cs 431/636 advanced rendering techniques

CS 431/636 Advanced Rendering TechniquesDr. David BreenUniversity Crossings 149Tuesday 6PM 8:50PMPresentation 24/7/09

Start UpAny questions from last time?Go over sampling image plane?Or intersection algorithms?

Slide CreditsLeonard McMillan, Seth Teller, Fredo Durand, Barb Cutler - MITG. Drew Kessler, Larry Hodges - Georgia Institute of TechnologyJohn Hart - University of IllinoisRick Parent - Ohio State University

More Geometry & Intersections

Ray/Plane IntersectionRay is defined by R(t) = Ro + Rd*t where t 0Ro = Origin of ray at (xo, yo, zo)Rd = Direction of ray [xd, yd, zd] (unit vector)

Plane is defined by [A, B, C, D]Ax + By + Cz + D = 0 for a point in the planeNormal Vector, N = [A, B, C] (unit vector)A2 + B2 + C2 = 1D = - N P0 (P0 - point in plane)

Ray/Plane (cont.)Substitute the ray equation into the plane equation:A(xo + xdt) + B(yo + ydt) + C(zo +zdt) + D = 0

Solve for t:t = -(Axo + Byo + Czo + D) / (Axd + Byd + Czd)t = -(N Ro - N P0 ) / (N Rd)

What Can Happen?

Ray/Plane SummaryIntersection point:(xi, yi, zi) = (xo + xdti, yo + ydti, zo + zdti)

Calculate N Rd and compare it to zero.Calculate ti and compare it to zero.Compute intersection point.Flip normal if N Rd is positive

Ray-Parallelepiped IntersectionAxis-alignedFrom (X1, Y1, Z1) to (X2, Y2, Z2)Ray P(t)=Ro+Rdty=Y2y=Y1x=X1x=X2RoRd

Nave ray-box IntersectionUse 6 plane equationsCompute all 6 intersectionCheck that points are inside box Ax+By+Cz+D 0

Factoring out computationPairs of planes have the same normalNormals have only one non-0 componentDo computations one dimension at a timeMaintain tnear and tfar (closest and farthest so far)

Test if parallelIf Rdx = 0, then ray is parallelIf Rox < X1 or Rox > x2 return falsey=Y2y=Y1x=X1x=X2RoRd

If not parallelCalculate intersection distance t1 and t2t1 = (X1-Rox)/Rdxt2 = (X2-Rox)/Rdx

Test 1Maintain tnear and tfarIf t1 > t2, swapif t1 > tnear, tnear = t1 if t2 < tfar, tfar = t2If tnear > tfar, box is missed

Test 2If tfar < 0, box is behindy=Y2y=Y1x=X1x=X2RoRdt1xt2xtfar

Algorithm recapDo for all 3 axesCalculate intersection distance t1 and t2Maintain tnear and tfarIf tnear > tfar, box is missedIf tfar < 0, box is behindIf box survived tests, return intersection at tnearIf tnear is negative, return tfar

Ray/Ellipsoid Intersection Ray/Cylinder Intersection Ellipsoid's surface is defined by the set of points {(xs, ys, zs)} satisfying the equation:

(xs/ a)2 + (ys/ b)2 + (zs/ c)2 - 1 = 0

Cylinder's surface is defined by the set of points {(xs, ys, zs)} satisfying the equation:

(xs)2 + (ys)2 - r2 = 0-z0 zs z0

Centers at origin

Substitute ray equation into surface equationsThis is a quadratic equation in t: At2 + Bt + C = 0Analyze as beforeSolve for t with quadratic formulaPlug t back into ray equation - DoneWell, not exactlyRay/Ellipsoid Intersection Ray/Cylinder Intersection

Ray/Cylinder Intersection Is intersection point Pi between -Z0 and Z0?If not, Pi is not validAlso need to do intersection test with z = -Z0 , Z0 planeIf (Pix)2 + (Piy)2 r2, youve intersected a capWhich valid intersection is closer?

Superquadrics

Bezier PatchPatch of order (n, m) can be defined in terms of a set of (n + 1)(m + 1) control points Pi+1,j+1 for integer indices i = 0 to n, j = 0 to m.

Tesselate Patches and Superquadrics

Tesselate Patches and Superquadrics

Utah TeapotModeled by 32 Bzier PatchesControl points available at http://www.holmes3d.net/graphics/teapot

SMF Triangle Meshesv -1 -1 -1v 1 -1 -1 v -1 1 -1 v 1 1 -1 v -1 -1 1 v 1 -1 1 v -1 1 1 v 1 1 1 f 1 3 4 f 1 4 2 f 5 6 8 f 5 8 7 f 1 2 6 f 1 6 5 f 3 7 8 f 3 8 4 f 1 5 7 f 1 7 3 f 2 4 8 f 2 8 6verticestrianglesDraw data structure

Triangle Meshes (.iv)

Transformations & Hierarchical Models

Add an extra dimensionin 2D, we use 3 x 3 matricesIn 3D, we use 4 x 4 matricesEach point has an extra value, wHomogeneous Coordinates

Most of the time w = 1, and we can ignore it

Homogeneous Coordinates

Translate (tx, ty, tz)Why bother with the extra dimension? Because now translations can be encoded in the matrix!Translate(c,0,0)xypp'c

Scale (sx, sy, sz)Isotropic (uniform) scaling: sx = sy = sz

You only have to implement uniform scalingScale(s,s,s)xpp'qq'y

RotationAbout z axisx'y'z'1=xyz1cos sin 00-sin cos 0000100001ZRotate()xyzpp'

Rotation

About x axis:

About y axis:

RotationAbout (kx, ky, kz), an arbitrary unit vector (Rodrigues Formula)

where c = cos & s = sin Rotate(k, )xyzk

How are transforms combined?(0,0)(1,1)(2,2)(0,0)(5,3)(3,1)Scale(2,2)Translate(3,1)TS =20

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=Scale then TranslateUse matrix multiplication: p' = T ( S p ) = ((TS) p)Caution: matrix multiplication is NOT commutative!

Non-commutative CompositionScale then Translate: p' = T ( S p ) = TS pTranslate then Scale: p' = S ( T p ) = ST p(0,0)(1,1)(4,2)(3,1)(8,4)(6,2)(0,0)(1,1)(2,2)(0,0)(5,3)(3,1)Scale(2,2)Translate(3,1)Translate(3,1)Scale(2,2)

TS =200020001100010311ST =20

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Non-commutative CompositionScale then Translate: p' = T ( S p ) = TS p20002031120

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==Translate then Scale: p' = S ( T p ) = ST p

Transformations in Ray Tracing

Transformations in ModelingPosition objects in a sceneChange the shape of objectsCreate multiple copies of objectsProjection for virtual camerasAnimations

Scene DescriptionSceneLightsCamera

ObjectsMaterialsBackground

Simple Scene Description FileCamera { center 0 0 10 direction 0 0 -1 up 0 1 0 }

Lights { numLights 1 DirectionalLight { direction -0.5 -0.5 -1 color 1 1 1 } }

Background { color 0.2 0 0.6 }

Materials { numMaterials }

Group { numObjects }

Hierarchical Models

Logical organization of scene

Group { numObjects 3 Group { numObjects 3 Box { } Box { } Box { } } Group { numObjects 2 Group { Box { } Box { } Box { } } Group { Box { } Sphere { } Sphere { } } } Plane { } }Simple Example with Groups

Group { numObjects 3 Group { numObjects 3 Box { } Box { } Box { } } Group { numObjects 2 Group { Box { } Box { } Box { } } Group { Box { } Sphere { } Sphere { } } } Plane { } }Adding Materials

Adding Transformations

Using TransformationsPosition the logical groupings of objects within the scene

Transformation in group

Directed Acyclic Graph is more efficient and useful

Processing Model TransformattionsGoalGet everything into world coordinatesTraverse graph/tree in depth-first orderConcatenate transformationsCan store intermediate transformationsApply/associate final transformation to primitive at leaf nodeWhat about cylinders, superquadrics, etc.?Transform ray!

Transform the RayMap the ray from World Space to Object SpacepWS = M pOSpOS = M-1 pWS

Transform RayNew origin:

New direction:originOS = M-1 originWSdirectionOS = M-1 (originWS + 1 * directionWS) - M-1 originWSoriginOSoriginWSdirectionOSdirectionWSObject SpaceWorld Space qWS = originWS + tWS * directionWS qOS = originOS + tOS * directionOSdirectionOS = M-1 directionWS

Transforming Points & DirectionsTransform point

Transform direction

Map intersection point and normal back to world coordinatesHomogeneous Coordinates: (x,y,z,w)W = 0 is a point at infinity (direction)

Transform NormalsWhy? Theyre used for shading

Transforming NormalsA surface normal is a property, not a geometric entity

Correct normal transformation matrix:

See http://www.cgafaq.info/wiki/Transforming_Normals

Given overlapping shapes A and B:

Union Intersection SubtractionConstructive Solid Geometry (CSG)

How can we implement CSG?Points on A, Outside of BPoints on B, Outside of APoints on B, Inside of APoints on A, Inside of B

Union Intersection Subtraction

Collect all the intersections

Union Intersection Subtraction

Implementing CSGTest "inside" intersections:Find intersections with A, test if they are inside/outside BFind intersections with B, test if they are inside/outside A

Overlapping intervals:Find the intervals of "inside" along the ray for A and BCompute union/intersection/subtraction of the intervals

Constructive Solid Geometry

Ray Tracing CSG Models

Rick ParentOhio State U.

CSGForm object as booleans of primitive objectsPrimitives: sphere, cube, cylinder, coneBoolean operators: union, intersection, differenceTree structure used to manage operationsLeaf nodes are primitive objectsIntermediate nodes specify combination operator

CSE 681UnionMin (tCmin, tBmin )BCRay intersects un