Unity不規則（規則的亦可）物體表面積和體積計算

這兩個計算，首先想到的是從mesh入手（當然，包圍盒可能更簡單，但是實在太不精确）。

Unity Mesh API：

這裡呢，我們需要的主要是vertices，也就是Mesh的頂點

每三個點是一個面，通過面積公式，計算出每個面的面積S=√[p(p-l1)(p-l2)(p-l3)]（p為半周長）

而體積的思路，依然是通過頂點，每個三角面和中心點組成一個三棱錐，計算出其所在平行六面體的體積，四面體的體積就是其1/6

詳細代碼如下：

/*******************************************************************************
* 版本聲明：v1.0.0
* 類 名 稱：AreaCalculation
* 建立日期：2021-03-31 17:08:28
* 作者名稱：末零
* 功能描述：根據網格擷取表面積和體積
* 修改記錄：
* 
******************************************************************************/

using System;
using UnityEngine;

namespace LastZero.Utility
{
	/// <summary>
    /// 表面積、體積的計算
    /// </summary>
	public static class MeshExtend
    {
        /// <summary>
        /// 擷取表面積
        /// </summary>
        /// <param name="obj">帶有MeshFilter的物體</param>
        /// <param name="callbackError">錯誤回調</param>
        /// <returns>表面積</returns>
        public static float GetArea(this Transform obj, Action callbackError = null)
        {
            Mesh mesh = obj.GetComponent<MeshFilter>().mesh;
            if (mesh == null)
            {
                Debug.LogWarning("There is no 'MeshFilter' component!");
                callbackError?.Invoke();
                return -1;
            }

            Vector3[] vertices = mesh.vertices;
            Vector3 lossyScale = obj.lossyScale;

            float area = 0;
            for (int i = 0; i < mesh.subMeshCount; i++)
            {
                int[] triangles = mesh.GetTriangles(i);
                for (int j = 0; j < triangles.Length; j+=3)
                {
                    area += CalculateTriangleArea(vertices[triangles[j]], vertices[triangles[j + 1]], vertices[triangles[j + 2]], lossyScale);
                }
            }

            return area;
        }

        /// <summary>
        /// 計算三角形面積
        /// </summary>
        /// <param name="point1">頂點1</param>
        /// <param name="point2">頂點2</param>
        /// <param name="point3">頂點3</param>
        /// <returns>面積</returns>
        private static float CalculateTriangleArea(Vector3 point1, Vector3 point2, Vector3 point3, Vector3 lossyScale)
        {
            //計算縮放
            point1 = new Vector3(point1.x * lossyScale.x, point1.y * lossyScale.y, point1.z * lossyScale.z);
            point2 = new Vector3(point2.x * lossyScale.x, point2.y * lossyScale.y, point2.z * lossyScale.z);
            point3 = new Vector3(point3.x * lossyScale.x, point3.y * lossyScale.y, point3.z * lossyScale.z);

            //計算邊長
            float l1 = (point2 - point1).magnitude;
            float l2 = (point3 - point2).magnitude;
            float l3 = (point1 - point3).magnitude;
            float p = (l1 + l2 + l3) * 0.5f;

            //計算面積  S=√[p(p-l1)(p-l2)(p-l3)]（p為半周長）
            return Mathf.Sqrt(p * (p - l1) * (p - l2) * (p - l3));
        }

        /// <summary>
        /// 擷取體積
        /// </summary>
        /// <param name="obj">帶有MeshFilter的物體</param>
        /// <param name="callbackError">錯誤回調</param>
        /// <returns></returns>
        public static float GetVolume(this Transform obj, Action callbackError = null)
        {
            Mesh mesh = obj.GetComponent<MeshFilter>().mesh;
            if (mesh == null)
            {
                Debug.LogWarning("There is no 'MeshFilter' component!");
                callbackError?.Invoke();
                return -1;
            }

            Vector3[] vertices = mesh.vertices;
            Vector3 lossyScale = obj.lossyScale;
            Vector3 o = GetCenter(vertices);

            float volume = 0;
            for (int i = 0; i < mesh.subMeshCount; i++)
            {
                int[] triangles = mesh.GetTriangles(i);
                for (int j = 0; j < triangles.Length; j += 3)
                {
                    volume += CalculateVolumeOfTriangle(vertices[triangles[j]], vertices[triangles[j + 1]], vertices[triangles[j + 2]], o, lossyScale);
                }
            }

            return Mathf.Abs(volume);
        }

        /// <summary>
        /// 擷取中心點
        /// </summary>
        /// <param name="points">頂點</param>
        /// <returns>中心點</returns>
        private static Vector3 GetCenter(Vector3[] points)
        {
            Vector3 center = Vector3.zero;
            for (int i = 0; i < points.Length; i++)
            {
                center += points[i];
            }
            center = center / points.Length;
            return center;
        }

        /// <summary>
        /// 計算一個面和中心點組成三棱錐的體積
        /// </summary>
        /// <param name="point1">頂點1</param>
        /// <param name="point2">頂點2</param>
        /// <param name="point3">頂點3</param>
        /// <param name="center">中心點</param>
        /// <returns></returns>
        private static float CalculateVolumeOfTriangle(Vector3 point1, Vector3 point2, Vector3 point3, Vector3 center, Vector3 lossyScale)
        {
            //計算縮放
            point1 = new Vector3(point1.x * lossyScale.x, point1.y * lossyScale.y, point1.z * lossyScale.z);
            point2 = new Vector3(point2.x * lossyScale.x, point2.y * lossyScale.y, point2.z * lossyScale.z);
            point3 = new Vector3(point3.x * lossyScale.x, point3.y * lossyScale.y, point3.z * lossyScale.z);

            //向量
            Vector3 v1 = point1 - center;
            Vector3 v2 = point2 - center;
            Vector3 v3 = point3 - center;

            //計算體積
            //首先我們求以這三個向量為鄰棱的平行六面體的面積
            //那就是（a×b）·c的絕對值
            //然後四面體的體積是平行六面體的六分之一
            //因為四面體的底是平行六面體的一半,而且要多乘一個三分之一
            float v = Vector3.Dot(Vector3.Cross(v1, v2), v3) / 6f;
            return v;
        }
    }
}