Coordinate

• 좌표공간

유니티 정의 positions

• 동차좌표 homogeneous coordinate
  • 동차좌표는 n차원의 직교좌표를 n+1차원으로 확장한 것이다
  • float3 => float4로 되면서 Homogeneous를 붙임
• Perspective-Correct Interpolation을 위해 Normalized Device Coordinate (Homogeneous)를 사용함.

    uint2  positionSS;  // Screen space pixel coordinates                       : [0, NumPixels)
    uint2  tileCoord;   // Screen tile coordinates                              : [0, NumTiles)
    float  deviceDepth; // Depth from the depth buffer                          : [0, 1] (typically reversed)
    float  linearDepth; // View space Z coordinate                              : [Near, Far]

OS  ----------------------- Object Space
 | UNITY_MATRIX_M * OS
WS  ----------------------- World Space
 | UNITY_MATRIX_V * WS
VS  ----------------------- View Space
 | UNITY_MATRIX_P * VS
CS  ----------------------- Clip Space (Homogeneous)
 | NDC    = CS * 0.5
 | NDC.x  =  NDC.x + NDC.w
 | NDC.y  =  NDC.y + NDC.w // DirectX
 | NDC.y  = -NDC.y + NDC.w // OpenGL
 | NDC.zw = CS.zw
NDC ---------------------- Normalized Device Coordinate (Homogeneous) [0..w]
 | pd = (NDC.xyz / NDC.w); // [0, 1] : perspective divide
 | ndc = pd * 2.0 - 1.0;   // [-1, 1]
 |
 |
ndc ---------------------- Normalized Device Coordinate (Nonhomogeneous) [-1..1]
 | float2 uv_Screen = positionNDC.xy / positionNDC.w;
 | float2 uv_Screen = GetNormalizedScreenSpaceUV(positionCS);
uv_Screen ---------------- [0, 1]
 | float2 positionScreen = uv_Screen * _ScreenParams.xy;
positionScreen ----------- [0, screenWidth] / [0, screenHeight]

Transform

• com.unity.render-pipelines.core/ShaderLibrary/SpaceTransforms.hlsl

---
title: Positions
---

flowchart LR
    ObjectPos --TransformObjectToWorld--> WorldPos
    ObjectPos --TransformObjectToHClip--> HClipPos

    WorldPos --TransformWorldToObject--> ObjectPos
    WorldPos --TransformWorldToView--> ViewPos
    WorldPos --TransformWorldToHClip--> HClipPos

    ViewPos --TransformViewToWorld--> WorldPos
    ViewPos --TransformWViewToHClip--> HClipPos

---
title: Directions
---

flowchart LR
    ObjectDir --TransformObjectToTangent--> TangentDir
    ObjectDir --TransformObjectToWorldDir--> WorldDir

    TangentDir --TransformTangentToObject--> ObjectDir
    TangentDir --TransformTangentToWorldDir --> WorldDir

    WorldDir --TransformWorldToTangentDir--> TangentDir
    WorldDir --TransformWorldToObjectDir--> ObjectDir
    WorldDir --TransformWorldToViewDir--> ViewDir
    WorldDir --TransformWorldToHClipDir--> HClipDir


    ViewDir --TransformViewToWorldDir--> WorldDir

---
title: Surface Normals
---

flowchart LR
    ObjectNormal --TransformObjectToWorldNormal--> WorldNormal
    
    WorldNormal --TransformWorldToViewNormal--> ViewNormal
    WorldNormal --TransformWorldToTangent--> TangentNormal
    WorldNormal --TransformWorldToObjectNormal--> ObjectNormal
    
    ViewNormal --TransformViewToWorldNormal--> WorldNormal
    
    TangentNormal --TransformTangentToWorld --> WorldNormal

UNITY_MATRIX

• localToWorldMatrix
  • 유니티 4까지는 GPU에 넘겨주기전에 스케일을 가공하여
  • renderer.localToWorldMatrix, transform.localToWorldMatrix가 달랐으나 지금은 같음.

Pserspective Camera

• https://chengkehan.github.io/ReconstructPositionFromDepth.html

// Find our current location in the camera's projection space.
Vector3 pt = Camera.main.projectionMatrix.MultiplyPoint(transform.position);

// Matrix4x4.MultiplyPoint
public Vector3 MultiplyPoint(Matrix4x4 mat, Vector3 v)
{
    Vector3 result;
    result.x = mat.m00 * v.x + mat.m01 * v.y + mat.m02 * v.z + mat.m03;
    result.y = mat.m10 * v.x + mat.m11 * v.y + mat.m12 * v.z + mat.m13;
    result.z = mat.m20 * v.x + mat.m21 * v.y + mat.m22 * v.z + mat.m23;
    float num = mat.m30 * v.x + mat.m31 * v.y + mat.m32 * v.z + mat.m33;
    num = 1 / num;
    result.x *= num;
    result.y *= num;
    result.z *= num;
    return result;
}

// z값 구하지 않으면
public Vector3 MultiplyPoint(Matrix4x4 mat, Vector3 v)
{
    Vector3 result;
    result.x = mat.m00 * v.x + mat.m01 * v.y + mat.m02 * v.z + mat.m03;
    result.y = mat.m10 * v.x + mat.m11 * v.y + mat.m12 * v.z + mat.m13;
    float num = mat.m30 * v.x + mat.m31 * v.y + mat.m32 * v.z + mat.m33;
    num = 1 / num;
    result.x *= num;
    result.y *= num;
    return result;
}

// 값을 대입하면
public Vector3 MultiplyPoint(Matrix4x4 mat, Vector3 v)
{
    Vector3 result;
    result.x = mat.m00 * v.x + 0 * v.y + 0 * v.z + 0;
    result.y = 0 * v.x + mat.m11 * v.y + 0 * v.z + 0;
    float num = 0 * v.x + 0 * v.y + -1 * v.z + 0;
    num = 1 / num;
    result.x *= num;
    result.y *= num;
    return result;
}

// 최종적으로
public Vector3 MultiplyPoint(Matrix4x4 mat, Vector3 v)
{
    Vector3 result;
    result.x = mat.m00 * v.x;
    result.y = mat.m11 * v.y;
    float num = -1 * v.z;
    num = 1 / num;
    result.x *= num;
    result.y *= num;
    return result;
}

(X, Y, linearEyeDepth)
positionNDC // [-1, 1]
X = positionNDC.x * linearEyeDepth / mat.m00
Y = positionNDC.x * linearEyeDepth / mat.m11


The zero-based row-column position:
|  _m00, _m01, _m02, _m03 |
|  _m10, _m11, _m12, _m13 |
|  _m20, _m21, _m22, _m23 |
|  _m30, _m31, _m32, _m33 |

The one-based row-column position:
|   _11,  _12,  _13,  _14 |
|   _21,  _22,  _23,  _24 |
|   _31,  _32,  _33,  _34 |
|   _41,  _42,  _43,  _44 |

UV

• texel(TExture + piXEL) coordinate

Direct X
(0,0)            (1,0)
  +-------+-------+
  |       |       |
  |       |       |
  +-------+-------+
  |       |       |
  |       |       |
  +-------+-------+
(0,1)            (1,1)


OpenGL / UnityEngine
(0,1)            (1,1)
  +-------+-------+
  |       |       |
  |       |       |
  +-------+-------+
  |       |       |
  |       |       |
  +-------+-------+
(0,0)            (1,0)

• 수학적으로 바라보면 모든 2D좌표계를 OpenGL방식으로하면 좌표계를 헷갈릴 걱정이 없다. 하지만, 프로그래밍 하는 입장에서는 DirectX방식이 좀 더 와닿을 것이다.

Ref

• Computergrafik - Vorlesung 6 - Coordinate Systems
• Unity - shader의 World matrix(unity_ObjectToWorld)를 수작업으로 구성
• Unity - shader의 Camera matrix(UNITY_MATRIX_V)를 수작업으로 구성
• Unity - unity_CameraWorldClipPlanes 내장 변수 의미
• Unity - shader의 원근 투영(Perspective projection) 행렬(UNITY_MATRIX_P)을 수작업으로 구성
• 렌더링 파이프라인의 좌표 공간들
• Look At Transformation Matrix in Vertex Shader
• transform.localToWorldMatrix
• Renderer.localToWorldMatrix
• Camera.worldToCameraMatrix
• Camera.projectionMatrix
• GL.GetGPUProjectionMatrix
• http://blog.hvidtfeldts.net/index.php/2014/01/combining-ray-tracing-and-polygons/