S04E16：正二十面体网格生成

说明

正二十面体，由 20 个小的正三角形面组成，每个顶点周围有 5 个顶点。

几何

第一步，还是先查找几何公式； 第二步，计算顶点坐标，由于正二十面体的各个顶点很有规律，所以计算还算简单

let phi = (1.0 + sqrtf(5)) * 0.5
let r2 = radius * radius
let den = (1.0 + (1.0 / pow(phi, 2.0)))
let h = sqrt(r2 / (den))
let w = h / phi

let points = [
    SIMD3<Float>(0.0, h, w),
    SIMD3<Float>(0.0, h, -w),
    SIMD3<Float>(0.0, -h, w),
    SIMD3<Float>(0.0, -h, -w),
    
    SIMD3<Float>(h, -w, 0.0),
    SIMD3<Float>(h, w, 0.0),
    SIMD3<Float>(-h, -w, 0.0),
    SIMD3<Float>(-h, w, 0.0),
    
    SIMD3<Float>(-w, 0.0, -h),
    SIMD3<Float>(w, 0.0, -h),
    SIMD3<Float>(-w, 0.0, h),
    SIMD3<Float>(w, 0.0, h)
]
// 因为 5 个面共用顶点，所以重复 5 次
meshPositions.append(contentsOf: points + points + points + points + points)

第三步，连线构建小三角形，与前面的正十二面体类似。

第四步，第五步，第六步，同正四面体与正八面体完全一致。

代码

/// 正二十面体，，radius 为外接球半径，res 三角形面剖分次数
public static func generateIcosahedron(radius: Float, res: Int = 0) throws -> MeshResource {
    let pointCount = 12
    var triangles = 20
    var vertices = pointCount * 5
    
    var descr = MeshDescriptor()
    var meshPositions: [SIMD3<Float>] = []
    var indices: [UInt32] = []
    var normals: [SIMD3<Float>] = Array(repeating: .zero, count: vertices)
    var textureMap: [SIMD2<Float>] = []
    
    let phi = (1.0 + sqrtf(5)) * 0.5
    let r2 = radius * radius
    let den = (1.0 + (1.0 / pow(phi, 2.0)))
    let h = sqrt(r2 / (den))
    let w = h / phi

    let points = [
        SIMD3<Float>(0.0, h, w),
        SIMD3<Float>(0.0, h, -w),
        SIMD3<Float>(0.0, -h, w),
        SIMD3<Float>(0.0, -h, -w),

        SIMD3<Float>(h, -w, 0.0),
        SIMD3<Float>(h, w, 0.0),
        SIMD3<Float>(-h, -w, 0.0),
        SIMD3<Float>(-h, w, 0.0),

        SIMD3<Float>(-w, 0.0, -h),
        SIMD3<Float>(w, 0.0, -h),
        SIMD3<Float>(-w, 0.0, h),
        SIMD3<Float>(w, 0.0, h)
    ]
    meshPositions.append(contentsOf: points + points + points + points + points)
    
    let index: [UInt32] = [
        0, 11, 5,
        0, 5, 1,
        0, 1, 7,
        0, 7, 10,
        0, 10, 11,

        1, 5, 9,
        5, 11, 4,
        11, 10, 2,
        10, 7, 6,
        7, 1, 8,

        3, 9, 4,
        3, 4, 2,
        3, 2, 6,
        3, 6, 8,
        3, 8, 9,

        4, 9, 5,
        2, 4, 11,
        6, 2, 10,
        8, 6, 7,
        9, 8, 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])
}