如何使用 T-SQL 生成 Mandelbrot？

Question

学习了一些关于 T-SQL 的知识，并认为一个有趣的练习是用它生成 Mandelbrot 集。

事实证明已经有人拥有了（而且最近出现了）。 我会让其他人将其作为答案发布，但我很好奇可以进行哪些优化。

或者，您会采取什么措施使代码更具可读性？

我将选择最具可读性（但相当紧凑）的版本作为接受的答案（太糟糕了，我们还没有代表赏金！），除非有人确实进行了出色的优化。

奖励点是那些教我一些关于 T-SQL 的答案。

-亚当

Answer 1

来自 thedailywtf.com

-- AUTHOR: GRAEME JOB
-- CREATED: 12-OCT-2008
-- BECAUSE: SINGLE SQL COMMAND < 50 LINES. JUST BECAUSE.
WITH 
      XGEN(X, IX) AS (              -- X DIM GENERATOR
            SELECT CAST(-2.2 AS FLOAT) AS X, 0 AS IX UNION ALL
            SELECT CAST(X + 0.031 AS FLOAT) AS X, IX + 1 AS IX
            FROM XGEN
            WHERE IX < 100
      ),
      YGEN(Y, IY) AS (              -- Y DIM GENERATOR
            SELECT CAST(-1.5 AS FLOAT) AS Y, 0 AS IY UNION ALL
            SELECT CAST(Y + 0.031 AS FLOAT) AS Y, IY + 1 AS IY
            FROM YGEN
            WHERE IY < 100
      ),
      Z(IX, IY, CX, CY, X, Y, I) AS (           -- Z POINT ITERATOR
            SELECT IX, IY, X, Y, X, Y, 0
            FROM XGEN, YGEN   
            UNION ALL
            SELECT IX, IY, CX, CY, X * X - Y * Y + CX AS X, Y * X * 2 + CY, I + 1
            FROM Z
            WHERE X * X + Y * Y < 16
            AND I < 100
      )
SELECT REPLACE(REPLACE(REPLACE(REPLACE(REPLACE(REPLACE(REPLACE(REPLACE(REPLACE(REPLACE(
      REPLACE(REPLACE(REPLACE(REPLACE(REPLACE(REPLACE(REPLACE(REPLACE(REPLACE(REPLACE(
      REPLACE(REPLACE(REPLACE(REPLACE(REPLACE(REPLACE(
      (X0+X1+X2+X3+X4+X5+X6+X7+X8+X9+X10+X11+X12+X13+X14+X15+X16+X17+X18+X19+
      X20+X21+X22+X23+X24+X25+X26+X27+X28+X29+X30+X31+X32+X33+X34+X35+X36+X37+X38+X39+
      X40+X41+X42+X43+X44+X45+X46+X47+X48+X49+X50+X51+X52+X53+X54+X55+X56+X57+X58+X59+
      X60+X61+X62+X63+X64+X65+X66+X67+X68+X69+X70+X71+X72+X73+X74+X75+X76+X77+X78+X79+
      X80+X81+X82+X83+X84+X85+X86+X87+X88+X89+X90+X91+X92+X93+X94+X95+X96+X97+X98+X99),
      'A',' '),   'B','.'),   'C',','),   'D',','),   'E',','),   'F','-'),   'G','-'),
      'H','-'),   'I','-'),   'J','-'),   'K','+'),   'L','+'),   'M','+'),   'N','+'),
      'O','%'),   'P','%'),   'Q','%'),   'R','%'),   'S','@'),   'T','@'),   'U','@'),
      'V','@'),   'W','#'),   'X','#'),   'Y','#'),   'Z',' ')
FROM (
      SELECT 'X' + CAST(IX AS VARCHAR) AS IX,
      IY,   SUBSTRING('ABCDEFGHIJKLMNOPQRSTUVWXYZ', ISNULL(NULLIF(I, 0), 1), 1) AS I
      FROM Z) ZT
PIVOT (
      MAX(I) FOR IX IN (
      X0,X1,X2,X3,X4,X5,X6,X7,X8,X9,X10,X11,X12,X13,X14,X15,X16,X17,X18,X19,
      X20,X21,X22,X23,X24,X25,X26,X27,X28,X29,X30,X31,X32,X33,X34,X35,X36,X37,X38,X39,
      X40,X41,X42,X43,X44,X45,X46,X47,X48,X49,X50,X51,X52,X53,X54,X55,X56,X57,X58,X59,
      X60,X61,X62,X63,X64,X65,X66,X67,X68,X69,X70,X71,X72,X73,X74,X75,X76,X77,X78,X79,
      X80,X81,X82,X83,X84,X85,X86,X87,X88,X89,X90,X91,X92,X93,X94,X95,X96,X97,X98,X99)
) AS PZT

_{（来源：thedailywtf.com）}

Answer 2

Create PROCEDURE dbo.mandlebrot
@left float,
@right float,
@Top float,
@Bottom float,
@Res float,
@MaxIterations Integer = 500
As
Set NoCount On

Declare @Grid Table (
    X float Not Null, 
    Y float Not Null,
    InSet Bit
   Primary Key (X, Y))

Declare @Xo float, @Yo float, @Abs float
Declare @PtX Float, @PtY Float
Declare @Iteration Integer Set @Iteration = 0
Select @Xo = @Left, @Yo = @Bottom

While @Yo <= @Top Begin
    While @Xo <= @Right Begin
        Select @PtX = @Xo, @PtY = @Yo
        While @Iteration < @MaxIterations 
            And (Square(@PtX) + Square(@PtY)) < 4.0 Begin
            Select @PtX = Square(@PtX) - Square(@PtY) + @Xo,
                   @PtY = 2* @PtX * @PtY + @Yo
            Select @Iteration, @PtX, @PtY
            Set @Iteration = @Iteration + 1
        End
        Insert @Grid(X, Y, InSet) 
        Values(@Xo, @Yo, Case 
            When @Iteration < @MaxIterations
                    Then 1 Else 0 End)
        Set @Xo = @Xo + @res
        Set @Iteration = 0
    End
    Select @Xo = @Left, 
           @Yo = @Yo + @Res
End

Select * From @Grid

Answer 3

希望这也能教会一些有关 T-SQL 的知识，它以基于集合的方法完成所有操作，这是 TSQL 的优势（即没有 while 循环）或变量：

SET NOCOUNT ON;

--populate
;WITH Numbers ([row]) AS
(
   SELECT TOP 100 CAST(ROW_NUMBER() OVER (ORDER BY NEWID()) AS FLOAT) [row]
   FROM sys.columns
)
SELECT A.row AS x, 
   B.row AS y, 
   0 AS iter, 
   A.row AS iterx, 
   B.row AS itery, 
   '.' AS symbol
INTO #GRID
FROM Numbers A, Numbers B
WHERE B.[row] <= 24
GO

-- scale
UPDATE #GRID
SET x = x * 3.0 / 100.0 - 2,
   y = y * 2.0 / 24.0 - 1,
   iterx = x * 3.0 / 100.0 - 2,
   itery = y * 2.0 / 24.0 - 1
GO

--iterate
UPDATE #GRID
SET iterx = iterx*iterx - itery*itery + x,
    itery = 2*iterx*itery + y,
    iter = iter+1
WHERE iterx*iterx+itery*itery <= 2*2
GO 257

UPDATE #GRID SET symbol = CHAR(64+(iter%26)) WHERE NOT iter = 257
GO

--print
WITH concatenated (y, c) AS 
(
   SELECT G2.y,
       (SELECT SUBSTRING(G.symbol, 1, 1) AS [data()] FROM #GRID G WHERE G.y = G2.y FOR XML PATH('')) c
   FROM (SELECT DISTINCT y FROM #GRID) AS G2
)
SELECT REPLACE(c, ' ', '') FROM concatenated ORDER BY y
GO


DROP TABLE #GRID

Answer 4

with points (x1,y1,x2,y2,depth) as
(
    select convert(float,-2.40), convert(float,-2.40), convert(float,2.40), convert(float,2.40), 8
    union all select x1,y1,(x1+x2)/2,(y1+y2)/2,depth-1 from points where depth>0
    union all select (x1+x2)/2,y1,x2,(y1+y2)/2,depth-1 from points where depth>0
    union all select x1,(y1+y2)/2,(x1+x2)/2,y2,depth-1 from points where depth>0
    union all select (x1+x2)/2,(y1+y2)/2,x2,y2,depth-1 from points where depth>0
),
mandelbrot(x1,y1,x2,y2,x,y,depth) as
(
    select x1,y1,x2,y2,convert(float,0),convert(float,0),20 from points where depth=0
    union all
    select x1,y1,x2,y2, x*x-y*y+x1, 2*x*y+y1,depth-1 from mandelbrot where depth > 0 and (x*x+y*y<4)
)
select geometry::STGeomFromText('POLYGON((' +
  convert(varchar,x1) + ' ' + convert(varchar,y1) + ',' +
  convert(varchar,x1) + ' ' + convert(varchar,y2) + ',' +
  convert(varchar,x2) + ' ' + convert(varchar,y2) + ',' +
  convert(varchar,x2) + ' ' + convert(varchar,y1) + ',' +
  convert(varchar,x1) + ' ' + convert(varchar,y1) + '))',0)
  from mandelbrot where depth = 0