SVG that draws the shape as a difference list. It takes a scaling factor as a parameter so we can generate scale-invariant SVG for line widths, arrow heads, and so on. We hardcode the line-width to a value that works well for diagrams around the same size as a unit circle. SVG uses screen coordinates, which we make a little less confusing with `yshows` rather than mysteriously negate \(y\) coordinates here and there. However, we do simply negate the angle of rotation when needed. We define a helper that exports a `Shape` to SVG given a desired number of pixels per unit length. in the diagram with 1.1 units of padding. We call the envelope function to size the SVG appropriately.

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lineWidth = 0.04 yshows = shows . negate svg pxPerUnit p = concat [ "<svg style='font-family:MJXZERO,MJXTEX-I;'" , " width=", show wPx , " height=", show hPx , " viewBox='", unwords (map show [x,y,w,h]), "'" , "><g font-size='0.8px'>", _svg p 1.0 "</g>" , "</svg>" ] where pad = 1.1 x0 = -(_envelope p -1) y0 = -(_envelope p i) w0 = _envelope p 1 - x0 h0 = _envelope p -i - y0 x = x0 - pad y = y0 - pad w = w0 + 2*pad h = h0 + 2*pad wPx = w * pxPerUnit hPx = h / w * wPx

We may name a `Shape` with a string so we can easily, say, connect two previously declared shapes. We implement this feature with the `_named` function. For a `Shape` \(D\), it returns a `Map` where each entry's key is the name of a component `Shape` of \(D\). The corresponding value is a tuple `(f, p)` where `p` is the component `Shape`, and `f` is a function that transforms coordinates with respect to the local origin of `p` to coordinates with respect to the local origin of \(D\). The following assigns a string name to a `Shape`:

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named s p = let p' = p { _named = (s, (id, p')) : _named p } in p'

== Unit Circle == The unit circle is a good introductory example. We define its local origin to be the center of the circle. We compute its trace by solving a quadratic to find the points of intersection between a line and a unit circle.

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unitCircle = Shape { _envelope = (1/) . magnitude , _trace = ptCirc , _svg = \zoom -> ("<circle fill='none' stroke='black' stroke-width='"++) . shows (lineWidth*zoom) . ("' r=1 />"++) , _named = mempty } where ptCirc v dv | disc < 0 = [] | otherwise = [(-b - sd) * aInv, (-b + sd) * aInv] where a = dot dv dv b = dot v dv c = dot v v - 1 disc = b^2 - a*c aInv = 1 / a sd = sqrt disc

== Regular Polygons == The \(n\)th roots of unity lie on the unit circle, and we can join them with edges to form a regular \(n\)-gon. We compute its envelope by finding the maximum normalized projection of each vertex on to the given direction. For large \(n\) it would be faster to test only the endpoints of the edge facing the given direction. We are similarly wasteful when computing the trace. We compute ray-segment intersections for every edge, and sort any results. To find the intersection of two lines, we solve equations of the following form for \(\lambda\) and \(\mu\): \[ \P + \lambda \v = \Q + \mu \w \] As \(i \v \cdot \v = 0\), we eliminate the \(\v\) term by dotting both sides with \(i\v\) to find: \[ \mu = \frac{i \v \cdot (\P - \Q)}{i \v \cdot \w} \] Similarly, dotting with \(i\w\) yields: \[ \lambda = \frac{i\w \cdot (\Q - \P)}{i\w \cdot \v} \] These solutions fail when \(i\v\cdot\w = 0\), that is, when the lines are parallel. (Our code liberally uses the identity \( i\v\cdot\w = -i\w\cdot\v \).)

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sort [] = [] sort (x:xt) = sort (filter (<= x) xt) ++ [x] ++ sort (filter (> x) xt) cyclogon n = Shape { _envelope = \dir -> foldr1 max $ (\d -> dot d dir / dot dir dir) <$> vs , _trace = \pt dir -> sort $ raySegment (pt, dir) =<< zip vs (tail vs ++ vs) , _svg = \zoom -> ("<polygon fill='none' stroke='black' stroke-width='"++) . shows (lineWidth*zoom) . ("' points='"++) . foldr (.) id (((' ':) .) . screenShow <$> vs) . ("' />"++) , _named = mempty } where vs = take n $ iterate (cis(tau/fromIntegral n) *) 1 screenShow (x :+ y) = (shows x) . (' ':) . (yshows y) raySegment (p, v) (w1, w2) | d == 0 || b < 0 || b > 1 = [] | otherwise = [a] where d = dot (i*w) v x = w1 - p w = w1 - w2 a = dot (i*w) x / d b = dot (i*x) v / d

== Struts == We define a horizontal strut to be an invisible 1D circle with no trace. A vertical strut is the analogous shape on the imaginary axis.

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hstrut = Shape { _envelope = \d@(dx :+ dy) -> abs dx/norm d , _trace = \_ _ -> [] , _svg = \zoom -> id , _named = mempty } vstrut = Shape { _envelope = \d@(dx :+ dy) -> abs dy/norm d , _trace = \_ _ -> [] , _svg = \zoom -> id , _named = mempty }

== Transforming Shapes == We can easily handle some well-known transformations. * Scaling: scale the envelope and trace by the same factor. * Translation: for the trace, we undo the translation on \(\P\) before computing the original trace; for the envelope, we compute the original envelope, then subtract the normalized projection of the translation vector on the given direction. * Rotation: for both the trace and envelope, undo the rotation on the given direction before computing the original function. SVG has primitives for all these transformations.

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onNamed f p = second (first (f .)) <$> _named p scale :: Double -> Shape -> Shape scale n prim = Shape { _envelope = \dir -> n * _envelope prim dir , _trace = \pt dir -> (n *) <$> _trace prim pt dir , _svg = \zoom -> ("<g transform='scale("++) . shows n . (")'>"++) . _svg prim (zoom / n) . ("</g>"++) , _named = onNamed (dilate n) prim } translate :: Com -> Shape -> Shape translate d@(dx :+ dy) prim = Shape { _envelope = \dir -> _envelope prim dir + dot d dir / dot dir dir , _trace = \pt dir -> _trace prim (pt - d) dir , _svg = \zoom -> ("<g transform='translate("++) . shows dx . (' ':) . yshows dy . (")'>"++) . _svg prim zoom . ("</g>"++) , _named = onNamed (d+) prim } translateX x = translate $ x :+ 0 translateY y = translate $ 0 :+ y rotateBy :: Double -> Shape -> Shape rotateBy theta p = Shape { _envelope = \dir -> _envelope p (dir * conjugate z) , _trace = \pt dir -> _trace p pt (dir * conjugate z) , _svg = \zoom -> ("<g transform='rotate("++) . shows (-theta / pi * 180) . (")'>"++) . _svg p zoom . ("</g>"++) , _named = onNamed (z*) p } where z = cis theta

We use a transformation to provide a handy function that returns a circle of any given radius. Hard-coding a dedicated `Shape` might perform better, but there's no need to optimize yet. We define `strutX` and `strutY` similarly. It might be more consistent to have `circle` take a diameter parameter rather than a radius, but this breaks tradition.

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circle n = scale n $ unitCircle strutX x = scale (x/2) hstrut strutY y = scale (y/2) vstrut

We could generalize the scaling and rotation cases. If \(T\) is an invertible linear transformation for a shape \(D\), then to compute envelope of \(T D\) on a vector \(\v\) we compute the envelope of \(D\) on \(T^{-1} \v\), and similarly for the trace. (The scaling case then simplifies considerably due to linearity.) Some care would be needed with SVG generation since we desire things like line width to be scale-invariant. Dividing the scaling parameter by the determinant of the matrix representing \(T\) ought to do the trick. == Composing Shapes == The `atop` function places one diagram atop another by lining up their local origins. The envelope of the combined diagrams is the maximum of their envelopes, while its trace is the union of their traces. As we represent sets with sorted lists, we combine the traces with merge sort. This associative operation is a good choice for turning `Shape` into a semigroup.

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mergeSort xs ys = case xs of [] -> ys x:xt -> case ys of [] -> xs y:yt | x <= y -> x:mergeSort xt ys | True -> y:mergeSort xs yt atop :: Shape -> Shape -> Shape atop p q = Shape { _envelope = \dir -> _envelope p dir `max` _envelope q dir , _trace = \pt dir -> _trace p pt dir `mergeSort` _trace q pt dir , _svg = \zoom -> _svg p zoom . _svg q zoom , _named = _named p <> _named q } instance Semigroup Shape where (<>) = atop

The pieces are in place for `beside`, which places one `Shape` next to another in a given direction so that their envelopes touch. We specialize a couple of directions so we can succinctly describe horizontal and vertical layouts.

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beside :: Com -> Shape -> Shape -> Shape beside dir x y = x <> translate (dilate (_envelope x dir + _envelope y (-dir)) dir) y (|||) = beside 1 (===) = beside -i hcat = foldr1 (|||) vcat = foldr1 (===)

For shapes like arrows, arrowheads, and labels, we have no need for the envelope and trace. We introduce the `ghost` function to help define `Shape` values that are thin wrappers around various SVG drawings.

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ghost f = Shape { _envelope = const 0 , _trace = \_ _ -> [] , _svg = f , _named = mempty } text :: String -> Shape text s = ghost \zoom -> ("<text fill='black'>"++) . (s++) . ("</text>"++) svgFilledPolygon pts = ("<polygon fill='black' points='"++) . foldr (.) id (map (\(x :+ y) -> (" "++) . shows x . (" "++) . yshows y) pts) . ("'/>"++) dart = ghost \zoom -> ("<g transform='scale("++) . shows (6*lineWidth/zoom) . (")'>"++) . svgFilledPolygon [0, t1, t2, conjugate t1] . ("</g>"++) where t1 = cis (2/5 * tau) - (1 :+ 0) t2 = (realPart t1 + 1/2):+0 dubDart = ghost \zoom -> ("<g transform='scale("++) . shows (6*lineWidth/zoom) . (")'>"++) . svgFilledPolygon [0, t1, t2, conjugate t1] . svgFilledPolygon [t2, t3, t4, conjugate t3] . ("</g>"++) where t1 = cis (2/5 * tau) - (1 :+ 0) t2 = (realPart t1 + 1/2):+0 t3 = t1 + t2 t4 = t2 + t2 lineWith :: String -> Com -> Com -> Shape lineWith attrs (x1 :+ y1) (x2 :+ y2) = ghost \zoom -> ("<line stroke-width='"++) . shows (lineWidth*zoom) . ("' x1="++) . shows x1 . (" y1="++) . yshows y1 . (" x2="++) . shows x2 . (" y2="++) . yshows y2 . (" stroke='black' "++) . (attrs++) . (" />"++) dashedAttrs = "stroke-dasharray=0.1" -- Assumes rad lies in [-tau..tau]. arcline :: Com -> Com -> Double -> Shape arcline a@(x1 :+ y1) b@(x2 :+ y2) rad = ghost \zoom -> ("<path stroke-width='"++) . shows (lineWidth*zoom) . ("' fill='none' stroke='black' d='M "++) . shows x1 . (" "++) . yshows y1 . (" A "++) . shows r . (" "++) . shows r . (" 0 "++) . shows (fromEnum $ abs rad >= pi) -- Large arc flag. . (' ':) . shows (fromEnum $ rad < 0) -- Sweep flag . (' ':) . shows x2 . (" "++) . yshows y2 . ("' />"++) where r = magnitude (b - a) / (2 * abs (sin (rad / 2)))

Next are utilities for drawing arrows between named diagrams. Here, we see the importance of changing coordinate systems: by the time we wish to draw arrows, the underlying objects may have undergone several transformations, so it would make no sense to use the original local coordinates of each endpoint.

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innerPoints aName bName p = do (fa, a) <- lookup aName $ _named p (fb, b) <- lookup bName $ _named p let d = fb 0 - fa 0 pa <- fa <$> maxTraceP a (0:+0) d pb <- fb <$> maxTraceP b (0:+0) (negate d) pure (pa, pb) anglePoints aName bName aRad bRad p = do (fa, a) <- lookup aName $ _named p (fb, b) <- lookup bName $ _named p pa <- fa <$> maxTraceP a (0:+0) (cis aRad) pb <- fb <$> maxTraceP b (0:+0) (cis bRad) pure (pa, pb) straightArrowWith tip lineAttrs aName bName p = maybe p (p <>) do (pa, pb) <- innerPoints aName bName p let hd = translate pb $ rotateBy (phase $ pb - pa) tip pure $ lineWith lineAttrs pa pb <> hd straightArrow = straightArrowWith dart "" existsArrow = straightArrowWith dart dashedAttrs curvedArrow rad aName bName aRad bRad p = maybe p (p <>) do (pa, pb) <- anglePoints aName bName aRad bRad p let hd = translate pb $ rotateBy (rad / 2) $ rotateBy (phase $ pb - pa) dart pure $ arcline pa pb rad <> hd

Lastly, we have a wrapper that inserts an SVG into this webpage. The `demo` variable refers to a `<div>` element at the top of this page. Each unit takes 20 pixels, which works well with demos with small numbers.

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draw p = do jsEval $ "repl.outdiv.insertAdjacentHTML('beforeend',`" ++ svg 20 p ++ "`);" pure ()

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draw $ (circle 3 === circle 1) ||| (circle 4 <> (circle 1 === circle 5))

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draw $ hcat $ scale 1.5 . rotateBy (tau/5) . cyclogon <$> [3..10]

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label = translate (-0.4 :+ -0.25) . text object s = label s <> (unitCircle |> named s) draw $ hcat [ object "Z" , translateY (-2) (label "h") <> translateY 2 (label "g") <> strutX 4 , object "X" , translateY 0.5 (label "f") <> strutX 4 , object "Y" ] |> straightArrow "X" "Y" |> curvedArrow (-tau/6) "Z" "X" (tau/8) (tau*3/8) |> curvedArrow (tau/6) "Z" "X" (-tau/8) (-tau*3/8)

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jsEval [r| const cells = convo.getElementsByClassName("cell"); const c2 = cursor.previousSibling; const c1 = c2.previousSibling; const c0 = c1.previousSibling; cells[1].replaceWith(c0, c1, c2); cursor.remove(); cursor = undefined; |]

*blynn@cs.stanford.edu*💡