说明
正八面体,就类似于两个金字塔,底对底叠起来的样子。
几何
第一步,还是找几何公式; 第二步,利用公式计算顶点坐标;
let a: Float = radius * sqrtf(2)//棱长
let rm = a / 2 // 中交球半径(过棱中点)
let points: [SIMD3<Float>] = [
SIMD3<Float>(0, radius, 0),
SIMD3<Float>(rm, 0, rm),
SIMD3<Float>(-rm, 0, rm),
SIMD3<Float>(-rm, 0, -rm),
SIMD3<Float>(rm, 0, -rm),
SIMD3<Float>(0, -radius, 0)
]
惟一不同的是,正八面体每个顶点周围有四个面(四条棱),所以每个顶点需要重复 4 次。其余步骤与正四面体一样,代码也是原封不动。
代码
/// 正八面体,radius 为外接球半径,res 三角面剖分次数
public static func generateOctahedron(radius: Float, res: Int = 0) throws -> MeshResource {
let pointCount = 6
var triangles = 8
var vertices = pointCount * 4
var descr = MeshDescriptor()
var meshPositions: [SIMD3<Float>] = []
var indices: [UInt32] = []
var normals: [SIMD3<Float>] = Array(repeating: .zero, count: vertices)
var textureMap: [SIMD2<Float>] = []
let a: Float = radius * sqrtf(2)//棱长
let rm = a / 2 // 中交球半径(过棱中点)
let points: [SIMD3<Float>] = [
SIMD3<Float>(0, radius, 0),
SIMD3<Float>(rm, 0, rm),
SIMD3<Float>(-rm, 0, rm),
SIMD3<Float>(-rm, 0, -rm),
SIMD3<Float>(rm, 0, -rm),
SIMD3<Float>(0, -radius, 0)
]
meshPositions.append(contentsOf: points + points + points + points)
let index: [UInt32] = [
0, 2, 1,
0, 3, 2,
0, 4, 3,
0, 1, 4,
5, 1, 2,
5, 2, 3,
5, 3, 4,
5, 4, 1
]
var countDict: [UInt32:Int] = [:]
for ind in index {
let count = countDict[ind] ?? 0
indices.append(ind + UInt32(pointCount * count))
countDict[ind] = count + 1
}
for i in 0..<triangles {
let ai = 3 * i
let bi = 3 * i + 1
let ci = 3 * i + 2
let i0 = indices[ai]
let i1 = indices[bi]
let i2 = indices[ci]
let v0 = meshPositions[Int(i0)]
let v1 = meshPositions[Int(i1)]
let v2 = meshPositions[Int(i2)]
let faceNormal = simd_normalize((v0 + v1 + v2) / 3)
normals[Int(i0)] = faceNormal
normals[Int(i1)] = faceNormal
normals[Int(i2)] = faceNormal
}
for _ in 0..<res {
let newTriangles = triangles * 4
let newVertices = vertices + triangles * 3
var newIndices: [UInt32] = []
var pos: SIMD3<Float>
for i in 0..<triangles {
let ai = 3 * i
let bi = 3 * i + 1
let ci = 3 * i + 2
let i0 = indices[ai]
let i1 = indices[bi]
let i2 = indices[ci]
let v0 = meshPositions[Int(i0)]
let v1 = meshPositions[Int(i1)]
let v2 = meshPositions[Int(i2)]
let faceNormal = normals[Int(i0)]
normals.append(contentsOf: [faceNormal, faceNormal, faceNormal])
// a
pos = (v0 + v1) * 0.5
meshPositions.append(pos)
// b
pos = (v1 + v2) * 0.5
meshPositions.append(pos)
// c
pos = (v2 + v0) * 0.5
meshPositions.append(pos)
let a = UInt32(ai + vertices)
let b = UInt32(bi + vertices)
let c = UInt32(ci + vertices)
newIndices.append(contentsOf: [
i0, a, c,
a, i1, b,
a, b, c,
c, b, i2
])
}
indices = newIndices
triangles = newTriangles
vertices = newVertices
}
for i in 0..<meshPositions.count {
let p = meshPositions[i]
let n = p
textureMap.append(SIMD2<Float>(abs(atan2(n.x, n.z)) / .pi, 1 - acos(n.y/radius) / .pi))
}
descr.positions = MeshBuffers.Positions(meshPositions)
descr.normals = MeshBuffers.Normals(normals)
descr.textureCoordinates = MeshBuffers.TextureCoordinates(textureMap)
descr.primitives = .triangles(indices)
return try MeshResource.generate(from: [descr])
}