•1Slide 1 Lecture 20Copyright Amitabh VarshneyModeling Overview• Goal: Represent 3D objects efficiently allowing for their easy design and modification• Modeling versus rendering primitives• Implicit, parametric, procedural modelingSlide 2 Lecture 20Copyright Amitabh VarshneyGetting Models• By hand• By program• From the world•2Slide 3 Lecture 20Copyright Amitabh Varshney3D Sensing with Laser• Project laser on object• Triangulate– Can be done with known laser position .– Or two cameras.• Project stripe and triangulate on all at once.Slide 4 Lecture 20Copyright Amitabh Varshney•Triangulation: Given a known direction of light, there is a linein the world where the light might strike the object. This lineprojects to a line in the image. Location of the point in the image tells us where along this line the light struck.•Epipolar constraint: Line and focal point form a plane. If we move the light up, we can get a different plane which doesn’t intersect this, so the new light direction doesn’t interfere.•3Slide 5 Lecture 20Copyright Amitabh VarshneyDigital MichaelangeloSlide 6 Lecture 20Copyright Amitabh VarshneyMany Other 3D Sensors• Time of flight sensors• Stereo• Medical imaging (eg., MRI, sonar)• Mechanical•4Slide 7 Lecture 20Copyright Amitabh Varshney3D Representations: Triangles• Excellent rendering primitive:– Edges are straight (linear)– Interior is flat: incremental scan-conversion of a few adds per pixel– Triangle visibility is constant (no self-occlusions)– Normals can be used to convey flat or curved shadingSlide 8 Lecture 20Copyright Amitabh VarshneyTriangles• Poor Modeling primitive:– Curved geometries require lots of triangles– Continuity/smoothness/blending is difficult– Shape design is hard since influence is completely local– Inside/outside tests are difficult for non-convex objects•5Slide 9 Lecture 20Copyright Amitabh VarshneyImplicit Modeling Primitives• Expressed by equations of the form f(x, y, z) = 0• Divide the space into inside/outside based on whether f(x, y, z) < 0 or > 0• Given an object, it is difficult to derive its implicit representation (there has been some recent progress in this direction though)• The class of known implicit functions is large enough to serve as a useful modeling primitive Slide 10 Lecture 20Copyright Amitabh VarshneyQuadrics• x2+ y2 + z2– 1 = 0 Sphere/Ellipsoid• x2+ y2 – z2– 1 = 0 Hyperboloid of one sheet• x2– y2 – z2– 1 = 0 Hyperboloid of two sheets• x2+ y2 – z2= 0 Elliptic Cone• x2+ y2 – z = 0 Elliptic paraboloid• – x2+ y2 – z = 0 Hyperbolic paraboloid•6Slide 11 Lecture 20Copyright Amitabh VarshneyQuadrics(x/a)2+ (y/b)2 + (z/c)2= 1 (x/a)2+ (y/b)2 – (z/c)2= 1 –(x/a)2– (y/b)2 + (z/c)2= 1(x/a)2+ (y/b)2 – (z/c)2= 0 (x/a)2+ (y/b)2 – (z/c) = 0 (x/a)2– (y/b)2 – (z/c)2= 0Ellipsoid Hyperboloid HyperboloidElliptic Cone Elliptic Paraboloid Hyperbolic ParaboloidImages courtesy, Ching-Kuang Shene, Michigan Technological UniversitySlide 12 Lecture 20Copyright Amitabh VarshneySuperquadricsImage Courtesy Montiel, Aguado, Zaluska, University of Surrey, UK•7Slide 13 Lecture 20Copyright Amitabh VarshneySuperquadrics(Superellipsoids)• Provide a lot of parameterized flexibility for modeling different kinds of objects( (x/a)2/s+ (y/b)2/s)s/t+ (z/c)2/t= 1For an ellipsoid, s = t = 1 • Similarly one can define superhyperboloids etc…Slide 14 Lecture 20Copyright Amitabh VarshneyMetaballs• Also known as blobby models• Useful for modeling soft contours: typically muscles for humans, animals• Have equations of the form:g(x, y, z) = kbkf(r – rk) – T = 0where f(r) is the density function, rkis the kthcenter (xk, yk, zk)• Surfaces are then generated for a given density T•8Slide 15 Lecture 20Copyright Amitabh VarshneyMetaballs• Density functions can be exponential:f ( r ) = e– aror quadratic: b ( 1 – 3 r2/d2), 0 r d/3f ( r ) = 3/2 b (1 – r/d)2, d/3 r d0, r > d2Slide 16 Lecture 20Copyright Amitabh VarshneyExample• Suppose we have a single, circularly symmetric Gaussian. What happens as we vary the threshold? We just get a circle, with varying radius.• Next, suppose we take the sum of two such Gaussians. What happens if we vary the threshold. Do we get 2 circles? Well, at first, when the threshold is high. But as we lower it, the circles join together smoothly. Think of the Gaussians as two hills. As we go from one hill to another, the ground stays higher than when we just leave the hill in another direction.• Blobby models are good for smoothly joining shapes.•9Slide 17 Lecture 20Copyright Amitabh VarshneyMetaballsImages courtesy Matt Ward.Slide 18 Lecture 20Copyright Amitabh VarshneyMetaballsImage courtesy Spencer ArtsMuscular structure is created using Metareyes plug-in and involved creating hundreds of metaballs.•10Slide 19 Lecture 20Copyright Amitabh VarshneyParametric Modeling• Uses equations of the form: x(u,v) = …y(u,v) = …• Useful for modeling surfaces where continuity is important• Allows for trade-off of local versus global influence for editing• Various representations:Bezier, B-splines, NURBS (Non-Uniform Rational B-Splines), trimmed NURBSSlide 20 Lecture 20Copyright Amitabh Varshney2D Example• Use control points to specify a curve.– Intuitive interface• Curve should be smooth.(Dave Mount)•11Slide 21 Lecture 20Copyright Amitabh VarshneyInterpolation vs. Approximation• Smooth curves (eg., polynomials) can be fit to control points.• But resulting curve can be unpredictable.(Dave Mount)Slide 22 Lecture 20Copyright Amitabh VarshneyBezier Curves• Approximate curves• Two points:(Dave Mount)•12Slide 23 Lecture 20Copyright Amitabh Varshney• Three points. Interpolate between pairs.• Then interpolate between them: (Dave Mount)Slide 24 Lecture 20Copyright Amitabh VarshneyProperties of Bezier Curves• Can extend to more points• Each pt on curve is a convex combination of control points.• Curve starts at first point and ends at last.• Tangent at 1st(last) point is direction to 2nd(to last).(Dave Mount)•13Slide 25 Lecture 20Copyright Amitabh VarshneyParametric Modelingwww.rhino3d.comSlide 26 Lecture 20Copyright Amitabh VarshneyParametric