Computer Graphics as Virtual Photography real camera photo Photographic Photography: Ray Tracing Basics I scene (captures processing print light) processing camera Computer 3D tone synthetic model Graphics: models image reproduction (focuses simulated lighting) Ray Tracing in the real world Backwards Ray Tracing Light is emitted from light source Turner Whitted Bounces off of the environment Light rays are traced backward from the eye (center of projection), through a Assumptions viewing plane, into scene to see what it Light travels in straight rays hits. Path of light changes based on object interaction. Can simulate using basic geometry. The pixel is then set to the color values Some light will reach and be focused by returned by the ray. camera onto film plane. This color is a result of the object hit by Lots of light will not! the ray. In image synthesis, we are only interested in the light that does Ray Tracing - Basics Insert Cheesy Ray Tracing Movie Here Sometimes you don’t hit an object 1
Ray Tracing - Basics Ray Tracing - Basics If you do hit an object, additional rays are spawned and sent into world to determine color at intersection point Shadow ray Reflected ray Transmitted ray Sometimes you do Ray Tracing - Basics Ray Tracing - Basics Shadow ray Shadow ray Ray spawned toward each light source to see if point is in shadow. Ray Tracing Ray Tracing Reflective Ray Transmitted ray 2
Recursive Ray Tracing Ray Tracing Ray Tracing incorporates into a single framework: Hidden surface removal Shadow computation Reflection of light Refraction of light Global Specular Interaction Extremely elegant and compact Ray Tracing Basics Ray Tracing Assignment Basic Ray Tracing -- Example For Checkpoint 2: Trace rays through camera model Using ray tracing for visible surface determination. Questions -- Break Whitted Introducing Ray Ray Tracing through the Camera Use mathematical description of a ray Issues and objects to determine intersection. Ray Geometry Parametric representation of a ray: Object-Ray Intersection Origin of ray, P o = (x o ,y o ,z o ) Projection Direction D = (dx, dy, dz) Ray ( ω ) = P o + ω D If D is normalized, then ω will be the distance from origin of the ray. 3
Ray-Object Intersection Ray-Sphere Intersection Most of the computation in ray tracing The Sphere is determining ray object-intersection A sphere can be defined by: When a ray intersects an object, we Center ( x c , y c , z c ) Radius r need to know: Equation of a point ( x s , y s , z s ) on a sphere: Point of intersection Normal of surface at point of intersection 2 2 2 2 ( x x ) ( y y ) ( z z ) r � + � + � = s c s c s c Ray-Sphere Intersection Ray-Sphere Intersection Using the Quadratic Formula Ray - Sphere Intersection Substituting ray equation for ( x s , y s , z s ) 2 − − B ± B 4 AC We get: ω = A ω 2 + B ω + C = 0 2 A where 2 2 2 A = dx + dy + dz Note: ω must be positive, otherwise the B = 2 ( dx ( x − x ) + dy ( y − y ) + dz ( z − z )) intersection is BEHIND the origin of the ray o c o c o c 2 2 2 2 C = ( x − x ) + ( y − y ) + ( z − z ) − r o c o c o c Ray-Sphere Intersection Ray-Sphere Intersection Note: If D is normalized If B 2 – 4C is: < 0 – no real root, no intersection A = dx 2 + dy 2 + dz 2 = 1 and = 0 – one root, ray intersects at sphere’s surface > 0 – two roots, ray goes through sphere. − ± 2 − B B 4 C ω = 2 Use least positive root 4
Ray-Sphere Intersection Ray-Plane Intersection A plane can be defined by: Once we found a ω i for the point of A normal vector and a point on the plane intersection, the actual point is: It has the equation (x i , y i , z i ) = (x 0 + dx * ω i , y 0 + dy * ω i , z 0 + dz * ω i ) The normal at the point of intersection is: Ax + By + Cz + F = 0 (x n , y n , z n ) = ((x i - x c )/r, (y i - y c )/r, (z i - z c )/r) where P n = (A, B, C) gives the normal and if (We divide by r to normalize!) normalized (A 2 + B 2 + C 2 = 1) , F will the shortest distance to the plane from the origin of world. Ray-Plane Intersection Ray-Plane Intesection If ( P n • D ) is Ray - Plane Intersection For plane with equation: 0 – then ray is parallel to plane, no intersection Ax + By + Cz + F = 0 If ω is Plug in equation for ray and we get < 0 – then the ray intersects behind the − ( Ax + By + Cz + F ) origin of the ray…ignore! ω = o o o = − ( P • P + F ) /( P • D ) n 0 n > 0 – calculate the point of intersection + + Adx Bdy Cdz Ray-Plane Intersection Ray-Polygon Intesection Find the plane in which the polygon Once we found a ω i for the point of sits intersection, the actual point is: Find the point of intersection between (x i , y i , z i ) = (x 0 + dx * ω i , y 0 + dy * ω i , z 0 + dz * ω i ) the ray and the plane And we already have the normal at the If point of intersection is found, see if point of intersection is: it lies within the boundaries of the P n = (A, B, C) polygon. 5
Ray-Polygon Intersection Ray-Polygon Intersection Find the point of intersection between the ray Find the plane in which the polygon sits and the plane A plane can be defined by: Done previously A normal vector and a point See if point of intersection lies within the And has the equation boundaries of the polygon. Ax + By + Cz + F = 0 One algorithm: Draw line from P i to each polygon vertex where P n = (A, B, C) gives the normal and if normalized Measure angles between lines (A 2 + B 2 + C 2 = 1), F will give the shortest distance to Recall: ( A • B ) = | A || B | cos θ the plane from the origin of the world. If sum of angles between lines is 360°, polygon contains P i Other Intersections Ray Tracing through the Camera To add other geometric primitives to Issues your ray tracer Ray Geometry Must mathematically derive the point of Object-Ray Intersection intersection between a ray and geometric Projection primitive. Questions? Projection Ray Tracing through a camera Set up your scene Determine position / orientation of objects in scene. Spawn a ray and send into scene Define ray direction (remember to normalize) Check for closest intersection Calculate and return color Display or save final image 6
Introducing Ray Graphics Pipeline Use mathematical description of a ray and objects to determine intersection. 3D Object 3D World 3D Eye 3D Eye 2D Eye 2D Screen Coordinates Coordinates Coordinates Coordinates Coordinates Coordinates Parametric representation of a ray: Object Viewing 3D Clipping Projection Window to Viewport Origin of ray, P o = (x o ,y o ,z o ) Transformation Transformation Mapping Direction D = (dx, dy, dz) Ray ( ω ) = P o + ω D If D is normalized, then ω will be the distance from origin of the ray. Graphics Pipeline Camera Transformations (u x ,u y ,u z ) are u u u -eye• u coordinates of unit u x y z 3D Object 3D World 3D Eye 3D Eye 2D Eye 2D Screen vector w.r.t. world space Coordinates Coordinates Coordinates Coordinates Coordinates Coordinates v v v -eye• v x y z M = Similar for v, n, -eye• n n n n Object Viewing 3D Clipping Projection Window to Viewport ( eye ) is the origin of x y z Transformation Transformation Mapping view space w.r.t world 0 0 0 1 space If ups are aligned, simply use negative eye location values in the fourth column Projection Spawning rays through camera Note: Projection not required as this will be Coordinate spaces done as part of the ray tracing process Can do in camera space or world space Camera space P p � � � � u x Must transform all objects/lights to camera � � � � P p space � v � � y � M = World space P p � � � � n z Must transform initial rays to world space � � � � 1 1 � � � � 7
Projection in Camera Space Converting to World Space The role of cameras can be described as projecting a 3D scene onto a 2D plane P p � � � � x u � � � � P p y v � � M 1 � � � = P p � � � � z n � � � � 1 1 � � � � Inverting a 4x4 Matrix Tips – World Space Need only transform the location of 1 st “pixel” Code samples from location on image plane and dx, dy, and dz as Graphics Gems you move across and down the plane Ken Perlin Available on Web Will link on DIARY Tips – Calculating Color Displaying your image Find point of intersection You don’t really need the full power of a 3D API to do ray tracing Good Safety tip – only consider intersections if they occur past the image Just need the ability to write color values plane. to pixels If intersection Some of the matrix operation routines may be helpful. Return color Of object 8
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