Morph cube into sphere

Question

I created a cube of small sphere models that act as dots. The dots have coordinates:

{0 <= x <= 9, 0 <= y <= -9, 0 <= z <= -9}

The interior of the cube is empty so dots only exist on the surface of the cube. The empty spaces are represented as points at (100, 100, 100) and when I do the draw loop I ignore points which match them, which is why in the code I'll post below will have that as a condition for doing certain things or not doing them.

The goal is to take the points of the cube, and apply transformations to them to map them onto a sphere.

This is the code to create the array for the cube positions then to create an array for the sphere positions:

// initialize cube array
points = new Matrix[10, 10, 10];

for (int i = 0; i < 10; i++)
{
    for (int j = 0; j < 10; j++)
    {
        for (int k = 0; k < 10; k++)
        {
            points[i, j, k] = Matrix.CreateTranslation(new Vector3(100, 100, 100));
        }
    }
}

for (int i = 0; i < 10; i++)
{
    for (int j = 0; j < 10; j++)
    {
        points[i, j, 0] = Matrix.CreateTranslation(new Vector3(i, -j, 0));
        points[i, j, 9] = Matrix.CreateTranslation(new Vector3(i, -j, -9));
    }
}

for (int j = 0; j < 10; j++)
{
    for (int k = 0; k < 10; k++)
    {
        points[0, j, k] = Matrix.CreateTranslation(new Vector3(0, -j, -k));
        points[9, j, k] = Matrix.CreateTranslation(new Vector3(9, -j, -k));
    }
}

for (int i = 0; i < 10; i++)
{
    for (int k = 0; k < 10; k++)
    {
        points[i, 0, k] = Matrix.CreateTranslation(new Vector3(i, 0, -k));
        points[i, 9, k] = Matrix.CreateTranslation(new Vector3(i, -9, -k));
    }
}
// end cube array initialization

// create sphere array
double d;
double theta;
double phi;
double r = 10;

spherePoints = new Matrix[10, 10, 10];
for (int i = 0; i < 10; i++)
{
    for (int j = 0; j < 10; j++)
    {
        for (int k = 0; k < 10; k++)
        {
            if (points[i, j, k] != Matrix.CreateTranslation(new Vector3(100, 100, 100)))
            {
                d = Math.Sqrt(Math.Pow(i, 2) + Math.Pow(-j, 2) + Math.Pow(-k, 2));
                theta = Math.Acos(-k / d);
                phi = Math.Atan2(-j, i);

                spherePoints[i, j, k] = Matrix.CreateTranslation(new Vector3((float)(r * Math.Sin(theta) * Math.Cos(phi)),
                                                                            (float)(r * Math.Sin(theta) * Math.Sin(phi)),
                                                                            (float)(r * Math.Cos(theta))));
            }
            else
                spherePoints[i, j, k] = Matrix.CreateTranslation(new Vector3(100, 100, 100));
        }
    }
}
// end creation of sphere array

Cube:

Not a sphere...:

From what I can tell I followed the formula exactly, but it seems to only generate an eighth of a sphere. There also appears to be weird grouping along the edges.

Answer 1

The problem is that you're only drawing your cube in one "quadrant" (or perhaps "octrant" would be more appropriate), so you're only getting 1/8 of your sphere.

Instead of having your cube go from [0,9], [-9,0], [-9,0], focus it on the origin.

Once your cube goes from [-5,5], [-5, 5], [-5,5], your spherical calculation will be fine.

Just to give a bit more understanding of how this is the issue:

• How many quadrants will your answers be in when you evaluate Acos(z/d), given that d is always positive and z is always negative?
• How many quadrants will your answers be in when you evaluate Atan(y, x), given that y is always negative and x is always negative?

Out of the 8 possible quadrant combinations, you're only filling one.