Lecture 10: Ray Tracing and Constructive Solid Geometry. Interactive Computer Graphics. Ray tracing with secondary rays. Ray tracing: Shadows

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1 Iteractive Computer Graphics Lecture 10: Ray Tracig ad Costructive Solid Geometry Ray tracig with secodary rays Ray tracig usig just primary rays produces images similar to ormal polygo rederig techiques Recursive ray tracig, with secodary rays produces more realistic images by addig shadows reflectios trasparecy Graphics Lecture 1: Slide 1 Graphics Lecture 1: Slide 2 Ray tracig to fid shadows Ray tracig: Shadows Light source Secodary rays travel towards each light source Illumiatio model with shadows: t [ k ( l ) + k ( v l ' ] I = k I + s I ) a a i i i The sum is take over each light source s i is a delta fuctio: d i s i Viewig plae Primary Ray s i 0 = 1 if light source is obscured if light source is ot obscured Graphics Lecture 1: Slide 3 Graphics Lecture 1: Slide 4 Ray tracig: Reflectio ad Refractio 1. Trace primary ray to determie earest itersectio 2. Cast ew ray i the directio of reflectio or refractio 3. Trace secodary rays like primary rays ad repeat 4. Stop recursio if ray hits light source ray hits backgroud maximum recursio depth is reached Recursive ray tracig Viewig plae Primary Ray Secodary Rays Graphics Lecture 1: Slide 5 Graphics Lecture 1: Slide 6

2 Recursive ray tracig tree Viewer primary ray Collectig the illumiatio Viewer C C reflectio ray trasmissio ray reflectio ray E D reflectio ray E D trasmissio ray F reflectio ray Backgroud F Itesity Backgroud Backgroud Itesity Itesity Graphics Lecture 1: Slide 7 Graphics Lecture 1: Slide 8 Recursive Ray tracig If o object itersects the ray, the ray tracig tree will be empty ad the pixel will be assiged the value of the backgroud. Itesities are accumulated startig from leaf odes upwards to the root ode Itesity from each ode i the tree is atteuated by the distace from the paret ode ad added to the itesity of the paret ode Ray tracig: Reflectios ad trasparecy Illumiatio model with shadows, reflectios ad trasparecy a a i i i t [ kd ( li ) + ks ( v li ' ] + krir ktit I = k I + s I ) + k r is the reflectio coefficiet of the reflected ray I r is the itesity of the reflected ray k t is the trasmissio coefficiet of the trasmitted ray I t is the itesity of the trasmitted ray Graphics Lecture 1: Slide 9 Graphics Lecture 1: Slide 10 Reflectios Reflectios primary ray v φ i φ out secodary ray v To calculate illumiatio as a result of reflectios calculate the directio of the secodary ray at the itersectio of the primary ray with the object. assume that is the surface ormal v is the directio of the primary ray v is the directio of the secodary ray as a result of reflectios v ' = v ( 2v ) Graphics Lecture 1: Slide 11 Graphics Lecture 1: Slide 12

3 Reflectios The v, v ad are uit ad coplaar so: v = α v + β Takig the dot product with yields the eq.: v = α v + β = v Requirig v to be a uit vector yields the secod eq.: 1 = v v = α αβv + β 2 Solvig both equatio yields: v ' = v ( 2v ) Reflectio Perfect reflectio implies that the ray is oly refracted i oe directio. Perfect reflectio is a good approximatio for mirror smooth or polished surfaces but perfect reflectio is a ot good approximatio for rough surfaces ueve surface Reflectio light ca be modeled as a large umber of rays scattered aroud the pricipal directio of reflectio usig a large umber of rays is computatioally impossible usig rays radomly distributed allows the creatio of realistic effects Graphics Lecture 1: Slide 13 Graphics Lecture 1: Slide 14 Reflectios Traslucet Objects primary ray v φ φ secodary ray v Reflected rays are distributed i a specular coe The agle of the refracted ray ca be determied by Sell s law: k 1 si( φ1) = k2 si( φ2) k 1 is a costat for medium 1 k 2 is a costat for medium 2 φ 1 is the agle betwee the icidet ray ad the surface ormal φ 2 is the agle betwee the refracted ray ad the surface ormal v Medium 2 (eg glass) Medium 1 φ 1 (eg air) φ 2 v Graphics Lecture 1: Slide 15 Graphics Lecture 1: Slide 16 Refractio Refractio I vector otatio Sell s law ca be writte: k1( v ) = k2( v' ) The directio of the refracted ray is 2 k 1 2 k2 v' = ( v) v v k 2 k1 This equatio oly has a solutio if ( 2 k2 ) 1 v > k1 This illustrates the physical pheomeo of the limitig agle: if light passes from oe medium to aother medium whose idex of refractio is low, the agle of the refracted ray is greater tha the agle of the icidet ray if the agle of the icidet ray is large, the agle of the refracted ray is larger tha 90 o the ray is reflected rather tha refracted 2 Graphics Lecture 1: Slide 17 Graphics Lecture 1: Slide 18

4 Refractio Refractio Perfect refractio implies that the ray is oly refracted i oe directio. Perfect refractio is a good approximatio for clear glass water but perfect refractio is a ot good approximatio for frosted glass dust Refractio light ca be modeled as a large umber of rays scattered aroud the pricipal directio of refractio usig a large umber of rays is computatioally impossible usig rays radomly distributed allows the creatio of realistic effects v φ 1 φ 2 v Secodary rays are refracted i a specular coe Graphics Lecture 1: Slide 19 Graphics Lecture 1: Slide 20 Mote Carlo Simulatios Mote Carlo simulatios are used to make estimates of variables that caot be computed i real time. For example trasmissio of light through frosted glass or reflectio from dull surfaces Mote Carlo Simulatios To determie the light arrivig i a specular coe we make a radom selectio of rays i that coe ad trace them recursively. The light itesity is set to the average of the secodary rays. The rays would usually be ormally distributed about the cetre Graphics Lecture 1: Slide 21 Graphics Lecture 1: Slide 22 Mote Carlo Simulatios Clearly the more samples we take the better the simulatio, but i a recursive ray tracig cotext computatioal demads will place a limit o this. Graphics Lecture 1: Slide 23 Graphics Lecture 1: Slide 24

5 Costructive Solid Geometry (CSG) Graphics Lecture 1: Slide 25 Graphics Lecture 1: Slide 26 Real ad virtual objects ca be represeted by solid models such as spheres, cyliders ad coes surface models such as triagles, quads ad polygos Surface models ca be redered either by object-order rederig (polygo rederig) image-order rederig (ray tracig) Solid models ca oly be redered by ray tracig Solid models are commoly used to describe mamade shapes computer aided desig computer assisted maufacturig Costructive Solid Geometry (CSG) CSG combies solid objects by usig three (four) differet boolea operatios itersectio ( ) uio (+) mius ( ) (complemet) I theory the mius operatio ca be replaced by a complemet ad itersectio operatio I practice the mius operatio is ofte more ituitive as it correspods to removig a solid volume Costructive Solid Geometry (CSG) Box Sphere Graphics Lecture 1: Slide 27 Graphics Lecture 1: Slide 28 Costructive Solid Geometry (CSG): Uio Costructive Solid Geometry (CSG): Itersectio Graphics Lecture 1: Slide 29 Graphics Lecture 1: Slide 30

6 Costructive Solid Geometry (CSG): Mius CSG trees: Primitive objects Graphics Lecture 1: Slide 31 Graphics Lecture 1: Slide 32 CSG tree of operatios CSG Fial Complex Object Uio Differece Differece Differece Graphics Lecture 1: Slide 33 Graphics Lecture 1: Slide 34 CSG trees CSG operatios are ot commutative: CSG trees CSG operatios are ot uique: + Graphics Lecture 1: Slide 35 Graphics Lecture 1: Slide 36

7 CSG trees: Degeeratio Problems Ray tracig CSG trees CSG trees must be redered by ray tracig CSG trees must be traversed i a depth first maer traversal starts at the leaf odes traversal of each ode yields a list of lie segmets of the ray that pass through the solid object the list of lies segmets is passed to paret ode ad processed accordigly Cube A Cube B = Plae Graphics Lecture 1: Slide 37 Graphics Lecture 1: Slide 38 Example: Ray tracig CSG trees Cocrete example: the viewpoit is at p v = (0, 0, -10) the ray passes through viewig plae at p i = (0, 0, 0). Spheres: Sphere A with ceter p s = (0, 0, 8) ad radius r = 5 Sphere B with ceter p s = (0, 0, 9) ad radius r = 3 Sphere C with ceter p s = (0, -3, 8) ad radius r = 2 Ray tracig CSG trees Calculate the itersectios of the ray for the followig objects, assumig the spheres defied before. a. A + B + C b. A B c. (B A) C d. A + (B C) Y A B Viewplae C Z Graphics Lecture 1: Slide 39 Graphics Lecture 1: Slide 40 Ray tracig CSG trees A + B + C: The ray eters the object at (0, 0, 3) ad exits the object at (0, 0, 13) A B: The ray eters the object at (0, 0, 3), leaves the object at (0, 0, 6), eters the object agai at (0, 0, 12) ad leaves the object at (0, 0, 13). (B A) C: B A produces a empty object list, the ray has therefore o itersectios. A + (B C): The ray eters the object at (0, 0, 3) ad exits the object at (0, 0, 13) Ray tracig CSG trees The itersectios of a ray ad a CSG object tree may be characterized by a list of the µ values of the ray equatio: µ, µ, K, µ ) ( 1 2 Each list of lie segmets will either cotai a odd umber of itersectio poits (the viewpoit is iside the solid object) a eve umber of itersectio poits (the viewpoit is outside the solid object) a empty list of itersectio poits Graphics Lecture 1: Slide 41 Graphics Lecture 1: Slide 42

8 Ray tracig CSG trees Ray tracig CSG trees: Uio + = Graphics Lecture 1: Slide 43 Graphics Lecture 1: Slide 44 Ray tracig CSG trees: Itersectios Ray tracig CSG trees: Mius = = Graphics Lecture 1: Slide 45 Graphics Lecture 1: Slide 46 Ray tracig CSG trees Extedig CSG trees CSG trees ca be prued durig ray tracig: if the left or right subtree of a itersectio operatio returs a empty list, the the other subtree eed ot be processed. if the left subtree of a mius operatio returs a empty list, the the right subtree eed ot be processed. CSG trees ca use boudig boxes/spheres to speed up rederig: each primitive that does ot belog the curretly processed boudig volume may be represeted by a empty itersectio list Graphics Lecture 1: Slide 47 Addig trasformatios as primitive operatios: scalig rotatio Scale traslatio Uio Graphics Lecture 1: Slide 48 Traslatio Sphere Itersectio Box Rotatio Cylider