Last Time Visibility Z Buffer and transparency A buffer Area subdivision BSP Trees Exact Cell Portal Project 2 11 04 04 University of Wisconsin CS559 Fall 2004 Today Lighting and Shading Part 1 11 04 04 University of Wisconsin CS559 Fall 2004 Where We Stand So far we know how to Transform between spaces Draw polygons Decide what s in front Next Deciding a pixel s intensity and color 11 04 04 University of Wisconsin CS559 Fall 2004 Normal Vectors The intensity of a surface depends on its orientation with respect to the light and the viewer The surface normal vector describes the orientation of the surface at a point Mathematically Vector that is perpendicular to the tangent plane of the surface What s the problem with this definition Just the normal vector or the normal Will use n or N to denote Normals are either supplied by the user or automatically computed 11 04 04 University of Wisconsin CS559 Fall 2004 Transforming Normal Vectors Normal vectors are directions Normal vectors are perpedicul ar to tangent v ectors n x p 0 There is a matrix form of this n t x p 0 Consider t he equation w ith a transform ed tangent n t T 1T x p 0 The right hand half is the transform ed point The new transpose normal must be equal to n t T 1 The new normal must then be n t T 1 t T 1 t n To transform a normal multiply it by the inverse transpose of the transformation matrix Recall rotation matrices are their own inverse transpose Don t include the translation Use nx ny nz 0 for homogeneous coordinates 11 04 04 University of Wisconsin CS559 Fall 2004 Local Shading Models Local shading models provide a way to determine the intensity and color of a point on a surface The models are local because they don t consider other objects We use them because they are fast and simple to compute They do not require knowledge of the entire scene only the current piece of surface Why is this good for hardware For the moment assume We are applying these computations at a particular point on a surface We have a normal vector for that point 11 04 04 University of Wisconsin CS559 Fall 2004 Local Shading Models What they capture Direct illumination from light sources Diffuse and Specular reflections Very Approximate effects of global lighting What they don t do 11 04 04 Shadows Mirrors Refraction Lots of other stuff University of Wisconsin CS559 Fall 2004 Standard Lighting Model Consists of three terms linearly combined Diffuse component for the amount of incoming light from a point source reflected equally in all directions Specular component for the amount of light from a point source reflected in a mirror like fashion Ambient term to approximate light arriving via other surfaces 11 04 04 University of Wisconsin CS559 Fall 2004 Diffuse Illumination kd I i L N Incoming light Ii from direction L is reflected equally in all directions No dependence on viewing direction Amount of light reflected depends on Angle of surface with respect to light source Actually determines how much light is collected by the surface to then be reflected Diffuse reflectance coefficient of the surface kd Don t want to illuminate back side Use 11 04 04 kd I i max L N 0 University of Wisconsin CS559 Fall 2004 Diffuse Example Where is the light Which point is brightest how is the normal at the brightest point related to the light 11 04 04 University of Wisconsin CS559 Fall 2004 Illustrating Shading Models Show the polar graph of the amount of light leaving for a given incoming direction Diffuse Show the intensity of each point on a surface for a given light position or direction Diffuse 11 04 04 University of Wisconsin CS559 Fall 2004 Specular Reflection Phong Reflectance Model L V R k s I i R V p Incoming light is reflected primarily in the mirror direction R Perceived intensity depends on the relationship between the viewing direction V and the mirror direction Bright spot is called a specularity Intensity controlled by The specular reflectance coefficient ks The Phong Exponent p controls the apparent size of the specularity Higher n smaller highlight 11 04 04 University of Wisconsin CS559 Fall 2004 Specular Example 11 04 04 University of Wisconsin CS559 Fall 2004 Illustrating Shading Models Show the polar graph of the amount of light leaving for a given incoming direction Specular Show the intensity of each point on a surface for a given light position or direction Specular 11 04 04 University of Wisconsin CS559 Fall 2004 Specular Reflection Improvement H L V L V L H N V k s I i H N p Compute based on normal vector and halfway vector H Always positive when the light and eye are above the tangent plane Not quite the same result as the other formulation need 2H 11 04 04 University of Wisconsin CS559 Fall 2004 Putting It Together I k a I a I i k d L N k s H N p Global ambient intensity Ia Gross approximation to light bouncing around of all other surfaces Modulated by ambient reflectance ka Just sum all the terms If there are multiple lights sum contributions from each light Several variations and approximations 11 04 04 University of Wisconsin CS559 Fall 2004 Color I r ka r I a r I i r kd r L N k s r H N n Do everything for three colors r g and b Note that some terms the expensive ones are constant Using only three colors is an approximation but few graphics practitioners realize it k terms depend on wavelength should compute for continuous spectrum Aliasing in color space Better results use 9 color samples 11 04 04 University of Wisconsin CS559 Fall 2004 Approximations for Speed The viewer direction V and the light direction L depend on the surface position being considered x Distant light approximation Assume L is constant for all x Good approximation if light is distant such as sun Distant viewer approximation Assume V is constant for all x Rarely good but only affects specularities 11 04 04 University of Wisconsin CS559 Fall 2004 Distant Light Approximation Distant light approximation Assume L is constant for all x Good approximation if light is distant such as sun Generally called a directional light source What aspects of surface appearance are affected by this approximation Diffuse Specular 11 04 04 University of Wisconsin CS559 Fall 2004 Local Viewer Approximation Specularities require the viewing direction V x c x Slightly expensive to compute Local viewer approximation uses a global V Independent of which point is being lit Use the view plane normal vector Error depends on the nature of the scene Is the diffuse component affected 11
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