three.js - How does the calculation of the light model work in a shader program?

Question

I am trying to read up this tutorial:

https://aerotwist.com/tutorials/an-introduction-to-shaders-part-2/

but I am not able to follow up. Basically the code creates a directional light by using shaders that run directly on the GPU. This is the code:

// same name and type as VS
varying vec3 vNormal;

void main() {

    // calc the dot product and clamp
    // 0 -> 1 rather than -1 -> 1
    vec3 light = vec3(0.5,0.2,1.0);

    // ensure it's normalized
    light = normalize(light);

    // calculate the dot product of
    // the light to the vertex normal
    float dProd = max(0.0, dot(vNormal, light));

    // feed into our frag colour
    gl_FragColor = vec4(dProd, dProd, dProd, 1.0);

} 

Specifically, the line that I don't understand is this one:

float dProd = max(0.0, dot(vNormal, light));

How does dot product of vNormal of a vertex and light create a directional light. Can anybody explain me diagrammatically. I am not able to get it . This looks bit of magic to me. I know in this vertex shader each vertex is being passed as an input which is called normal because it is represent in terms of "1" and that shared variable is then used in the above fragment shader code. But apart from this I didn't understand how it works.

P.S: I could have asked to the blog writer but he is on 2 weeks holiday as I know. So I thought someone with some physics or three.js experience might be able to tell me.

Answer 1

Lambertian reflectance model

To model the reflection of light in computer graphics is used a Bidirectional reflectance distribution function (BRDF). BRDF is a function that gives the relation between the light reflected along an outgoing direction and the light incident from an incoming direction.

A perfect diffuse surface has a BRDF that has the same value for all incident and outgoing directions. This substantially reduces the computations and thus it is commonly used to model diffuse surfaces as it is physically plausible, even though there are no pure diffuse materials in the real world. This BRDF is called Lambertian reflection because it obeys Lambert's cosine law.

Lambertian reflection is often used as a model for diffuse reflection. This technique causes all closed polygons (such as a triangle within a 3D mesh) to reflect light equally in all directions when rendered The diffusion coefficient is calculated from the angle between the normal vector and the light vector.

f_Lambertian = max( 0.0, dot( N, L )

where N is the normal vector of the surface, and L is the vector towards to the light source.

How it works

In general The dot product of 2 vectors is equal the cosine of the angle between the 2 vectors multiplied by the magnitude (lenght) of both vectors.

dot( A, B ) == length( A ) * length( B ) * cos( angle_A_B ) 

This follows, that the dot product of 2 unit vectors is equal the cosine of the angle between the 2 vectors, because the length of a unit vector is 1.

uA = normalize( A )
uB = normalize( B )
cos( angle_A_B ) == dot( uA, uB )

If we take a look at the cos(x) function between the angles -90° and 90° then we can see that it has a maximum of 1 at an angle of 0° and It goes down to 0 at the angles of 90° and -90°.

This behavior is exactly that what we want for the reflection model. When the nromal vetor of the surface and the diretion to the light source are in the same direction (the angle between is 0°) then we want a maximium of reflection. In contrast, if the vectors a orthonormalized (the angle in between is 90°) then we want a minimum of reflection and we want a smooth and continuous functional running between the two borders of 0° and 90°.

If the light model is calculated in the vertex shader, the reflection is calculated for each corner of the primitive. In between the primitives the reflections are interpolate according to its barycentric coordinates. See the resulting reflections on a spherical surface:

See also:

• GLSL fixed function fragment program replacement
• Phong lighting: add specular lighting separately or with ambient and diffuse?

WebGL example: Lambertian diffuse reflection on a sphere

(function loadscene() {
  
  var gl, progDraw, vp_size;
  var bufSphere = {};
  
  function render(delteMS){

      Camera.create();
      Camera.vp = vp_size;
          
      gl.viewport( 0, 0, vp_size[0], vp_size[1] );
      gl.enable( gl.DEPTH_TEST );
      gl.clearColor( 0.0, 0.0, 0.0, 1.0 );
      gl.clear( gl.COLOR_BUFFER_BIT | gl.DEPTH_BUFFER_BIT );

      // set up draw shader
      ShaderProgram.Use( progDraw.prog );
      ShaderProgram.SetUniformM44( progDraw.prog, "u_projectionMat44", Camera.Perspective() );
      ShaderProgram.SetUniformM44( progDraw.prog, "u_viewMat44", Camera.LookAt() );
      var modelMat = IdentityMat44()
      modelMat = RotateAxis( modelMat, CalcAng( delteMS, 13.0 ), 0 );
      modelMat = RotateAxis( modelMat, CalcAng( delteMS, 17.0 ), 1 );
      ShaderProgram.SetUniformM44( progDraw.prog, "u_modelMat44", modelMat );
      ShaderProgram.SetUniformF3( progDraw.prog, "u_color", [1.0, 0.5, 0.0] );
      ShaderProgram.SetUniformF3( progDraw.prog, "u_lightDir", [-4.0, 0.0, -1.0] )
      
      // draw scene
      VertexBuffer.Draw( bufSphere );

      requestAnimationFrame(render);
  }
  
  function resize() {
      //vp_size = [gl.drawingBufferWidth, gl.drawingBufferHeight];
      vp_size = [window.innerWidth, window.innerHeight]
      canvas.width = vp_size[0];
      canvas.height = vp_size[1];
  }
  
  function initScene() {
  
      canvas = document.getElementById( "canvas");
      gl = canvas.getContext( "experimental-webgl" );
      if ( !gl )
        return null;
  
      progDraw = {}
      progDraw.prog = ShaderProgram.Create( 
        [ { source : "draw-shader-vs", stage : gl.VERTEX_SHADER },
          { source : "draw-shader-fs", stage : gl.FRAGMENT_SHADER }
        ] );
      if ( !progDraw.prog )
          return null;
      progDraw.inPos = gl.getAttribLocation( progDraw.prog, "inPos" );
      
      // create sphere
      var layer_size = 16, circum_size = 32;
      var rad_circum = 1.0;
      var rad_tube = 0.5;
      var sphere_pts = [];
      sphere_pts.push( 0.0, 0.0, -1.0 );
      for ( var i_l = 1; i_l < layer_size; ++ i_l ) {
          var angH = (1.0 - i_l / layer_size) * Math.PI;
          var h = Math.cos( angH );
          var r = Math.sin( angH );
          for ( var i_c = 0; i_c < circum_size; ++ i_c ) {
              var circumX = Math.cos(2 * Math.PI * i_c / circum_size);
              var circumY = Math.sin(2 * Math.PI * i_c / circum_size);
              sphere_pts.push( r * circumX, r * circumY, h );
          }
      }
      sphere_pts.push( 0.0, 0.0, 1.0 );
      var sphere_inx = [];
      for ( var i_c = 0; i_c < circum_size; ++ i_c ) {
          sphere_inx.push( i_c+1, 0, (i_c+1) % circum_size + 1 )
      } 
      for ( var i_l = 0; i_l < layer_size-2; ++ i_l ) {
          var l1 = i_l * circum_size + 1;
          var l2 = (i_l+1) * circum_size + 1
          for ( var i_c = 0; i_c < circum_size; ++ i_c ) {
              var i_n = (i_c+1) % circum_size;
              sphere_inx.push( l1+i_c, l1+i_n, l2+i_c, l1+i_n, l2+i_n, l2+i_c );
          }
      }
      for ( var i_c = 0; i_c < circum_size; ++ i_c ) {
          var i_start = 1 + (layer_size-2) * circum_size;
          var i_n = (i_c+1) % circum_size;
          sphere_inx.push( i_start + i_c, i_start + i_n, sphere_pts.length/3-1 );
      }
      bufSphere = VertexBuffer.Create(
          [ { data : sphere_pts, attrSize : 3, attrLoc : progDraw.inPos } ],
          sphere_inx );
        
      window.onresize = resize;
      resize();
      requestAnimationFrame(render);
  }
  
  function Fract( val ) { 
      return val - Math.trunc( val );
  }
  function CalcAng( deltaTime, intervall ) {
      return Fract( deltaTime / (1000*intervall) ) * 2.0 * Math.PI;
  }
  function CalcMove( deltaTime, intervall, range ) {
      var pos = self.Fract( deltaTime / (1000*intervall) ) * 2.0
      var pos = pos < 1.0 ? pos : (2.0-pos)
      return range[0] + (range[1] - range[0]) * pos;
  }    
  function EllipticalPosition( a, b, angRag ) {
      var a_b = a * a - b * b
      var ea = (a_b <= 0) ? 0 : Math.sqrt( a_b );
      var eb = (a_b >= 0) ? 0 : Math.sqrt( -a_b );
      return [ a * Math.sin( angRag ) - ea, b * Math.cos( angRag ) - eb, 0 ];
  }
  
  glArrayType = typeof Float32Array !="undefined" ? Float32Array : ( typeof WebGLFloatArray != "undefined" ? WebGLFloatArray : Array );
  
  function IdentityMat44() {
    var m = new glArrayType(16);
    m[0]  = 1; m[1]  = 0; m[2]  = 0; m[3]  = 0;
    m[4]  = 0; m[5]  = 1; m[6]  = 0; m[7]  = 0;
    m[8]  = 0; m[9]  = 0; m[10] = 1; m[11] = 0;
    m[12] = 0; m[13] = 0; m[14] = 0; m[15] = 1;
    return m;
  };
  
  function RotateAxis(matA, angRad, axis) {
      var aMap = [ [1, 2], [2, 0], [0, 1] ];
      var a0 = aMap[axis][0], a1 = aMap[axis][1]; 
      var sinAng = Math.sin(angRad), cosAng = Math.cos(angRad);
      var matB = new glArrayType(16);
      for ( var i = 0; i < 16; ++ i ) matB[i] = matA[i];
      for ( var i = 0; i < 3; ++ i ) {
          matB[a0*4+i] = matA[a0*4+i] * cosAng + matA[a1*4+i] * sinAng;
          matB[a1*4+i] = matA[a0*4+i] * -sinAng + matA[a1*4+i] * cosAng;
      }
      return matB;
  }
  
  function Cross( a, b ) { return [ a[1] * b[2] - a[2] * b[1], a[2] * b[0] - a[0] * b[2], a[0] * b[1] - a[1] * b[0], 0.0 ]; }
  function Dot( a, b ) { return a[0]*b[0] + a[1]*b[1] + a[2]*b[2]; }
  function Normalize( v ) {
      var len = Math.sqrt( v[0] * v[0] + v[1] * v[1] + v[2] * v[2] );
      return [ v[0] / len, v[1] / len, v[2] / len ];
  }
  
  var Camera = {};
  Camera.create = function() {
      this.pos    = [0, 1.5, 0.0];
      this.target = [0, 0, 0];
      this.up     = [0, 0, 1];
      this.fov_y  = 90;
      this.vp     = [800, 600];
      this.near   = 0.5;
      this.far    = 100.0;
  }
  Camera.Perspective = function() {
      var fn = this.far + this.near;
      var f_n = this.far - this.near;
      var r = this.vp[0] / this.vp[1];
      var t = 1 / Math.tan( Math.PI * this.fov_y / 360 );
      var m = IdentityMat44();
      m[0]  = t/r; m[1]  = 0; m[2]  =  0;                              m[3]  = 0;
      m[4]  = 0;   m[5]  = t; m[6]  =  0;                              m[7]  = 0;
      m[8]  = 0;   m[9]  = 0; m[10] = -fn / f_n;                       m[11] = -1;
      m[12] = 0;   m[13] = 0; m[14] = -2 * this.far * this.near / f_n; m[15] =  0;
      return m;
  }
  Camera.LookAt = function() {
      var mz = Normalize( [ this.pos[0]-this.target[0], this.pos[1]-this.target[1], this.pos[2]-this.target[2] ] );
      var mx = Normalize( Cross( this.up, mz ) );
      var my = Normalize( Cross( mz, mx ) );
      var tx = Dot( mx, this.pos );
      var ty = Dot( my, this.pos );
      var tz = Dot( [-mz[0], -mz[1], -mz[2]], this.pos ); 
      var m = IdentityMat44();
      m[0]  = mx[0]; m[1]  = my[0]; m[2]  = mz[0]; m[3]  = 0;
      m[4]  = mx[1]; m[5]  = my[1]; m[6]  = mz[1]; m[7]  = 0;
      m[8]  = mx[2]; m[9]  = my[2]; m[10] = mz[2]; m[11] = 0;
      m[12] = tx;    m[13] = ty;    m[14] = tz;    m[15] = 1; 
      return m;
  } 
  
  var ShaderProgram = {};
  ShaderProgram.Create = function( shaderList ) {
      var shaderObjs = [];
      for ( var i_sh = 0; i_sh < shaderList.length; ++ i_sh ) {
          var shderObj = this.CompileShader( shaderList[i_sh].source, shaderList[i_sh].stage );
          if ( shderObj == 0 )
              return 0;
          shaderObjs.push( shderObj );
      }
      var progObj = this.LinkProgram( shaderObjs )
      if ( progObj != 0 ) {
          progObj.attribIndex = {};
          var noOfAttributes = gl.getProgramParameter( progObj, gl.ACTIVE_ATTRIBUTES );
          for ( var i_n = 0; i_n < noOfAttributes; ++ i_n ) {
              var na