Introduction to Computer Graphics with Webgl Classical Viewing

Introduction to Computer Graphics with WebGL

Ed Angel

Classical Viewing

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Classical Viewing

• Viewing requires three basic elements - One or more objects - A viewer with a projection surface - Projectors that go from the object(s) to the projection surface • Classical views are based on the relationship among these elements - The viewer picks up the object and orients it how she would like to see it • Each object is assumed to constructed from flat principal faces - Buildings, polyhedra, manufactured objects

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Planar Geometric Projections

• Standard projections project onto a plane • Projectors are lines that either - converge at a center of projection - are parallel • Such projections preserve lines - but not necessarily angles • Nonplanar projections are needed for applications such as map construction

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1 Classical Projections

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Taxonomy of Planar Geometric Projections

planar geometric projections

parallel perspective

1 point 2 point 3 point multiview axonometric oblique orthographic

isometric dimetric trimetric

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Perspective vs Parallel

• Computer graphics treats all projections the same and implements them with a single pipeline

• Classical viewing developed different techniques for drawing each type of projection

• Fundamental distinction is between parallel and perspective viewing even though mathematically parallel viewing is the limit of perspective viewing

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2 Perspective Projection

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Parallel Projection

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Orthographic Projection

Projectors are orthogonal to projection surface

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3 Multiview Orthographic Projection • Projection plane parallel to principal face • Usually form front, top, side views

isometric (not multiview orthographic view) front in CAD and architecture, we often display three multiviews plus isometric side top

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Axonometric Projections

Allow projection plane to move relative to object

classify by how many angles of a corner of a projected cube are the same θ 1 none: trimetric θ 2 θ 3 two: dimetric three: isometric

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Types of Axonometric Projections

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4 Oblique Projection

Arbitrary relationship between projectors and projection plane

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Advantages and Disadvantages • Can pick the angles to emphasize a particular face - Architecture: plan oblique, elevation oblique • Angles in faces parallel to projection plane are preserved while we can still see “around” side

• In physical world, cannot create with simple camera; possible with bellows camera or special lens (architectural)

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Perspective Projection

Projectors coverge at center of projection

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5 Three-Point Perspective

• No principal face parallel to projection plane • Three vanishing points for cube

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Two-Point Perspective

• On principal direction parallel to projection plane • Two vanishing points for cube

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One-Point Perspective

• One principal face parallel to projection plane • One vanishing point for cube

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Positioning the Camera

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From the Beginning

• In the beginning: - fixed function pipeline - Model-View and Projection Transformation - Predefined frames: model, object, camera, clip, ndc, window • After deprecation - pipeline with programmable shaders - no transformations - clip, ndc window frames • MV.js reintroduces original capabilities

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Computer Viewing

• There are three aspects of the viewing process, all of which can be implemented in the pipeline, - Positioning the camera • Setting the model-view matrix - Selecting a lens • Setting the projection matrix - Clipping • Setting the view volume

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1 The WebGL Camera

• In WebGL, initially the object and camera frames are the same - Default model-view matrix is an identity • The camera is located at origin and points in the negative z direction • WebGL also specifies a default view volume that is a cube with sides of length 2 centered at the origin - Default projection matrix is an identity

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Default Projection

Default projection is orthogonal

clipped out 2

z=0

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Repositioning the Camera Frame • If we want to visualize objects with both positive and negative z values we can either - Move the camera in the positive z direction • Translate the camera frame - Move the objects in the negative z direction • Translate the world frame • Both of these views are equivalent and are determined by the model-view matrix - Need a translation (translate(0.0,0.0,-d);) d > 0

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2 Moving Camera back from Origin

frames after translation by –d d > 0 default frames

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Moving the Camera

• We can move the camera to any desired position by a sequence of rotations and translations • Example: side view - Rotate the camera - Move it away from origin - Model-view matrix C = TR

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WebGL code

• Remember that last transformation specified is first to be applied

// Using MV.js

var t = translate (0.0, 0.0, -d); var ry = rotateY(90.0); var m = mult(t, ry);

var m = mult(translate (0.0, 0.0, -d), rotateY(90.0););

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3 lookAt lookAt(eye, at, up)

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The lookAt Function

• The GLU library contained the function gluLookAt to form the required model-view matrix through a simple interface • Note the need for setting an up direction • Replaced by lookAt() in MV.js - Can concatenate with modeling transformations • Example: isometric view of cube aligned with axes var eye = vec3(1.0, 1.0, 1.0); var at = vec3(0.0, 0.0, 0.0); var up = vec3(0.0, 1.0, 0.0); var mv = lookAt(eye, at, up);

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Other Viewing APIs

• The lookAt function is only one possible API for positioning the camera • Others include - View reference point, view plane normal, view up (PHIGS, GKS-3D) - Yaw, pitch, roll - Elevation, azimuth, twist - Direction angles

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Projection

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Basic Vertex Shader

attribute vec4 vPosition; attribute vec4 vColor; varying vec4 fColor; uniform mat4 modelViewMatrix; uniform mat4 projectionMatrix;

void main() { gl_Position = projectionMatrix*modelViewMatrix*vPosition; fColor = vColor; }

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Projection Normalization

• The default projection in the eye (camera) frame is orthogonal • For points within the default view volume

xp = x yp = y zp = 0 • Most graphics systems use view normalization - All other views are converted to the default view by transformations that determine the projection matrix - Allows use of the same pipeline for all views

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1 Homogeneous Coordinate Representation default orthographic projection

xp = x pp = Mp yp = y zp = 0 wp = 1 M =

In practice, we can let M = I and set the z term to zero later

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Simple Perspective

• Center of projection at the origin • Projection plane z = d, d < 0

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Perspective Equations

Consider top and side views

xp = yp = zp = d

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2 Homogeneous Coordinate Form

consider q = Mp where M =

p = ⇒ q =

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Perspective Division

• However w ≠ 1, so we must divide by w to return from homogeneous coordinates • This perspective division yields

x = y = p p zp = d the desired perspective equations

• We specify the desired clipping volume with MV.js functions that are equivalent to deprecated OpenGL functions

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WebGL Projection

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WebGL Orthogonal Viewing ortho(left,right,bottom,top,near,far)

near and far measured from camera

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ortho1.html

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1 Viewing a Cube

camera

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Sphere Equations

x = rsinθ cosφ y = rsinθ sinφ z = rcosθ

θ constant: circles of constant longitude € φ constant: circles of constant latitude

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Rotating Camera var render = function() { gl.clear( gl.COLOR_BUFFER_BIT | gl.DEPTH_BUFFER_BIT); eye = vec3(radius*Math.sin(theta)*Math.cos(phi), radius*Math.sin(theta)*Math.sin(phi), radius*Math.cos(theta)); modelViewMatrix = lookAt(eye, at , up); projectionMatrix = ortho(left, right, bottom, ytop, near, far); gl.uniformMatrix4fv( modelViewMatrixLoc, false, flatten(modelViewMatrix) ); gl.uniformMatrix4fv( projectionMatrixLoc, false, flatten(projectionMatrix) ); gl.drawArrays( gl.TRIANGLES, 0, numVertices ); requestAnimFrame(render); }

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2 WebGL Perspective frustum(left,right,bottom,top,near,far)

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Using Field of View

With frustum it is often difficult to get the desired view perpective(fovy, aspect, near, far) often provides a better interface

front plane

aspect = w/h

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Computing Matrices

• Compute in JS file, send to vertex shader with gl.uniformMatrix4fv • Dynamic: update in render() or shader

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3 Rotating Camera var render = function(){ gl.clear( gl.COLOR_BUFFER_BIT | gl.DEPTH_BUFFER_BIT); eye = vec3(radius*Math.sin(theta)*Math.cos(phi), radius*Math.sin(theta)*Math.sin(phi), radius*Math.cos(theta)); modelViewMatrix = lookAt(eye, at , up); projectionMatrix = perspective(fovy, aspect, near, far); gl.uniformMatrix4fv( modelViewMatrixLoc, false, flatten(modelViewMatrix) ); gl.uniformMatrix4fv( projectionMatrixLoc, false, flatten(projectionMatrix) ); gl.drawArrays( gl.TRIANGLES, 0, NumVertices ); requestAnimFrame(render); }

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Orthogonal Projection Matrices

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Normalization

• Rather than derive a different projection matrix for each type of projection, we can convert all projections to orthogonal projections with the default view volume

• This strategy allows us to use standard transformations in the pipeline and makes for efficient clipping

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Pipeline View

model-view projection perspective transformation transformation division 4D → 3D nonsingular

clipping projection

3D 2D against default cube →

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1 Notes

• We stay in four-dimensional homogeneous coordinates through both the modelview and projection transformations - Both these transformations are nonsingular - Default to identity matrices (orthogonal view)

• Normalization lets us clip against simple cube regardless of type of projection

• Delay final projection until end - Important for hidden-surface removal to retain depth information as long as possible

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Orthogonal Normalization

ortho(left,right,bottom,top,near,far)

normalization ⇒ find transformation to convert specified clipping volume to default

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Orthogonal Matrix

• Two steps - Move center to origin T(-(left+right)/2, -(bottom+top)/2,(near+far)/2)) - Scale to have sides of length 2 S(2/(left-right),2/(top-bottom),2/(near-far))

# 2 right − left & 0 0 − % right − left right − left ( % ( 2 top + bottom % 0 0 − ( P = ST = % top − bottom top − bottom ( % 2 far + near ( 0 0 % near − far far − near ( % ( $ 0 0 0 1 '

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2 Final Projection

• Set z =0 • Equivalent to the homogeneous coordinate transformation

Morth =

• Hence, general orthogonal projection in 4D is

P = MorthST

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Oblique Projections

• The OpenGL projection functions cannot produce general parallel projections such as

• However if we look at the example of the cube it appears that the cube has been sheared • Oblique Projection = Shear + Orthogonal Projection

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General Shear

top view side view

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3 Shear Matrix

xy shear (z values unchanged)

H(θ,φ) =

Projection matrix

General case:

P = Morth H(θ,φ)

P = Morth STH(θ,φ)

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Equivalency

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Effect on Clipping

• The projection matrix P = STH transforms the original clipping volume to the default clipping volume

object top view z = 1

DOP DOP x = -1 x = 1 far plane z = -1 clipping near plane volume distorted object (projects correctly) 1 Computer Graphics with WebGL © Ed Angel, 2014 2

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Perspective Projection Matrices

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Simple Perspective

Consider a simple perspective with the COP at the origin, the near clipping plane at z = -1, and a 90 degree field of view determined by the planes x = ±z, y = ±z

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Perspective Matrices

Simple projection matrix in homogeneous coordinates

M =

Note that this matrix is independent of the far clipping plane

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1 Generalization

N =

after perspective division, the point (x, y, z, 1) goes to x’’ = x/z y’’ = y/z Z’’ = -(α+β/z) which projects orthogonally to the desired point regardless of α and β 4 Computer Graphics with WebGL © Ed Angel, 2014

Picking α and β

If we pick

α =

β =

the near plane is mapped to z = -1 the far plane is mapped to z =1 and the sides are mapped to x = ± 1, y = ± 1 Hence the new clipping volume is the default clipping volume

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Normalization Transformation

distorted object projects correctly

original clipping volume original object new clipping volume

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2 Normalization and Hidden-Surface Removal • Although our selection of the form of the perspective matrices may appear somewhat arbitrary, it was chosen so that if z1 > z2 in the original clipping volume then the for the transformed points z1’ > z2’ • Thus hidden surface removal works if we first apply the normalization transformation • However, the formula z’’ = -(α+β/z) implies that the distances are distorted by the normalization which can cause numerical problems especially if the near distance is small

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WebGL Perspective

frustum allows for a nonsymmetric viewing frustum (although perspective does not)

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OpenGL Perspective Matrix

• The normalization in frustum requires an initial shear to form a right viewing pyramid, followed by a scaling to get the normalized perspective volume. • Finally, the perspective matrix results in needing only a final orthogonal transformation

P = NSH

our previously defined shear and scale perspective matrix

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3 Why do we do it this way?

• Normalization allows for a single pipeline for both perspective and orthogonal viewing • We stay in four dimensional homogeneous coordinates as long as possible to retain three- dimensional information needed for hidden-surface removal and shading • We simplify clipping

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Perspective Matrices

# 2* near right − left & 0 0 % right − left right − left ( frustum % ( 2* near top + bottom % 0 0 ( P = % top − bottom top − bottom ( % far + near 2* far * near ( 0 0 − − % far − near far − near ( % ( $ 0 0 −1 0 '

# near & perspective 0 0 0 % right ( € % ( near % 0 0 0 ( P = % top ( % far + near 2* far * near ( 0 0 − − % far − near far − near ( % ( $ 0 0 −1 0 '

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Meshes

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Meshes

• Polygonal meshes are the standard method for defining and displaying surfaces - Approximations to curved surfaces - Directly from CAD packages - Subdivision • Most common are quadrilateral and triangular meshes - Triangle strips and fans

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Height Fields • For each (x, z) there is a unique y • Sampling leads to an array of y values

• If yij = f(xi, zj) then the four points (xi, yij, zj), (x(i+1), y(i+1)j, zj), (xi, yi(j+1), zj+1), (x(i+1), y(i+1)(j+1), z(j+1) form a quadrilateral • Display as quadrilateral or triangular mesh using strips

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1 Honolulu Plot Using Line Strips

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Lines on Back and Hidden Faces

sombrero or Mexican hat function (sin πr)/(πr)

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Using Polygons

• We can use four adjacent data points to form a quadrilateral and thus two triangles which can be shaded • Note using a single color for all polygons will not work well - need lighting and shading to see the 3D nature of the data • But what if we want to see the grid? - avoids the need for lighting calculations • We can display each quadrilateral twice - First as two filled white triangles - Second as a black line loop

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2 Hat Using Triangles and Lines

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Polygon Offset

• Even though we draw the polygon first followed by the lines, small numerical errors cause some of fragments on the line to be display behind the corresponding fragment on the triangle • Polygon offset (gl.polygonOffset) moves fragments slightly away from camera • Apply to triangle rendering

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Hat with Polygon Offset

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3 Other Mesh Issues

• How do we construct a mesh from disparate data (unstructured points) • Technologies such as laser scans can produced tens of millions of such points • Delaunay triangulation • Can we use one triangle strip for an entire 2D mesh? • Mesh simplification

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Lighting and Shading

Why we need shading

Suppose we build a model of a blue sphere using many polygons and color it using a single color. We get something like

But we want something that looks like

Shading

• Why does the image of a real sphere look like

• Light-material interactions cause each point to have a different color or shade • Need to consider - Light sources - Material properties - Location of viewer - Surface orientation

1 Scattering

• Light strikes A - Some scattered - Some absorbed • Some of scattered light strikes B - Some scattered - Some absorbed • Some of this scattered light strikes A and so on

Rendering Equation

• The infinite scattering and absorption of light can be described by the rendering equation - Cannot be solved in general - Ray tracing is a special case for perfectly reflecting surfaces • Rendering equation is global and includes - Shadows - Multiple scattering from object to object

Global Effects

shadow

multiple reflection translucent surface

2 Local vs Global Rendering

• Correct shading requires a global calculation involving all objects and light sources - Incompatible with pipeline model which shades each polygon independently (local rendering)

• However, in computer graphics, especially real time graphics, we are happy if things “look right” - Exist many techniques for approximating global effects

Light-Material Interaction

• Light that strikes an object is partially absorbed and partially scattered (reflected)

• The amount reflected determines the color and brightness of the object - A surface appears red under white light because the red component of the light is reflected and the rest is absorbed

• The reflected light is scattered in a manner that depends on the smoothness and orientation of the surface

Light Sources

General light sources are difficult to work with because we must integrate light coming from all points on the source

3 Simple Light Sources

• Point source - Model with position and color - Distant source = infinite distance away (parallel) • Spotlight - Restrict light from ideal point source • Ambient light - Same amount of light everywhere in scene - Can model contribution of many sources and reflecting surfaces

Surface Types

• The smoother a surface, the more reflected light is concentrated in the direction a perfect mirror would reflected the light • A very rough surface scatters light in all directions

smooth surface rough surface

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The Phong Model

Phong Model

• A simple model that can be computed rapidly • Has three components - Diffuse - Specular - Ambient • Uses four vectors - To source - To viewer - Normal - Perfect reflector

Ideal Reflector

• Normal is determined by local orientation • Angle of incidence = angle of relection • The three vectors must be coplanar

r = 2 (l · n ) n - l

1 Lambertian Surface

• Perfectly diffuse reflector • Light scattered equally in all directions • Amount of light reflected is proportional to the vertical component of incoming light

- reflected light ~cos θi

- cos θi = l · n if vectors normalized

- There are also three coefficients, kr, kb, kg that show how much of each color component is reflected • We can also add a 1/d2 term to account for the distance from the light source

Specular Surfaces

• Most surfaces are neither ideal diffusers nor perfectly specular (ideal reflectors) • Smooth surfaces show specular highlights due to incoming light being reflected in directions concentrated close to the direction of a perfect reflection

specular highlight

Modeling Specular Relections

• Phong proposed using a term that dropped off as the angle between the viewer and the ideal reflection increased

α Ir ~ ks I cos φ

φ reflected shininess coef intensity incoming intensity absorption coef

2 The Shininess Coefficient

• Values of α between 100 and 200 correspond to metals • Values between 5 and 10 give surface that look like plastic

cosα φ

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Applying the Phong Model

Ambient Light

• Ambient light is the result of multiple interactions between (large) light sources and the objects in the environment

• Amount and color depend on both the color of the light(s) and the material properties of the object

• Add ka Ia to diffuse and specular terms

intensity of ambient light reflection coef

Distance Terms

• The light from a point source that reaches a surface is inversely proportional to the square of the distance between them • We can add a factor of the form 1/(a + bd +cd2) to the diffuse and specular terms • The constant and linear terms soften the effect of the point source

1 Light Sources

• In the Phong Model, we add the results from each light source • Each light source has separate diffuse, specular, and ambient terms to allow for maximum flexibility even though this form does not have a physical justification • Separate red, green and blue components • Hence, 9 coefficients for each point source

- Idr, Idg, Idb, Isr, Isg, Isb, Iar, Iag, Iab

Material Properties

• Material properties match light source properties - Nine absorbtion coefficients

• kdr, kdg, kdb, ksr, ksg, ksb, kar, kag, kab - Shininess coefficient α

Adding Up the Components

For each light source and each color component, the Phong model can be written (without the distance terms) as

α I =kd Id l · n + ks Is (v · r ) + ka Ia

For each color component we add contributions from all sources

2 Modified Phong Model

• The specular term in the Phong model is problematic because it requires the calculation of a new reflection vector and view vector for each vertex • Blinn suggested an approximation using the halfway vector that is more efficient

The Halfway Vector

• h is normalized vector halfway between l and v

h = ( l + v )/ | l + v |

Using the halfway vector

• Replace (v · r )α by (n · h )β • β is chosen to match shininess • Note that halfway angle is half of angle between r and v if vectors are coplanar • Resulting model is known as the modified Phong or Phong-Blinn lighting model - Specified in OpenGL standard

3 Example

Only differences in these teapots are the parameters in the modified Phong model

Computation of Vectors

• l and v are specified by the application • Can computer r from l and n • Problem is determining n • For simple surfaces it can be determined but how we determine n differs depending on underlying representation of surface • WebGL leaves determination of normals to application

Plane Normals

• Equation of plane: ax+by+cz+d = 0

• We know that plane is determined by three points p0, p2, p3 or normal n and p0 • Normal can be obtained by

n = (p2-p0) × (p1-p0)

4 Normal to Sphere

• Implicit function f(x,y.z)=0 • Normal given by gradient • Sphere f(p)=p·p-1 n = [∂f/∂x, ∂f/∂y, ∂f/∂z]T=p

Computing Reflection Direction

• Angle of incidence = angle of reflection • Normal, light direction and reflection direction are coplaner • Want all three to be unit length

r = 2(l • n)n − l