out

source image
============
.........33.........
........3223........
.......321123.......
......321..123......
.....321....123.....
....321......123....
...3217......8123...
..321..7....8..123..
.321....7..8....123.
321......78......123
645......87......546
.645....8..7....546.
..645..8....7..546..
...6458......7546...
....645......546....
.....645....546.....
......645..546......
.......645546.......
........6446........
.........66.........

rows
====
7......546???????????
77......546??????????
77.....546.??????????
77.....54..??????????
77.....46..??????????
77....54...??????????
77....56....?????????
777...44....?????????
777...56.....????????
777..756......???????
77777744.......??????
777..756......???????
777...56.....????????
777...44....?????????
77....56....?????????
77....54...??????????
77.....46..??????????
77.....54..??????????
77.....546.??????????
77......546??????????
78......546??????????
78......546??????????
78.....546.??????????
78.....54..??????????
78.....46..??????????
78....54...??????????
78....56....?????????
788...44....?????????
788...56.....????????
788..856......???????
78888844.......??????
788..856.......??????
788...56......???????
788...44.....????????
78....56.....????????
78....54....?????????
78.....46...?????????
78.....54...?????????
78.....546..?????????
78......5466?????????
77......1233?????????
77......1233?????????
77.....123..?????????
77.....12...?????????
77.....23...?????????
77....12....?????????
77....13.....????????
777...22.....????????
777...13......???????
777..713.......??????
77777722........?????
777..713.......??????
777...13......???????
777...22.....????????
77....13.....????????
77....12....?????????
77.....23...?????????
77.....12...?????????
77.....123..?????????
77......1233?????????
78......1233?????????
78......1233?????????
78.....123..?????????
78.....12...?????????
78.....23...?????????
78....12....?????????
78....13.....????????
788...22.....????????
788...13......???????
788..813.......??????
78888822.......??????
788..813......???????
788...13.....????????
788...22....?????????
78....13....?????????
78....12...??????????
78.....23..??????????
78.....12..??????????
78.....123.??????????
78......123??????????

image after transforming into rows and back
===========================================
.........33.........
........3223........
.......321123.......
......321..123......
.....321....123.....
....321......123....
...3217......8123...
..321..7....8..123..
.321....7..8....123.
321......78......123
645......87......546
.645....8..7....546.
..645..8....7..546..
...6458......7546...
....645......546....
.....645....546.....
......645..546......
.......645546.......
........6446........
.........66.........

Rotate.c++

#include <math.h>
#include <memory.h>

#include <iostream>
#include <vector>

using namespace std;

template <int WIDTH, int HEIGHT, class Cell = char, Cell empty = '?'>
struct Image
{
    Cell raw[WIDTH][HEIGHT];

    Image() {
        memset(raw, '.', WIDTH * HEIGHT);
    }

    template <int w1, int h1>
    void blit(char * img) {
        for (int i = 0; i < w1; i++)
            for (int j = 0; j < h1; j++)
                raw [i][j] = img[j * w1 + i];
    }

    Cell getRadial(float angle, float shift) {
        int x = WIDTH / 2, y = HEIGHT / 2;

        x += sin(angle) * shift;
        y += cos(angle) * shift;

        if (x < 0 || x >= WIDTH 
         || y < 0 || y >= HEIGHT)
        {
            return empty;   
        }

        return raw[x][y];
    }

    void putRadial(Cell cell, float angle, float shift) {
        int x = WIDTH / 2, y = HEIGHT / 2;

        x += sin(angle) * shift;
        y += cos(angle) * shift;

        if (x <  0 || x >= WIDTH 
         || y <  0 || y >= HEIGHT)
        {
            return; 
        }

        raw[x][y] = cell;
    }

    void dump(string header) {

        cout << header << endl;
        
        for (int i = 0; i < header.size(); i++)
            cout << "=";

        cout << endl;

        for (int i = 0; i < WIDTH; i++) {
            for (int j = 0; j < HEIGHT; j++) {
                cout << raw[i][j];
            }
            cout << "\n";
        }

        cout << endl;
    }

    string row(float angle, int rad, float step = 1) {
        string accum;

        for (int i = 0; i <= rad; i++) {
            accum += getRadial(angle, i * step);
        }

        return accum;
    }
};

int main(int argc, char const *argv[])
{
    Image<20, 20> image;
    Image<20, 20> image2;

    image.blit<20, 20>(
        ".........36........."
        "........3246........"
        ".......321546......."
        "......321..546......"
        ".....321....546....."
        "....321......546...."
        "...3217......8546..."
        "..321..7....8..546.."
        ".321....7..8....546."
        "321......78......546"
        "321......87......546"
        ".321....8..7....546."
        "..321..8....7..546.."
        "...3218......7546..."
        "....321......546...."
        ".....321....546....."
        "......321..546......"
        ".......321546......."
        "........3246........"
        ".........36........."
    );

    image.dump("source image");

    cout << "rows\n====" << endl;

    vector<string> rows;

    for (float angle = 0; angle < M_PI * 2; angle += M_PI / 40) {
        rows.push_back(image.row(angle, 15));

        cout << image.row(angle, 20) << "\n";
    }

    cout << endl;

    int i = 0;
    for (float angle = 0; angle < M_PI * 2; angle += M_PI / 40, i++) {
        for (int j = 0; j < 15; j++) {
            image2.putRadial(rows[i][j], angle, j);
        }
    }

    image2.dump("image after transforming into rows and back");

    return 0;
}