Chapter 3-3D Modeling

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1 Chapter 3-3D Modeling Polygon Meshes Geometric Primitives Interpolation Curves Levels Of Detail (LOD) Constructive Solid Geometry (CSG) Extrusion & Rotation Volume- and Point-based Graphics 1

2 The 3D rendering pipeline (our version for this class) 3D models in model coordinates 3D models in world coordinates 2D Polygons in camera coordinates Pixels in image coordinates Scene graph Camera Rasterization Animation, Interaction Lights 2

3 Representations of (Solid) 3D Objects Complex 3D objects need to be constructed from a set of primitives Representation schema is a mapping of 3D objects --> primitives Primitives should be efficiently supported by graphics hardware Desirable properties of representation schemata: Representative power: Can represent many (or all) possible 3D objects Representation is a mapping: Unique representation for any 3D object Representation mapping is injective: Represented 3D object is unique Representation mapping is surjective: Each possible representation value is valid Representation is precise, does not make use of approximations Representation is compact in terms of storage space Representation enables simple algorithms for manipulation and rendering Most popular on modern graphics hardware: Boundary representations (B-Reps) using vertices, edges and faces. 3

4 Polygon Meshes Describe the surface of an object as a set of polygons Mostly use triangles, since they are trivially convex and flat Current graphics hardware is optimized for triangle meshes 4

5 3D Polygons and Planes A polygon in 3D space should be flat, i.e. all vertices in one 2D plane Trivially fulfilled for triangles Mathematical descriptions of a 2D plane in 3D space (hyperplane) Method 1: Point p and two non-parallel vectors v and w x = p + sv! + tw "! Method 2: Three non-collinear points (take one point and the difference vectors to the other two) Method 3: Point p and normal vector n for the plane n! (x p "# ) = 0 Method 4: Single plane equation using the dot product Ax 1 + Bx 2 + Cx 3 + D = 0 (A, B, C) is the normal vector of the plane A, B, C, D real numbers All description methods easily convertible from one to the other (E.g. using cross product to compute normal vector) 5

6 Example: Triangle and Associated Plane Three points (corners of the unit cube) p = (1,0,0) y q = (1,1,0) z 1 1 r q 1 p Plane equation: 1x 1 + 0x 2 + 0x 3 + D = 0 Inserting any point gives: x 1 1 = 0 Normal vector x r = (1,0,1) Two in-plane vectors: v! = q p = " \$ \$ # Normal vector: v! w "! = # % % % \$ % ' ' & v 2 w 3 v 3 w 2 v 3 w 1 v 1 w 3 v 1 w 2 v 2 w 1 w!" = r p = & ( ( ( ' = # % % \$ " \$ \$ # % ' ' & & # 1 ( ( = % 0 % ' \$ 0 & ( ( ' 6

7 Example (contd.): Triangle Front and Back Face Three points (corners of the unit cube) Normal vector z 1 1 y q r 1 p x p = (1,0,0) q = (1,0,1) r = (1,1,0) Normal vector:! "! # v w = % % \$ Plane equation: ( 1)x 1 + 0x 2 + 0x 3 + D = 0 Inserting any point gives: x 1 +1 = 0 Different order of vertices: mathematically negative (clockwise) & ( ( ' For an arbitrary point: Left hand side of plane equation gives value > 0: Point is in front of plane Left hand side of plane equation gives value < 0: Point is behind plane 7

8 Right Hand Rule for Polygons A rule of thumb to determine the front side (= direction of the normal vector) for a polygon Please note: The relationship between vertex order and normal vector is just a convention! Q: How can we see this from the previous slides? Source: 8

9 Face-Vertex Meshes 9

10 Möbius Strip: Non-Orientable Surface Complete object: Does not have a front and back side! M. C. Escher: Moebius Strip II 10

11 Polygon Meshes: Optional data Color per vertex or per face: produces colored models Normal per face: Easy access to front/back information (for visibility tests) Normal per vertex: Standard computation accelerated (average of face normals) Allows free control over the normals use weighted averages of normals mix smooth and sharp edges (VRML/X3D: crease angles) wait for shading chapter ;-) Texture coordinates per vertex wait for texture chapter ;-) 11

12 Polygon Meshes: other descriptions Other representations for polygon meshes exist optimized for analyzing and modifying topology optimized for accessing large models optimized for fast rendering algorithms optimized for graphics hardware! Example: triangle strip needs N+2 points for N polygons implicit definition of the triangles optimized on graphics hardware OpenGL / JOGL: gl.glbegin(gl2.gl_triangle_strip); gl.glvertex3d(-1, -1, 1);

13 Approximating Primitives by Polygon Meshes Trivial for non-curved primitives... The curved surface of a cylinder, sphere etc. must be represented by polygons somehow (Tesselation). Not trivial, only an approximation and certainly not unique! GLU utility functions for tesselation exist Goal: small polygons for strong curvature, larger ones for areas of weak curvature This means ideally constant polygon size for a sphere Where do we know this problem from??? Something playful

14 Chapter 3-3D Modeling Polygon Meshes Geometric Primitives Interpolation Curves Levels Of Detail (LOD) Constructive Solid Geometry (CSG) Extrusion & Rotation Volume- and Point-based Graphics 14

15 Geometric Primitives Simplest way to describe geometric objects Can be used directly by some renderers (e.g., Ray tracing) Can be transformed into polygons easily (Tesselation) Can be transformed into Voxels easily Useful for creating simple block world models! Supported in many frameworks of different levels VRML/X3D, Java 3D OpenGL, WebGL, JOGL 15

16 Box Described by (width, length, height) Origin usually in the center 8 points, 12 edges, 6 rectangles, 12 triangles 16

17 Pyramid, Tetrahedron (Tetraeder) Basis of pyramid = rectangle given by (width, length, height) 5 points, 8 edges, 6 triangles! Basis of tetrahedron = triangle given by (width, length, height) 4 points, 6 edges, 4 triangles, 17

18 Generalization: Polyhedra Polyhedron (Polyeder): Graphical object where a set of surface polygons separates the interior from the exterior Most frequently used and best supported by hardware: surface triangles Representation: Table of Vertex coordinates Additional information, like surface normal vector for polygons Regular polyhedra: Five Platonic regular polyhedra exist Tetrahedron (Tetraeder) Hexahedron, Cube (Hexaeder, Würfel) Oktahedron (Oktaeder) Dodekahedron (Dodekaeder) Icosahedron (Ikosaeder) 18

19 Cylinder, cone, truncated cone Cylinder given by (radius, height) Number of polygons dep. on tesselation! Cone given by (radius, height) Number of polygons dep. on tesselation! Truncated cone given by (r1, r2, height) Number of polygons dep. on tesselation! Q: Which of these would you rather have if you only had one available? 19

20 Sphere, Torus Sphere is described by (radius) Torus is defined by (radius1, radius2) Number of polygons dep. on tesselation 20

21 Geometric Primitives: Summary Not all of these exist in all graphics packages Some packages define additional primitives (dodecahedron, teapot...;-)! Practically the only way to model in a text editor Can give quite accurate models Extremely lean! Very few polygons! Think of application areas even in times of powerful PC graphics cards! 21

22 Chapter 3-3D Modeling Polygon Meshes Geometric Primitives Interpolation Curves Levels Of Detail (LOD) Constructive Solid Geometry (CSG) Extrusion & Rotation Volume- and Point-based Graphics 22

23 Interpolation Curves, Splines Original idea: Spline used in ship construction to build smooth shapes: Elastic wooden band Fixed in certain positions and directions Mathematically simulated by interpolation curves Piecewise described by polynomials Different types exist Natural splines Bézier curves B-Splines Control points may be on the line or outside of it. All on the line for a natural spline (0,0) X-Achse Y-Achse 23

24 Bézier Curves (and de-casteljau Algorithm) Bézier curves first used in automobile construction (1960s, Pierre Bézier Renault, Paul de Casteljau Citroën) Degree 1: straight line interpolated between 2 points Degree 2: quadratic polynomial P4 Degree 3: cubic Bézier curve, described by cubic polynomial Curve is always contained in convex hull of points Algorithm (defines line recursively): Choose t between 0 and 1 P2 I1: Divide line between P1 and P2 as t : (1 t) I2, I3: Repeat for all Ps (one segment less!) I1 J1, J2: Repeat for I1, I2, I3 (same t) K: Repeat for J1, J2 (single point!) Bézier curve: all points K for t between 0 and 1 see (Dominik Menke) P1 J1 I2 K I3 J2 P3 24

25 Bézier Patches Combine 4 Bézier curves along 2 axes Share 16 control points Results in a smooth surface Entire surface is always contained within the convex hull of all control points Border line is fully determined by border control points Several patches can be combined connect perfectly if border control points are the same. Advantage: move just one control point to deform a larger surface... Other interpolation surfaces based on other curves Generalization of Bézier idea: B-splines Further generalization: Non-uniform B-splines Non-uniform rational B-splines (NURBS) (supported by OpenGL GLU) 25

26 Interpolation in OpenGL (Bezier Example) Utah teapot Martin Newell, vertices 32 bicubic Bézier surface patches 26

27 Chapter 3-3D Modeling Polygon Meshes Geometric Primitives Interpolation Curves Levels Of Detail (LOD) Constructive Solid Geometry (CSG) Extrusion & Rotation Volume- and Point-based Graphics 27

28 Levels of Detail Assume you have a very detailed model from close distance, you need all polygons from a far distance, it only fills a few pixels How can we avoid drawing all polygons? 28

29 Mesh reduction Original: ~5.000 polygons Reduced model: ~1.000 polygons ==> about 80% reduction! Very strong reductions possible, depending on initial mesh! Loss of shape if overdone 29

30 A method for polygon reduction Rossignac and Borell, 1992, Vertex clustering Subdivide space into a regular 3D grid For each grid cell, melt all vertices into one Choose center of gravity of all vertices as new one Triangles within one cell disappear Triangles across 2 cells become edges (i.e. disappear) Triangles across 3 cells remain Good guess for the minimum size of a triangle edge length roughly = cell size Yields constant vertex density in space Does not pay attention to curvature! 30

31 Billboard A flat object which is always facing you Very cheap in terms of polygons (2 triangles) Needs a meaningful texture Example (from SketchUp): guy in the initial empty world rotates about his vertical axis to always face you 31

32 Chapter 3-3D Modeling Polygon Meshes Geometric Primitives Interpolation Curves Levels Of Detail (LOD) Constructive Solid Geometry (CSG) Extrusion & Rotation Volume- and Point-based Graphics 32

33 Constructive Solid Geometry Basic idea: allow geometric primitives and all sorts of boolean operations for combining them Can build surprisingly complex objects Good for objects with holes (often the simplest way) Basic operations: Or: combine the volume of 2 objects And: intersect the volume of 2 objects Not: all but the volume of an object Xor: all space where 1 object is, but not both Think about: wheels of this car tea mug coke bottle (Problems??) 33

34 CSG: a complex Example rounded_cube = cube And sphere cross = cyl1 Or cyl2 Or cyl3 result = rounded_cube And (Not cross)! Think: Are CSG operations associative?!...commutative? 34

35 Chapter 3-3D Modeling Polygon Meshes Geometric Primitives Interpolation Curves Levels Of Detail (LOD) Constructive Solid Geometry (CSG) Extrusion & Rotation Volume- and Point-based Graphics 35

36 Extrusion (sweep object) Move a 2D shape along an arbitrary path Possibly also scale in each step 36

37 Rotation Rotate a 2D shape around an arbitrary axis Can be expressed by extrusion along a circle! How can we model a vase? How a Coke bottle? 37

38 Chapter 3-3D Modeling Polygon Meshes Geometric Primitives Interpolation Curves Levels Of Detail (LOD) Constructive Solid Geometry (CSG) Extrusion & Rotation Volume- and Point-based Graphics 38

39 Voxel data Voxel = Volume + Pixel, i.e., voxel = smallest unit of volume Regular 3D grid in space Each cell is either filled or not Memory increases (cubic) with precision! Easily derived from CSG models Also the result of medical scanning devices MRI, CT, 3D ultrasonic! Volume rendering = own field of research Surface reconstruction from voxels 39

40 Point-based graphics Objects represented by point samples of their surface ( Surfels ) Each point has a position and a color Surface can be visually reconstructed from these points! purely image-based rendering no mesh structure very simple source data (x,y,z,color) Point-data is acquired e.g., by 3D cameras Own rendering techniques Own pipeline ==> own lecture ;-) 40

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