Surface and Solid Geometry. 3D Polygons

Transcription

1 Surface and Solid Geometry D olygons Once we know our plane equation: Ax + By + Cz + D = 0, we still need to manage the truncation which leads to the polygon itself Functionally, we will need to do this to know if a point lies in a polygon or not, for example

2 D olygons To do this, we can project the D polygon into D and see if the point is in the d z using the inside test (xi,yi,zi) y x (xi,yi ) D olygons roject d to d based on largest of A,B,C z (xi,yi,zi) y This example: Z (or C) is principal component of N, normal so project on to xy-plane x (xi,yi )

3 Z-buffer For D with multiple polygons, must deal with visibility image plane roject vertices into image plane and with projected vertex include depth in additional depth- or Z-buffer 5 Z-buffer Scan convert each projected polygon: For each polygon in scene project verticies for each pixel inside poly calculate z if z < closest draw into frame buffer update z buffer 6

4 Z-buffer How do we calculate z for buffer, quickly? z = (-Ax -By -D)/C from plane eq. Same scanline (x + ): z' = (-A(x + ) - By -D)/C or z' = z - A/C z z' 7 Z-buffer How do we calculate z for buffer, quickly? For the next scanline, following edge coherence: x s+ = x s + /m, y s+ = y s + plus z s = (-Ax s -By s -D)/C, z s+ = z s - (A/m + B)/C z s+ z z' 8

5 Z-buffer Z-buffering is very common approach, also often accelerated with hardware OpenGL embeds this approach D olygon GRAHICS IELINE Image ixels 9 0 5

6 Surface Geometry Surfaces Interpolating points for parameters, u and v Bi-cubic patch 6

7 Subdivision Surfaces Refinement Types of Subdivision Interpolating Schemes - Limit Surfaces/Curve will pass through original set of data points. Approximating Schemes - Limit Surface will not necessarily pass through the original set of data points. 7

8 8 Subdivision Surfaces Approach Limit Curve Surface through an Iterative Refinement rocess. Refinement Refinement Refinement Interpolation example A rimer: Chaiken s Algorithm (approximating surface) Q Q 5 Q Q Q Q Apply Iteratively Limit Curve Surface 0 Q 0 Q Q Q Q Q 5 Subdivision Surfaces

9 9 Surface Example Surface evolution with subdivision level Limit surface Catmull-Clark Subdivision (978) n v i n f FACE f f v v e EDGE VERTEX j j j j i i f n e n v n n v

10 Subdivision used in production Traditionally B-spline patches have been used in production for character animation. Difficult to control B-spline patch density in character modelling. Subdivision in Character Animation Tony Derose, Michael Kass, Tien Troung (SIGGRAH 98) (Geri s Game, ixar 998) Solid Constructive Geometry 0

11 Solid geometry Inherently D, geometric elements describe sets of spaces enclosed by D boundaries For example, a solid sphere is the simplest solid element. Other simple primitives include the cube, cylinder, cone, and torus Solid geometry Other objects are defined by combinations of primitives. An entire math has been explored related to the combination of solid primitives called constructive solid geometry (CSG) With CSG, complex shapes may be generated from operations formed on primitives

12 Solid geometry Boolean operators are defined tools for combining solid geometry for CSG These perform group operations on the points included in the solid primitives They are: - Union - Subtraction - Intersection These bool-op's are implemented in Maya Solid geometry Boolean operator: Union combines two elements into a single one

13 Solid geometry Boolean operator: Union combines two elements into a single one 5 Solid geometry Boolean operator: Subtract take the difference between two elements 6

14 Solid geometry Boolean operator: Subtract take the difference between two elements 7 Solid geometry Boolean operator: Subtract take the difference between two elements 8

15 Solid geometry Boolean operator: Subtract 9 Solid geometry Boolean operator: Intersection finds the common points in the given primitives 0 5

16 Solid Constructive Geometry 6

17 CSG Tree Solid geometry Graph for hierarchy of Boolean operations Often used for CAD and Mech Eng Solid geometry Adams & Dutre 00 7

18 Surfaces splitting Split along the intersection curve A A C A A B Label each part of the object (inside/outside) A C A A C 5 Example 6 8

19 Intersection curve Compute approximate intersection of both mesh sets 7 Split and label operations Interior faces Exterior faces 8 9

20 Reconstruction Depending on boolean operation: Merge operation along the intersection curve Intersection Union A - B B - A 9 WHY? Solid Constructive Geometry 0 0

21 Solid Constructive Geometry Solid Constructive Geometry