// time is a value from 0 to 1;
// 0 - begin of the effect,
// 1 - whole page has been flipped
void update(float time)
{
    const Point A, B', C, D;  // corners of the animated page (except B')
    const float t = sqrt(time);
    const Point P1 = lerp(B', A, t);
    const Point P2 = lerp(C, D, t);
    const Line line = CreateLineFromPoints(P1, P2);
    const Point N = GetNormalVectorFromLine(line);
    const float maxRadius = 0.5 * pageWidth * pi;
    float R;
    if (time <= 0.5)
    {
        // for the first half of the effect keep a fixed value of radius
        R = maxRadius;
    }
    else
    {
        // in the second half decrease the radius from its max value to 0
        const float s = sqrt( 2 * (1 - time) );
        // note that "2 * (1 - time)" is between 0 and 1, so is "s"

        R = (0, maxRadius, s);      // <=>  R = maxRadius * s;
    }

    const float invR = 1 / R;
    const float halfCylinderCircumference = π * R;    // π ≈ 3.14159
    Vertex v;
    for v in grid.vertices
    {
        const float dist = distance(v, line);
         if (dist > 0)
        {
               const Point p = v - N * dist;
              if (dist <= halfCylinderCircumference)
              {
                     const float alpha = dist * invR;
                     const float sinAlpha = R * sin(alpha);
                     v.x = p.x + N.x * sinAlpha;
                     v.y = p.y + N.y * sinAlpha;
                     v.z = (1 - cos(alpha)) * R;
              }
             else
             {
                   v.x = p.x - N.x * (dist - halfCylinderCircumference);
                   v.y = p.y - N.y * (dist - halfCylinderCircumference);
                   v.z = 2 * R;
             }
        }
    }
}