Normal sentences

Question

Write a program or function which, given an input string and a standard deviation σ, outputs that string along the normal distribution curve with mean 0 and standard deviation σ.

Normal distribution curve

The y coordinate of each character c is:

where σ is given as input, and where x is the x axis coordinate of c.

• The character in the center of the string has x = 0. If the string's length is even, either of the two middle characters can be chosen as the center.
• Characters are separated by steps of 0.1 (e.g. the character to the left of the center one has x = -0.1, the one to the right of the middle one has x = 0.1, etc.).

Printing the string

• Lines, like characters, are separated by steps of 0.1.
• Each character is printed on the line with the y value that is closest to its own y value (if the value is precisely in between the values of two lines, choose the one with the biggest value (just like how round usually returns 1.0 for 0.5)).
• For example, if the y coordinate of the center value (i.e. the maximum value) is 0.78 and the y coordinate of the first character is 0.2, then there will 9 lines: the center character being printed on line 0 and the first character being printed on line 8.

Inputs and outputs

• You may take both inputs (the string and σ) as program arguments, through STDIN, function arguments or anything similar in your language.
• The string will only contain printable ASCII characters. The string can be empty.
• σ > 0.
• You may print the output to STDOUT, in a file, or return it from a function (as long as it is a string and not say a list of strings for each line).
• A trailing new line is acceptable.
• Trailing spaces are acceptable as long as they don't make the line exceed the last line in length (so no trailing space is acceptable on the last line).

Test cases

σ    String

0.5  Hello, World!

     , W     
   lo   or   
  l       l  
 e         d 
H           !



0.5  This is a perfectly normal sentence

                tly                
              ec    n              
             f       o             
            r         r            
           e           m           
          p             a          
        a                l         
      s                    se      
This i                       ntence



1.5  Programming Puzzles & Code Golf is a question and answer site for programming puzzle enthusiasts and code golfers.

                                                d answer site for p                                               
                                      uestion an                   rogramming                                     
                      Code Golf is a q                                        puzzle enthusia                     
Programming Puzzles &                                                                        sts and code golfers.



0.3  .....................

          .          
         . .         

        .   .        

       .     .       


      .       .      

     .         .     
    .           .    
   .             .   
...               ...

Scoring

This is code-golf,

                 nsw                 
                a   er               
              t                      
             s         i             
            e           n            
           t                         
         or               by         
       sh                   te       
so the                        s wins.

Answer 1

Python 3 with SciPy, 239 233 bytes

from scipy import stats,around,arange
def f(s,t):
 l=len(t);p=[];y=around(stats.norm.pdf((arange(l)-l//2)*.1,scale=s),1)*10
 for i in range(l):p+=[[' ']*(max(y)-y[i])];p[i]+=[t[i]]+[' ']*(y[i]-y[0])
 for j in zip(*p):print(*j,sep='')

A function that takes input via argument of standard deviation s and string t, and prints the result to STDOUT.

How it works

from scipy import stats,around,arange  Import the statistics, rounding and range functions
                                       from SciPy
def f(s,t):                            Function with input standard deviation s and string
                                       t
l=len(t);p=[]                          Define the much-used length of t as l and initialise
                                       the print values list p
arange(l)                              Generate a list of integer x values in [0,l)...
...-l//2*.1                            ...and scale such that 0 is at the middle character
                                       and the x-step is 0.1
stats.norm.pdf(...,scale=s)            Generate a list containing the y values for each x
                                       value by calling the normal probability
                                       density function scaled with s...
y=around(...,1)                        ...round all values to 1 decimal place...
...*10                                 ...and multiply by 10 to give the vertical index of
                                       each character
for i in range(l):...                  For all characters in t...
p+=[[' ']*(max(y)-y[i])]               ..add the number of lines below the character as
                                       spaces...
p[i]+=[t[i]]+[' ']*(y[i]-y[0])         ...add the character and the number of lines above
                                       the character as spaces

This leaves p containing a list for each desired output line, but transposed.

for j in zip(*p):...                   For every output line in the transpose of p...
print(*j,sep='')                       ...print the output line

Try it on Ideone

Answer 2

Ruby: 273 254 Bytes

->n,s{j,o,g,r,l=-(n.size/2),[],0,{}
n.gsub(/./){(r[((2*Math::PI)**-0.5*10*Math.exp(-(j/1e1)**2/2/s/s)/s).round]||="")<<$&
j+=1}
r.sort.map{|y, c|o<<(l ?$/*(y-l-1):"")+(" "*g)+(c[0,(h=c.size)/2])+(" "*(n.size-g*2-h))+(c[h/2,h])
g+=h/2
l=y}
puts o.reverse}

A huge thanks to Kevin Lau for saving 18 bytes!