在unity向量空间内绘制几何(2)：计算球体的表面坐标

上一篇文章：在unity向量空间内通过将极坐标转换为直角坐标，绘制阿基米德螺线，对数螺线与玫瑰线等几何图形，研究了如何通过极坐标寻找2D平面中一些几何图形上的连续坐标点。

今天进一步研究一下如何获取一个3D球面表面上的连续坐标点。关键词：极坐标，黑客帝国，五毛。

Deus Ex Machina

在电影Matrix黑客帝国第三部“矩阵革命”的最后部分，有一段Neo独闯机器城与机器世界首领交涉的情节。

（电影截图：Neo独面机器世界主宰）

这个机器首领的名字叫做Deus Ex Machina，拉丁文翻译成英文是God out of the machine，是机器城市的中枢，控制着未来世界中的所有机器。外形简单来说就是一个胖球加一堆刺。

下文将会尝试在Unity向量空间内，通过将球面的极坐标公式转化成代码模拟重构这个“上帝机器”。

通过极坐标求球体表面坐标点

首先先了解三维空间内的球面的极坐标公式，假如有以下三维空间坐标系：

那么，球面sphere上的任意一点的三维坐标x,y,z,可以用以下极坐标公式表示：
x=r∗sin⁡θ∗cos⁡ϕx=r*\sin\theta*\cos\phix=r∗sinθ∗cosϕ
y=r∗sin⁡θ∗sin⁡ϕy=r*\sin\theta*\sin\phiy=r∗sinθ∗sinϕ
y=r∗cos⁡θy=r*\cos\thetay=r∗cosθ
(公式-1)

有了上一篇文章的基础，我们用相似的思想把公式转换为一个可以返回连续xy坐标点的C#函数：

 public Vector3[] GenerateSSphereVectro3(int radius,Vector3 centre,int thetaPortion,int phiPortion){Vector3[] coordinates=new Vector3[thetaPortion*phiPortion];float thetaMultiplier=360f/(float)thetaPortion;float phiMultiplier=360f/(float)phiPortion;for(int i=0,index=0;i<thetaPortion;i++){for(int j=0;j<phiPortion;j++){coordinates[index]=new Vector3(radius*Mathf.Sin(i*thetaMultiplier)*Mathf.Cos(j*phiMultiplier),radius*Mathf.Sin(i*thetaMultiplier)*Mathf.Sin(j*phiMultiplier),radius*Mathf.Cos(i*thetaMultiplier));index+=1;}}return coordinates;}

参数说明：
int radius:球体半径。
Vector3 centre:球体圆心。
int thetaPortion:将θ\thetaθ角平均分成的份数。
int phiPortion:将ϕ\phiϕ角平均分成的份数。

返回值说明：
Vector3 coordinates:一组分布在球面的三维坐标向量。数量为thetaPortion*phiPortion。

函数解析：
1，根据传进的thetaPortion与phiPortion参数声明一组三维坐标数组

Vector3[] coordinates=new Vector3[thetaPortion*phiPortion];

2,根据传进的thetaPortion与phiPortion参数计算坐标点在极坐标公式中相差的角度。

float thetaMultiplier=360f/(float)thetaPortion;
float phiMultiplier=360f/(float)phiPortion;

3，通过两个for循环，依次寻找球面上的每个坐标点。

for(int i=0,index=0;i<thetaPortion;i++){for(int j=0;j<phiPortion;j++){coordinates[index]=new Vector3(radius*Mathf.Sin(i*thetaMultiplier)*Mathf.Cos(j*phiMultiplier),radius*Mathf.Sin(i*thetaMultiplier)*Mathf.Sin(j*phiMultiplier),radius*Mathf.Cos(i*thetaMultiplier));index+=1;}
}

在Unity Editor内生成锥形体

（图1：CreateCone生成的锥形体）

Unity自带的几何模型中没有圆锥体。以下是生成圆锥体的代码，搬的砖，源码地址http://wiki.unity3d.com/index.php?title=CreateCone，作者Wolfram Kresse。将此脚本放在/Assets/Editor 文件夹下。然后上方菜单的GameObject/CreatOther里会出现cone。

using UnityEngine;
using UnityEditor;
using System.Collections;// an Editor method to create a cone primitive (so far no end caps)
// the top center is placed at (0/0/0)
// the bottom center is placed at (0/0/length)
// if either one of the radii is 0, the result will be a cone, otherwise a truncated cone
// note you will get inevitable breaks in the smooth shading at cone tips
// note the resulting mesh will be created as an asset in Assets/Editor
// Author: Wolfram Kresse
public class CreateCone : ScriptableWizard {public int numVertices = 10;public float radiusTop = 0f;public float radiusBottom = 1f;public float length = 1f;public float openingAngle = 0f; // if >0, create a cone with this angle by setting radiusTop to 0, and adjust radiusBottom according to length;public bool outside = true;public bool inside = false;public bool addCollider = false;[MenuItem ("GameObject/Create Other/Cone")]static void CreateWizard(){ScriptableWizard.DisplayWizard("Create Cone", typeof(CreateCone));}void OnWizardCreate(){GameObject newCone=new GameObject("Cone");if(openingAngle>0&&openingAngle<180){radiusTop=0;radiusBottom=length*Mathf.Tan(openingAngle*Mathf.Deg2Rad/2);}string meshName = newCone.name + numVertices + "v" + radiusTop + "t" + radiusBottom + "b" + length + "l" + length + (outside?"o":"") + (inside?"i":"");string meshPrefabPath = "Assets/Editor/" + meshName + ".asset";Mesh mesh = (Mesh)AssetDatabase.LoadAssetAtPath(meshPrefabPath, typeof(Mesh));if(mesh==null){mesh=new Mesh();mesh.name=meshName;// can't access Camera.current//newCone.transform.position = Camera.current.transform.position + Camera.current.transform.forward * 5.0f;int multiplier=(outside?1:0)+(inside?1:0);int offset=(outside&&inside?2*numVertices:0);Vector3[] vertices=new Vector3[2*multiplier*numVertices]; // 0..n-1: top, n..2n-1: bottomVector3[] normals=new Vector3[2*multiplier*numVertices];Vector2[] uvs=new Vector2[2*multiplier*numVertices];int[] tris;float slope=Mathf.Atan((radiusBottom-radiusTop)/length); // (rad difference)/heightfloat slopeSin=Mathf.Sin(slope);float slopeCos=Mathf.Cos(slope);int i;for(i=0;i<numVertices;i++){float angle=2*Mathf.PI*i/numVertices;float angleSin=Mathf.Sin(angle);float angleCos=Mathf.Cos(angle);float angleHalf=2*Mathf.PI*(i+0.5f)/numVertices; // for degenerated normals at cone tipsfloat angleHalfSin=Mathf.Sin(angleHalf);float angleHalfCos=Mathf.Cos(angleHalf);vertices[i]=new Vector3(radiusTop*angleCos,radiusTop*angleSin,0);vertices[i+numVertices]=new Vector3(radiusBottom*angleCos,radiusBottom*angleSin,length);if(radiusTop==0)normals[i]=new Vector3(angleHalfCos*slopeCos,angleHalfSin*slopeCos,-slopeSin);elsenormals[i]=new Vector3(angleCos*slopeCos,angleSin*slopeCos,-slopeSin);if(radiusBottom==0)normals[i+numVertices]=new Vector3(angleHalfCos*slopeCos,angleHalfSin*slopeCos,-slopeSin);elsenormals[i+numVertices]=new Vector3(angleCos*slopeCos,angleSin*slopeCos,-slopeSin);uvs[i]=new Vector2(1.0f*i/numVertices,1);uvs[i+numVertices]=new Vector2(1.0f*i/numVertices,0);if(outside&&inside){// vertices and uvs are identical on inside and outside, so just copyvertices[i+2*numVertices]=vertices[i];vertices[i+3*numVertices]=vertices[i+numVertices];uvs[i+2*numVertices]=uvs[i];uvs[i+3*numVertices]=uvs[i+numVertices];}if(inside){// invert normalsnormals[i+offset]=-normals[i];normals[i+numVertices+offset]=-normals[i+numVertices];}}mesh.vertices = vertices;mesh.normals = normals;      mesh.uv = uvs;// create triangles// here we need to take care of point order, depending on inside and outsideint cnt=0;if(radiusTop==0){// top conetris=new int[numVertices*3*multiplier];if(outside)for(i=0;i<numVertices;i++){tris[cnt++]=i+numVertices;tris[cnt++]=i;if(i==numVertices-1)tris[cnt++]=numVertices;elsetris[cnt++]=i+1+numVertices;}if(inside)for(i=offset;i<numVertices+offset;i++){tris[cnt++]=i;tris[cnt++]=i+numVertices;if(i==numVertices-1+offset)tris[cnt++]=numVertices+offset;elsetris[cnt++]=i+1+numVertices;}}else if(radiusBottom==0){// bottom conetris=new int[numVertices*3*multiplier];if(outside)for(i=0;i<numVertices;i++){tris[cnt++]=i;if(i==numVertices-1)tris[cnt++]=0;elsetris[cnt++]=i+1;tris[cnt++]=i+numVertices;}if(inside)for(i=offset;i<numVertices+offset;i++){if(i==numVertices-1+offset)tris[cnt++]=offset;elsetris[cnt++]=i+1;tris[cnt++]=i;tris[cnt++]=i+numVertices;}}else{// truncated conetris=new int[numVertices*6*multiplier];if(outside)for(i=0;i<numVertices;i++){int ip1=i+1;if(ip1==numVertices)ip1=0;tris[cnt++]=i;tris[cnt++]=ip1;tris[cnt++]=i+numVertices;tris[cnt++]=ip1+numVertices;tris[cnt++]=i+numVertices;tris[cnt++]=ip1;}if(inside)for(i=offset;i<numVertices+offset;i++){int ip1=i+1;if(ip1==numVertices+offset)ip1=offset;tris[cnt++]=ip1;tris[cnt++]=i;tris[cnt++]=i+numVertices;tris[cnt++]=i+numVertices;tris[cnt++]=ip1+numVertices;tris[cnt++]=ip1;}}mesh.triangles = tris;      AssetDatabase.CreateAsset(mesh, meshPrefabPath);AssetDatabase.SaveAssets();}MeshFilter mf=newCone.AddComponent<MeshFilter>();mf.mesh = mesh;newCone.AddComponent<MeshRenderer>();if(addCollider){MeshCollider mc=newCone.AddComponent<MeshCollider>();mc.sharedMesh=mf.sharedMesh;}Selection.activeObject = newCone;}
}

接着实例化thetaPortion*phiPortion个圆锥体，把球面的三维坐标依次赋值给它们的position，调整角度，为了达成动画效果，构造一些空物体作为圆锥体们的父物体。并在Update里面让父物体们进行旋转。

     if(rotateFlag){for(int i=0;i<thetaPortion;){parents[i].transform.Rotate(Vector3.forward,rotateSpeed);i+=2;}for(int i=1;i<thetaPortion;){parents[i].transform.Rotate(Vector3.back,rotateSpeed);i+=2;}}

(图2：立方体版本）

（图3：圆锥体版本）

放进去一些雷电的特效，闪电的中心点在圆心，闪电的终点随机寻找各个圆锥体的尖部。如果有对这个特效感兴趣的，可在AssetStore内搜索Procedural Examples，免费。

 IEnumerator lightingMove(){for(int i=0;i<100;i++){for(int j=0;j<21;j++){lightingEmitters[j].transform.position=Ball[Random.Range(0,1296)];yield return new WaitForSeconds(0.2f);}}}

（图4，加上雷电后）

加些音效，利用DoTween写个摄像机移动路径，录制视频，完成。

 private Vector3[] circlePoints=new Vector3[200];// Use this for initializationvoid Start () {for(int i=0;i<200;i++){circlePoints=Circle(200,18,new Vector3(0f,5f,0f));}transform.position=circlePoints[0];transform.DOPath(circlePoints,130f);}// Update is called once per framevoid Update () {transform.LookAt(Vector3.zero);}

（图5/6，最终五毛特效视频截图）

做好的五毛特效视频地址：http://v.youku.com/v_show/id_XMTU5MTI5OTU0OA==.html

维护日志:
2020-8-2:review