Unity Mesh(三) Mesh畫球
阿新 • • 發佈:2019-01-23
關於畫球一開始真是一點思路都沒有,樓主也查了好多資料,比較有代表性的是兩篇帖子。
一篇是Jasper Flick的帖子,一個很厲害的人:
這一篇的思路是根據柏拉圖體,正八面體分割成的球。
第二篇是OpenGL或者XNA回答的思路,是根據柏拉圖體正二十面體畫的
如果你能直接看懂上面兩篇中的任何一篇,那麼樓主下面寫的對於你來說都是廢話,你可以直接不用看了。
一、思路
簡單的說下,首先是畫出一個正八面體,這個我們上一篇文章:Unity Mesh(二) Mesh畫立方體和八面體,已經寫了怎麼畫正八面體,然後我們的思路是取每條邊的重點,細分三角形,比如那正八面體的一個面來說,我們拆分一次的情況如圖所示:
兩次的情況如圖:
以此類推,根據前面兩篇我們可以瞭解到,Mesh畫圖形必須知道三角形的頂點和三角形的點順序,這樣的話我們需要知道的引數有三個,三角形的頂點數,三角形的個數,三角形的頂點順序。
下面我們根據這三個資料的要求,一次進行計算說明。
二、計算三角形的個數
我們以正八面體中的一個面為例,我們拆分一次會有四個小三角形(這裡不算總共,我們Mesh畫以最小單位的三角形畫),拆分兩次會有16個三角形,這樣的話我們假設拆分次數為s(subdivisions),這樣每個面會有4^s個三角形,我們的正八面體有八個面,我們一共就有8*4^s個三角形,也就是2^(2s+3)個三角形。 結論:三角形的個數是:2^(2s+3)三、計算三角形的頂點數
1次拆分:1+2+1=4 2次拆分:1+2+3+2+1=9 3次拆分:1+2+3+4+5+4+3+2+1=25 以一個面來說我們用r(row)表示三角形的行數,r=2^s就代表一個面的三角形的行數,這樣我們四分之一的正八面體有的頂點數為(r+1)^2. 現在我們知道了1/4個正八面體有(r+1)^2.個頂點,但是整個合併不是4*(r+1)^2個頂點,因為我們還要捨去重合的點,為了更好的理解,我在Untiy裡放了4個1/4個正八面體來演示合併後的頂點的計算流程。
如圖所示,是四個四分之一正八面體,每個單個都有(r+1)^2個頂點,但是當上面兩個合併時:
我們發現重合了兩條我們計算的邊,那麼重複計算的就是這兩條邊上的頂點,看如下圖的演示: 所以,當我們兩個四分之一的正八面體合併時,重複了2r+1個頂點,那麼就可以理解為我們二分之一個正八面體有(r+1)^2-(2r+1)個頂點。 圖示:
最後我們再將這兩個合併,我們會發現重合了四條邊:
由此可得我們正八面體拆分後一共有4(r+1)^2-2(2r+1)-4r個頂點
這是我們常規的演算法下我們需要的頂點數,我們可以用0次拆分6個頂點,1次拆分18個頂點進行驗證這個公式,樓主自己驗證過是沒問題的。
四、頂點優化
上面一步中我們已經計算出了正八面體拆分後的頂點數,但是這個還是不夠的,我們還得新增,在我們計算三角形順序的時候我們會遇到這樣的問題:這樣我們三角形的順序就變為 0,1,2, 0,2,3 0,3,4 0,4,5 這樣的話,我們的指令碼寫起了就方便多了。 0次拆分,我們新增1個頂點。 1次拆分,我們新增3個頂點。 2次拆分,我們新增7個頂點。
相當於每有一個橫著的四邊形,我們新增一個,於是我們添加了2r-1個頂點。 綜合之前的,我們mesh的Verticles的大小為4(r+1)^2-8r-2+(2r-1)=4(r+1)^2-3(2r+1).
五、三角形的頂點順序
三角形的頂點順序,首先我們三角形的頂點陣列的大小應該就是三角形的個數*3,然後我們從頂點或者底部的點出發,遞迴遍歷即可,這裡我是直接參考的Jasper Flick的方法。 private static int CreateLowerStrip(int steps, int vTop, int vBottom, int t, int[] triangles)
{
for (int i = 1; i < steps; i++)
{
triangles[t++] = vBottom;
triangles[t++] = vTop - 1;
triangles[t++] = vTop;
triangles[t++] = vBottom++;
triangles[t++] = vTop++;
triangles[t++] = vBottom;
}
triangles[t++] = vBottom;
triangles[t++] = vTop - 1;
triangles[t++] = vTop;
return t;
}
六、畫球
這些條件都有了,我們來畫球,詳細的程式碼如下:using UnityEngine;
using System.Collections;
[RequireComponent(typeof(MeshFilter), typeof(MeshRenderer))]
public class DrawOctahedronSphere1 : MonoBehaviour
{
public Material mat;
private static Vector3[] directions = {
Vector3.left,
Vector3.back,
Vector3.right,
Vector3.forward
};
void Start()
{
DrawSphere(1, 1);
}
public void DrawSphere(int subdivisions = 0, float radius = 1)
{
gameObject.GetComponent<MeshRenderer>().material = mat;
Mesh mesh = GetComponent<MeshFilter>().mesh;
mesh.Clear();
int resolution = 1 << subdivisions;
Vector3[] vertices = new Vector3[(resolution + 1) * (resolution + 1) * 4 - 3*(resolution * 2 + 1)];
int[] triangles = new int[(1 << (subdivisions * 2 + 3)) * 3];
CreateOctahedron(vertices, triangles, resolution);
Debug.LogError(triangles.Length + " " + vertices.Length);
foreach (var item in triangles)
{
Debug.Log(item);
}
foreach (var item in vertices)
{
Debug.Log(item);
}
mesh.vertices = vertices;
mesh.triangles = triangles;
}
private static void CreateOctahedron(Vector3[] vertices, int[] triangles, int resolution)
{
int v = 0, vBottom = 0, t = 0;
vertices[v++] = Vector3.down;
for (int i = 1; i <= resolution; i++)
{
float progress = (float)i / resolution;
Vector3 from, to;
vertices[v++] = to = Vector3.Lerp(Vector3.down, Vector3.forward, progress);
for (int d = 0; d < 4; d++)
{
from = to;
to = Vector3.Lerp(Vector3.down, directions[d], progress);
t = CreateLowerStrip(i, v, vBottom, t, triangles);
v = CreateVertexLine(from, to, i, v, vertices);
vBottom += i > 1 ? (i - 1) : 0;
}
vBottom = v - 1 - i * 4;
}
for (int i = resolution - 1; i >= 1; i--)
{
float progress = (float)i / resolution;
Vector3 from, to;
vertices[v++] = to = Vector3.Lerp(Vector3.up, Vector3.forward, progress);
for (int d = 0; d < 4; d++)
{
from = to;
to = Vector3.Lerp(Vector3.up, directions[d], progress);
t = CreateUpperStrip(i, v, vBottom, t, triangles);
v = CreateVertexLine(from, to, i, v, vertices);
vBottom += i + 1;
}
vBottom = v - 1 - i * 4;
}
vertices[vertices.Length - 1] = Vector3.up;
for (int i = 0; i < 4; i++)
{
triangles[t++] = vBottom;
triangles[t++] = v;
triangles[t++] = ++vBottom;
}
}
private static int CreateVertexLine(Vector3 from, Vector3 to, int steps, int v, Vector3[] vertices)
{
for (int i = 1; i <= steps; i++)
{
vertices[v++] = Vector3.Lerp(from, to, (float)i / steps);
}
return v;
}
private static int CreateLowerStrip(int steps, int vTop, int vBottom, int t, int[] triangles)
{
for (int i = 1; i < steps; i++)
{
triangles[t++] = vBottom;
triangles[t++] = vTop - 1;
triangles[t++] = vTop;
triangles[t++] = vBottom++;
triangles[t++] = vTop++;
triangles[t++] = vBottom;
}
triangles[t++] = vBottom;
triangles[t++] = vTop - 1;
triangles[t++] = vTop;
return t;
}
private static int CreateUpperStrip(int steps, int vTop, int vBottom, int t, int[] triangles)
{
triangles[t++] = vBottom;
triangles[t++] = vTop - 1;
triangles[t++] = ++vBottom;
for (int i = 1; i <= steps; i++)
{
triangles[t++] = vTop - 1;
triangles[t++] = vTop;
triangles[t++] = vBottom;
triangles[t++] = vBottom;
triangles[t++] = vTop++;
triangles[t++] = ++vBottom;
}
return t;
}
}
這裡預設大小為1,拆分一次的效果如圖: 拆分四次的效果如圖
基本上四次就夠了,按照Jasper的計算,6次是上限,也許到這裡,你會問,這不是個球啊,是的,這裡還差一步,我們沒有設定頂點的法線,球的法線,是從球心到頂點的,於是我們加上法線的程式碼:
private static void Normalize(Vector3[] vertices, Vector3[] normals)
{
for (int i = 0; i < vertices.Length; i++)
{
normals[i] = vertices[i] = vertices[i].normalized;
}
}
還有就是球的大小用radius*verticle就好,我們把拆分次數和半徑寫到控制面板上,最後我們優化後的完整程式碼如下:
using UnityEngine;
using System.Collections;
[RequireComponent(typeof(MeshFilter), typeof(MeshRenderer))]
public class DrawOctahedronSphere : MonoBehaviour
{
public Material mat;
public int subdivisions;
public int radius;
private static Vector3[] directions = {
Vector3.left,
Vector3.back,
Vector3.right,
Vector3.forward
};
void Start()
{
DrawSphere(subdivisions, radius);
}
public void DrawSphere(int subdivisions = 0, float radius = 1)
{
if (subdivisions > 4)
{
subdivisions = 4;
}
gameObject.GetComponent<MeshRenderer>().material = mat;
Mesh mesh = GetComponent<MeshFilter>().mesh;
mesh.Clear();
int resolution = 1 << subdivisions;
Vector3[] vertices = new Vector3[(resolution + 1) * (resolution + 1) * 4 - 3 * (resolution * 2 + 1)];
int[] triangles = new int[(1 << (subdivisions * 2 + 3)) * 3];
CreateOctahedron(vertices, triangles, resolution);
if (radius != 1f)
{
for (int i = 0; i < vertices.Length; i++)
{
vertices[i] *= radius;
}
}
Vector3[] normals = new Vector3[vertices.Length];
Normalize(vertices, normals);
mesh.vertices = vertices;
mesh.triangles = triangles;
mesh.normals = normals;
}
private static void CreateOctahedron(Vector3[] vertices, int[] triangles, int resolution)
{
int v = 0, vBottom = 0, t = 0;
vertices[v++] = Vector3.down;
for (int i = 1; i <= resolution; i++)
{
float progress = (float)i / resolution;
Vector3 from, to;
vertices[v++] = to = Vector3.Lerp(Vector3.down, Vector3.forward, progress);
for (int d = 0; d < 4; d++)
{
from = to;
to = Vector3.Lerp(Vector3.down, directions[d], progress);
t = CreateLowerStrip(i, v, vBottom, t, triangles);
v = CreateVertexLine(from, to, i, v, vertices);
vBottom += i > 1 ? (i - 1) : 0;
}
vBottom = v - 1 - i * 4;
}
for (int i = resolution - 1; i >= 1; i--)
{
float progress = (float)i / resolution;
Vector3 from, to;
vertices[v++] = to = Vector3.Lerp(Vector3.up, Vector3.forward, progress);
for (int d = 0; d < 4; d++)
{
from = to;
to = Vector3.Lerp(Vector3.up, directions[d], progress);
t = CreateUpperStrip(i, v, vBottom, t, triangles);
v = CreateVertexLine(from, to, i, v, vertices);
vBottom += i + 1;
}
vBottom = v - 1 - i * 4;
}
vertices[vertices.Length - 1] = Vector3.up;
for (int i = 0; i < 4; i++)
{
triangles[t++] = vBottom;
triangles[t++] = v;
triangles[t++] = ++vBottom;
}
}
private static int CreateVertexLine(Vector3 from, Vector3 to, int steps, int v, Vector3[] vertices)
{
for (int i = 1; i <= steps; i++)
{
vertices[v++] = Vector3.Lerp(from, to, (float)i / steps);
}
return v;
}
private static int CreateLowerStrip(int steps, int vTop, int vBottom, int t, int[] triangles)
{
for (int i = 1; i < steps; i++)
{
triangles[t++] = vBottom;
triangles[t++] = vTop - 1;
triangles[t++] = vTop;
triangles[t++] = vBottom++;
triangles[t++] = vTop++;
triangles[t++] = vBottom;
}
triangles[t++] = vBottom;
triangles[t++] = vTop - 1;
triangles[t++] = vTop;
return t;
}
private static int CreateUpperStrip(int steps, int vTop, int vBottom, int t, int[] triangles)
{
triangles[t++] = vBottom;
triangles[t++] = vTop - 1;
triangles[t++] = ++vBottom;
for (int i = 1; i <= steps; i++)
{
triangles[t++] = vTop - 1;
triangles[t++] = vTop;
triangles[t++] = vBottom;
triangles[t++] = vBottom;
triangles[t++] = vTop++;
triangles[t++] = ++vBottom;
}
return t;
}
private static void Normalize(Vector3[] vertices, Vector3[] normals)
{
for (int i = 0; i < vertices.Length; i++)
{
normals[i] = vertices[i] = vertices[i].normalized;
}
}
}
最終效果如圖:
我夢寐以求的球啊,你終於出來了,也許你們有更方便的演算法,也許有些步驟寫的不詳細,你們可以自己嘗試體驗下,樓主笨笨的。算了九張草稿紙,才算理解。 這是樓主的草稿紙:
慢慢噴吧,歡迎多多指教!