I have received an email asking to explain how I generate a procedural sphere as per this post. Rather than respond to emails personally, I have decide in future to answer the questions on the blog, presenting the information to a wider audience for comment, correction, improvement and perhaps a little learning.
The base of my generated sphere is a cube. Depending on how smooth the generated sphere needs to be, I recursively split each side into 4 equally sized children. This is achieved my finding the center of the face, popping that point out so that it is the correct radius from the center, then making 4 faces using the original points and this new point. (There is some ASCII art showing this in the code below)
void CSphere::Initialize(float p_fRadius, int p_iMaxDepth)
{
// TLB----TRB
// /| /|
// / | / |
// TLF----TRF |
// | BLB--|BRB
// | / | /
// | / |/
// BLF----BRF
//Putting the Vertices of the initial Cube at p_fRadius is not correct as the distance of
//p_fRadius, p_fRadius, p_fRadius from the Origin is greater than p_fRadius.
CVector3 l_Vertices[8];
l_Vertices[TOP_LEFT_FRONT] = MoveToRadiusDistance(CVector3(-p_fRadius, p_fRadius, -p_fRadius), p_fRadius);
l_Vertices[TOP_RIGHT_FRONT] = MoveToRadiusDistance(CVector3( p_fRadius, p_fRadius, -p_fRadius), p_fRadius);
l_Vertices[BOTTOM_RIGHT_FRONT] = MoveToRadiusDistance(CVector3( p_fRadius, -p_fRadius, -p_fRadius), p_fRadius);
l_Vertices[BOTTOM_LEFT_FRONT] = MoveToRadiusDistance(CVector3(-p_fRadius, -p_fRadius, -p_fRadius), p_fRadius);
l_Vertices[TOP_LEFT_BACK] = MoveToRadiusDistance(CVector3(-p_fRadius, p_fRadius, p_fRadius), p_fRadius);
l_Vertices[TOP_RIGHT_BACK] = MoveToRadiusDistance(CVector3( p_fRadius, p_fRadius, p_fRadius), p_fRadius);
l_Vertices[BOTTOM_RIGHT_BACK] = MoveToRadiusDistance(CVector3( p_fRadius, -p_fRadius, p_fRadius), p_fRadius);
l_Vertices[BOTTOM_LEFT_BACK] = MoveToRadiusDistance(CVector3(-p_fRadius, -p_fRadius, p_fRadius), p_fRadius);
//Initialize the faces of the cube (The face structure just stores the vertices for the face corners and has render functionality, not applicable to this explanation, and its depth in the face tree)
m_pFaces[FRONT].Initialise(FRONT, l_Vertices[TOP_LEFT_FRONT], l_Vertices[TOP_RIGHT_FRONT], l_Vertices[BOTTOM_RIGHT_FRONT], l_Vertices[BOTTOM_LEFT_FRONT], DEPTH0);
m_pFaces[RIGHT].Initialise(RIGHT, l_Vertices[TOP_RIGHT_FRONT], l_Vertices[TOP_RIGHT_BACK], l_Vertices[BOTTOM_RIGHT_BACK], l_Vertices[BOTTOM_RIGHT_FRONT], DEPTH0);
m_pFaces[BACK].Initialise(BACK, l_Vertices[TOP_RIGHT_BACK], l_Vertices[TOP_LEFT_BACK], l_Vertices[BOTTOM_LEFT_BACK], l_Vertices[BOTTOM_RIGHT_BACK], DEPTH0);
m_pFaces[LEFT].Initialise(LEFT, l_Vertices[TOP_LEFT_BACK], l_Vertices[TOP_LEFT_FRONT], l_Vertices[BOTTOM_LEFT_FRONT], l_Vertices[BOTTOM_LEFT_BACK], DEPTH0);
m_pFaces[TOP].Initialise(TOP, l_Vertices[TOP_LEFT_BACK], l_Vertices[TOP_RIGHT_BACK], l_Vertices[TOP_RIGHT_FRONT], l_Vertices[TOP_LEFT_FRONT], DEPTH0);
m_pFaces[BOTTOM].Initialise(BOTTOM, l_Vertices[BOTTOM_LEFT_FRONT], l_Vertices[BOTTOM_RIGHT_FRONT], l_Vertices[BOTTOM_RIGHT_BACK], l_Vertices[BOTTOM_LEFT_BACK], DEPTH0);
//Subdivide each patch to the lowest resolution
m_pPatches[FRONT].SubDivide(p_fRadius, p_iMaxDepth);
m_pPatches[RIGHT].SubDivide(p_fRadius, p_iMaxDepth);
m_pPatches[BACK].SubDivide(p_fRadius, p_iMaxDepth);
m_pPatches[LEFT].SubDivide(p_fRadius, p_iMaxDepth);
m_pPatches[TOP].SubDivide(p_fRadius, p_iMaxDepth);
m_pPatches[BOTTOM].SubDivide(p_fRadius, p_iMaxDepth);
}
where
bool CSphereFace::SubDivide(float p_fRadius, int p_iMaxDepth)
{
if (m_iDepth >= p_iMaxDepth)
return false;
//Create the Additional Vertices
//
// NW---------------D-------------NE
// | | |
// | | |
// | | |
// | | |
// A--------------Center-----------C
// | | |
// | | |
// | | |
// | | |
// SW----------------B-------------SE
//
//
g_vAdditionalVertices[A] = m_vBaseVertices[eNorthWest] + ((m_vBaseVertices[eSouthWest] - m_vBaseVertices[eNorthWest]) / 2.0f);
g_vAdditionalVertices[B] = m_vBaseVertices[eSouthWest] + ((m_vBaseVertices[eSouthEast] - m_vBaseVertices[eSouthWest]) / 2.0f);
g_vAdditionalVertices[C] = m_vBaseVertices[eNorthEast] + ((m_vBaseVertices[eSouthEast] - m_vBaseVertices[eNorthEast]) / 2.0f);
g_vAdditionalVertices[D] = m_vBaseVertices[eNorthWest] + ((m_vBaseVertices[eNorthEast] - m_vBaseVertices[eNorthWest]) / 2.0f);
//Create Child Nodes
m_pChildren = new CSphereFace[4];
m_pChildren[eNorthWest].Initialise(eNorthWest, m_vBaseVertices[eNorthWest], g_vAdditionalVertices[D], m_vBaseVertices[eCentre], g_vAdditionalVertices[A], m_iDepth + 1);
m_pChildren[eNorthEast].Initialise(eNorthEast, g_vAdditionalVertices[D], m_vBaseVertices[eNorthEast], g_vAdditionalVertices[C], m_vBaseVertices[eCentre], m_iDepth + 1);
m_pChildren[eSouthWest].Initialise(eSouthWest, g_vAdditionalVertices[A], m_vBaseVertices[eCentre], g_vAdditionalVertices[B], m_vBaseVertices[eSouthWest], m_iDepth + 1);
m_pChildren[eSouthEast].Initialise(eSouthEast, m_vBaseVertices[eCentre], g_vAdditionalVertices[C], m_vBaseVertices[eSouthEast], g_vAdditionalVertices[B], m_iDepth + 1);
m_pChildren[eNorthWest].SubDivide(p_fRadius, p_iMaxDepth);
m_pChildren[eNorthEast].SubDivide(p_fRadius, p_iMaxDepth);
m_pChildren[eSouthWest].SubDivide(p_fRadius, p_iMaxDepth);
m_pChildren[eSouthEast].SubDivide(p_fRadius, p_iMaxDepth);
return true;
}
and
CVector3 MoveToRadiusDistance(CVector3 p_Vector, float p_fRadius)
{
//Get the Normalized Vector, of this vertex from the origin (center of the sphere) and pop it out to the correct radius,
return p_Vector.Normalize() * p_fRadius;
}
That is pretty much the basics of how I create the sphere from recursively subdividing a cube. If I overlooked anything please comment and I will correct the post to reflect any gap in information or any mistake.
