※この記事のUnityのバージョンはUnity5.3.2 p4です
はじめに
4次元の立方体をUnityで表示させてみました.
考え方
4次元の立方体はそのままでは画面に出すことができないので、3次元空間への写像を作ってそれを画面に出します.
写像は以下のような計算で求めることができます.(レイと3次元空間の各交点を結ぶと3次元の物体になります)
コード
以上をコードにすると以下のようになると思います.
HyperCubeProjector.cs
using UnityEngine;
using System.Collections;
public class HyperCubeProjector : MonoBehaviour
{
public float k = 16f;
public Vector4 cameraRay = new Vector4(1f, 0f, 0f, 1f);
public Vector4 cubePosition = new Vector4(4f, 0f, 0f, 0f);
// 4次元立方体の頂点
private Vector4[] v = new Vector4[]
{
new Vector4(+1f, +1f, +1f, +1f),
new Vector4(+1f, +1f, +1f, -1f),
new Vector4(+1f, +1f, -1f, +1f),
new Vector4(+1f, +1f, -1f, -1f),
new Vector4(+1f, -1f, +1f, +1f),
new Vector4(+1f, -1f, +1f, -1f),
new Vector4(+1f, -1f, -1f, +1f),
new Vector4(+1f, -1f, -1f, -1f),
new Vector4(-1f, +1f, +1f, +1f),
new Vector4(-1f, +1f, +1f, -1f),
new Vector4(-1f, +1f, -1f, +1f),
new Vector4(-1f, +1f, -1f, -1f),
new Vector4(-1f, -1f, +1f, +1f),
new Vector4(-1f, -1f, +1f, -1f),
new Vector4(-1f, -1f, -1f, +1f),
new Vector4(-1f, -1f, -1f, -1f)
};
void OnGUI()
{
// 4次元立方体を画面に出す
Vector3[] vertices = Projection(this.k, this.cameraRay, this.v, this.cubePosition);
DrawHyperCube(vertices);
}
private void DrawLine(Vector3 p1, Vector3 p2, Color color)
{
Debug.DrawLine(this.transform.position + p1, this.transform.position + p2, color);
}
private void DrawHyperCube(Vector3[] vertices)
{
Color colorX = Color.red;
Color colorY = Color.green;
Color colorZ = Color.blue;
Color colorW = Color.yellow;
// X軸に平行な辺
DrawLine(vertices[0], vertices[8], colorX);
DrawLine(vertices[1], vertices[9], colorX);
DrawLine(vertices[2], vertices[10], colorX);
DrawLine(vertices[3], vertices[11], colorX);
DrawLine(vertices[4], vertices[12], colorX);
DrawLine(vertices[5], vertices[13], colorX);
DrawLine(vertices[6], vertices[14], colorX);
DrawLine(vertices[7], vertices[15], colorX);
// Y軸に平行な辺
DrawLine(vertices[0], vertices[4], colorY);
DrawLine(vertices[1], vertices[5], colorY);
DrawLine(vertices[2], vertices[6], colorY);
DrawLine(vertices[3], vertices[7], colorY);
DrawLine(vertices[8], vertices[12], colorY);
DrawLine(vertices[9], vertices[13], colorY);
DrawLine(vertices[10], vertices[14], colorY);
DrawLine(vertices[11], vertices[15], colorY);
// Z軸に平行な辺
DrawLine(vertices[0], vertices[2], colorZ);
DrawLine(vertices[1], vertices[3], colorZ);
DrawLine(vertices[4], vertices[6], colorZ);
DrawLine(vertices[5], vertices[7], colorZ);
DrawLine(vertices[8], vertices[10], colorZ);
DrawLine(vertices[9], vertices[11], colorZ);
DrawLine(vertices[12], vertices[14], colorZ);
DrawLine(vertices[13], vertices[15], colorZ);
// W軸に平行な辺
DrawLine(vertices[0], vertices[1], colorW);
DrawLine(vertices[2], vertices[3], colorW);
DrawLine(vertices[4], vertices[5], colorW);
DrawLine(vertices[6], vertices[7], colorW);
DrawLine(vertices[8], vertices[9], colorW);
DrawLine(vertices[10], vertices[11], colorW);
DrawLine(vertices[12], vertices[13], colorW);
DrawLine(vertices[14], vertices[15], colorW);
}
// 4次元立方体の3次元空間への写像を作る
private static Vector3[] Projection(float k, Vector4 n, Vector4[] v, Vector4 cubePos)
{
Vector3[] p = new Vector3[v.Length]; // 3次元空間とレイの交点
for (int i = 0; i < v.Length; i++)
{
Vector4 ray = v[i] + cubePos;
float L = k * n.sqrMagnitude/ Vector4.Dot(n, ray);
p[i] = L * ray;
}
return p;
}
}
実行結果
4次元立方体をくるくる回す
Projection()の中身を以下のように変えると立方体が回転します
HyperCubeProjector.cs
// 4次元立方体の3次元空間への写像を作る
private static Vector3[] Projection(float k, Vector4 n, Vector4[] v, Vector4 cubePos)
{
// X-W成分を回転させる行列
Matrix4x4 rot = Matrix4x4.identity;
rot.m00 = +Mathf.Cos(Time.time);
rot.m03 = -Mathf.Sin(Time.time);
rot.m30 = +Mathf.Sin(Time.time);
rot.m33 = +Mathf.Cos(Time.time);
Vector3[] p = new Vector3[v.Length]; // 3次元空間とレイの交点
for (int i = 0; i < v.Length; i++)
{
Vector4 ray = rot * v[i] + cubePos;
float L = k * n.sqrMagnitude/ Vector4.Dot(n, ray);
p[i] = L * ray;
}
return p;
}