College of Computer and Information Science, Northeastern UniversityFebruary 3, CS G140 Graduate Computer Graphics Prof. Harriet Fell Spring 2006 Lecture 3 - January 30, 2006

College of Computer and Information Science, Northeastern UniversityFebruary 3, Today’s Topics Homogeneous Coordinates Viewing in 2D 3D Transformations Viewing – from Shirley et al. Chapter 7 Recursive Ray Tracing  Reflection  Refraction

College of Computer and Information Science, Northeastern UniversityFebruary 3, Translation y x y x (a, b) This is not a linear transformation. The origin moves. (x, y)  (x+a,y+b)

College of Computer and Information Science, Northeastern UniversityFebruary 3, Homogeneous Coordinates x x y z y Embed the xy-plane in R 3 at z = 1. (x, y)  (x, y, 1)

College of Computer and Information Science, Northeastern UniversityFebruary 3, D Linear Transformations as 3D Matrices Any 2D linear transformation can be represented by a 2x2 matrix or a 3x3 matrix

College of Computer and Information Science, Northeastern UniversityFebruary 3, D Linear Translations as 3D Matrices Any 2D translation can be represented by a 3x3 matrix. This is a 3D shear that acts as a translation on the plane z = 1.

College of Computer and Information Science, Northeastern UniversityFebruary 3, Translation as a Shear x x y z y

College of Computer and Information Science, Northeastern UniversityFebruary 3, D Affine Transformations An affine transformation is any transformation that preserves co-linearity (i.e., all points lying on a line initially still lie on a line after transformation) and ratios of distances (e.g., the midpoint of a line segment remains the midpoint after transformation). With homogeneous coordinates, we can represent all 2D affine transformations as 3D linear transformations. We can then use matrix multiplication to transform objects.

College of Computer and Information Science, Northeastern UniversityFebruary 3, y x Rotation about an Arbitrary Point y x  

College of Computer and Information Science, Northeastern UniversityFebruary 3,  y x Rotation about an Arbitrary Point   T(-c x, -c y ) R(  ) T(c x, c y )

College of Computer and Information Science, Northeastern UniversityFebruary 3, Windowing Transforms (a,b) (A,B) (c,d) (C,D) (C-c,D-d) (A-a,B-b) translate scale translate

College of Computer and Information Science, Northeastern UniversityFebruary 3, D Transformations Remember: A 3D linear transformation can be represented by a 3x3 matrix.

College of Computer and Information Science, Northeastern UniversityFebruary 3, D Affine Transformations

College of Computer and Information Science, Northeastern UniversityFebruary 3, D Rotations

College of Computer and Information Science, Northeastern UniversityFebruary 3, Canonical View Volume (-1,-1,-1) (1,1,1) x y z

College of Computer and Information Science, Northeastern UniversityFebruary 3, (3,2) (0,0) (0,3) Pixel Coordinates y x x = -0.5 y = 4.5 y = x = 4.5

College of Computer and Information Science, Northeastern UniversityFebruary 3, y x (-1,1) (1,-1) reflect Canonical View to Pixels y x (-1,-1) (1,1) x (-n x /2,n y /2) y (n x /2,-n y /2) scale y x (-0.5, n y --0.5) (n x -0.5, -0.5) translate

College of Computer and Information Science, Northeastern UniversityFebruary 3, Orthographic Projection ( l,b,n ) ( r,t,f ) x y z left bottom near right top far

College of Computer and Information Science, Northeastern UniversityFebruary 3, Orthographic Projection Math

College of Computer and Information Science, Northeastern UniversityFebruary 3, Orthographic Projection Math

College of Computer and Information Science, Northeastern UniversityFebruary 3, Arbitrary View Positions g aze t he view-up e ye w = -g ||g|| u = t x w ||t x w|| v = w x u

College of Computer and Information Science, Northeastern UniversityFebruary 3, Arbitrary Position Geometry e v u w (u=r,v=t,w=f) (u=l,v=b,w=n) o z y x

College of Computer and Information Science, Northeastern UniversityFebruary 3, Arbitrary Position Transformation Move e to the origin and align (u, v, w) with (x, y, z). Compute M = M o M v. For each line segment (a,b) p = Ma, q = Mb, drawline(p,q).

College of Computer and Information Science, Northeastern UniversityFebruary 3, Perspective Projection ( l,b,n ) ( r,t,f ) x y z

College of Computer and Information Science, Northeastern UniversityFebruary 3, Lines to Lines

College of Computer and Information Science, Northeastern UniversityFebruary 3, Perspective Geometry g View plane y ysys n z y s = (n/z)y e

College of Computer and Information Science, Northeastern UniversityFebruary 3, Perspective Transformation The perspective transformation should take Where p(n) = n p(f) = f and n  z 1 > z 2  f implies p(z 1 ) > p(z 2 ). P(z) = n + f – fn/z satisfies these requirements.

College of Computer and Information Science, Northeastern UniversityFebruary 3, Perspective Transformation is not a linear transformation. is a linear transformation.

College of Computer and Information Science, Northeastern UniversityFebruary 3, The Whole Truth about Homogeneous Coordinates A point in 2-space corresponds to a line through the origin in 3-space minus the origin itself. A point in 3-space corresponds to a line through the origin in 4-space minus the origin itself.

College of Computer and Information Science, Northeastern UniversityFebruary 3, Homogenize

College of Computer and Information Science, Northeastern UniversityFebruary 3, Perspective Transformation Matrix Compute M = M o M p M v. For each line segment (a,b) p = Ma, q = Mb, drawline(homogenize(p), homogenize ( q)).

College of Computer and Information Science, Northeastern UniversityFebruary 3, Viewing for Ray-Tracing Simplest Views (0, 0, 0) (wd, ht, 0) screen (wd/s, ht/2, d)

A Viewing System for Ray-Tracing g aze e ye near distance = |g| (e x, e y, e z ) = w h perp (w/2, h/2, n) (0, 0, 0) (w, h, 0) |p|j -nk-nk You need a transformation that sends (e x, e y, e z ) to (w/2, h/2, n) g to –nk p to |p|j

College of Computer and Information Science, Northeastern UniversityFebruary 3, Time for a Break

College of Computer and Information Science, Northeastern UniversityFebruary 3, Recursive Ray Tracing Adventures of the 7 Rays - Watt Specular Highlight on Outside of Shere

College of Computer and Information Science, Northeastern UniversityFebruary 3, Adventures of the 7 Rays - Watt Specular Highlight on Inside of Sphere Recursive Ray Tracing

College of Computer and Information Science, Northeastern UniversityFebruary 3, Adventures of the 7 Rays - Watt Reflection and Refraction of Checkerboard Recursive Ray Tracing

College of Computer and Information Science, Northeastern UniversityFebruary 3, Adventures of the 7 Rays - Watt Refraction Hitting Background Recursive Ray Tracing

College of Computer and Information Science, Northeastern UniversityFebruary 3, Adventures of the 7 Rays - Watt Local Diffuse Plus Reflection from Checkerboard Recursive Ray Tracing

College of Computer and Information Science, Northeastern UniversityFebruary 3, Adventures of the 7 Rays - Watt Local Diffuse in Complete Shadow Recursive Ray Tracing

College of Computer and Information Science, Northeastern UniversityFebruary 3, Adventures of the 7 Rays - Watt Local Diffuse in Shadow from Transparent Sphere Recursive Ray Tracing

College of Computer and Information Science, Northeastern UniversityFebruary 3, Recursive Ray-Tracing How do we know which rays to follow? How do we compute those rays? How do we organize code so we can follow all those different rays?

select center of projection(cp) and window on view plane; for (each scan line in the image ) { for (each pixel in scan line ) { determine ray from the cp through the pixel; pixel = RT_trace(ray, 1);}} // intersect ray with objects; compute shade at closest intersection // depth is current depth in ray tree RT_color RT_trace (RT_ray ray; int depth){ determine closest intersection of ray with an object; if (object hit) { compute normal at intersection; return RT_shade (closest object hit, ray, intersection, normal, depth);} else return BACKGROUND_VALUE; }

// Compute shade at point on object, // tracing rays for shadows, reflection, refraction. RT_color RT_shade ( RT_object object,// Object intersected RT_ray ray,// Incident ray RT_point point,// Point of intersection to shade RT_normal normal,// Normal at point int depth )// Depth in ray tree { RT_color color; // Color of ray RT_ray rRay, tRay, sRay;// Reflected, refracted, and shadow ray color = ambient term ; for ( each light ) { sRay = ray from point to light ; if ( dot product of normal and direction to light is positive ){ compute how much light is blocked by opaque and transparent surfaces, and use to scale diffuse and specular terms before adding them to color;}}

if ( depth < maxDepth ) {// return if depth is too deep if ( object is reflective ) { rRay = ray in reflection direction from point; rColor = RT_trace(rRay, depth + 1); scale rColor by specular coefficient and add to color; } if ( object is transparent ) { tRay = ray in refraction direction from point; if ( total internal reflection does not occur ) { tColor = RT_trace(tRay, depth + 1); scale tColor by transmission coefficient and add to color; } return color;// Return the color of the ray }

College of Computer and Information Science, Northeastern UniversityFebruary 3, Computing R I N R   I R L+R L + R = (2 L  N) N R = (2 L  N) N - L ININ ININ

College of Computer and Information Science, Northeastern UniversityFebruary 3, Reflections, no Highlight

College of Computer and Information Science, Northeastern UniversityFebruary 3, Second Order Reflection

College of Computer and Information Science, Northeastern UniversityFebruary 3, Refelction with Highlight

College of Computer and Information Science, Northeastern UniversityFebruary 3, Nine Red Balls

College of Computer and Information Science, Northeastern UniversityFebruary 3, Refraction N -N I T TT II

College of Computer and Information Science, Northeastern UniversityFebruary 3, Refraction and Wavelength N -N I T TT II

College of Computer and Information Science, Northeastern UniversityFebruary 3, Computing T N -N I M T TT II sin(  I ) cos(  I ) I = - sin(  I )M + cos(  I )N T = sin(  T )M - cos(  T )N How do we compute M, sin(  T ), and cos(  T )? We look at this in more detail

College of Computer and Information Science, Northeastern UniversityFebruary 3, Computing T -N T TT II x sin(  T ) cos(  T )

Computing T -N T TT II = x sin(  T ) cos(  T ) II I Parallel to I

College of Computer and Information Science, Northeastern UniversityFebruary 3, Total Internal Reflection When is cos(  T ) defined? Then there is no transmitting ray and we have total internal reflection.

College of Computer and Information Science, Northeastern UniversityFebruary 3, Index of Refraction The speed of all electromagnetic radiation in vacuum is the same, approximately 3×108 meters per second, and is denoted by c. Therefore, if v is the phase velocity of radiation of a specific frequency in a specific material, the refractive index is given byphase velocity

College of Computer and Information Science, Northeastern UniversityFebruary 3, Indices of Refraction Material  at λ=589.3 nm vacuum1 (exactly) helium air at STP water ice1.31 liquid water (20°C)1.333 ethanol1.36 glycerine rock salt1.516 glass (typical)1.5 to 1.9 cubic zirconia2.15 to 2.18 diamond

College of Computer and Information Science, Northeastern UniversityFebruary 3, One Glass Sphere

College of Computer and Information Science, Northeastern UniversityFebruary 3, Five Glass Balls

College of Computer and Information Science, Northeastern UniversityFebruary 3, A Familiar Scene

College of Computer and Information Science, Northeastern UniversityFebruary 3, Bubble

College of Computer and Information Science, Northeastern UniversityFebruary 3, Milky Sphere

College of Computer and Information Science, Northeastern UniversityFebruary 3, Lens - Carl Andrews 1999