1. disediakan oleh Suriati bte Sadimon GMM, FSKSM, UTM 2004 SURFACE

2. disediakan oleh Suriati bte Sadimon GMM, FSKSM, UTM 2004 Taxonomy of surfaces for CAD and CG 1.Plane surface - the most elementary of the surface type - defined by four curves/ lines or by three points or a line and a point

3. disediakan oleh Suriati bte Sadimon GMM, FSKSM, UTM 2004 Taxonomy of surfaces for CAD and CG 2. Simple basic surface - Sphere, Cube, Cone, and Cylinder

4. disediakan oleh Suriati bte Sadimon GMM, FSKSM, UTM 2004 Taxonomy of surfaces for CAD and CG 3. Ruled surface produced by linear interpolation between two bounding geometric elements. (curves, c1 and c2) Bounding curves must both be either geometrically open (line, arc) or closed (circle, ellipse). a surface is generated by moving a straight line with its end points resting on the curves.

5. disediakan oleh Suriati bte Sadimon GMM, FSKSM, UTM 2004 Taxonomy of surfaces for CAD and CG 3. Ruled surface (cont) C1 C2 C1 C2

6. disediakan oleh Suriati bte Sadimon GMM, FSKSM, UTM 2004 Taxonomy of surfaces for CAD and CG 3. Tabulated cylinder Defined by projecting a shape curve along a line or a vector Shape curve Vector

7. disediakan oleh Suriati bte Sadimon GMM, FSKSM, UTM 2004 Taxonomy of surfaces for CAD and CG 4. Surface of revolution Generated when a curve is rotated about an axis Requires – a shape curve (must be continuous) a specified angle an axis defined in 3D modelspace. The angle of rotation can be controlled Useful when modelling turned parts or parts which possess axial symmetry

8. disediakan oleh Suriati bte Sadimon GMM, FSKSM, UTM 2004 Taxonomy of surfaces for CAD and CG 4. Surface of revolution (cont)  curve axis

9. disediakan oleh Suriati bte Sadimon GMM, FSKSM, UTM 2004 Taxonomy of surfaces for CAD and CG 5. Swept surface A shape curve is swept along a path defined by an arbitrary curve. Extension of the surface of revolution (path a single curve) and tabulated surface (path a vector)

10. disediakan oleh Suriati bte Sadimon GMM, FSKSM, UTM 2004 Taxonomy of surfaces for CAD and CG 5. Swept surface (cont) Shape curve Path- an arbitrary curve

11. disediakan oleh Suriati bte Sadimon GMM, FSKSM, UTM 2004 Taxonomy of surfaces for CAD and CG 6. Sculptured surface Sometimes referred to as a “curve mesh” surface. coon’s patch among the most general of the surface types unrestricted geometric Generated by interpolation across a set of defining shape curves

12. disediakan oleh Suriati bte Sadimon GMM, FSKSM, UTM 2004 Taxonomy of surfaces for CAD and CG 6. Sculptured surface (cont) Or A set of cross-sections curves are established. The system will interpolate the crosssections to define a smooth surface geometry. This technique called lofting or blending

13. disediakan oleh Suriati bte Sadimon GMM, FSKSM, UTM 2004 NURBS Surface –P(u,v) =   w i,j N i,k (u) j N j,l (v) p i,j   w i,j N i,k (u) j N j,l (v) –u, v = knot values in u and v direction (u k-1  u  u n+1,v k-1  v  v n+1 ) –p i,j - control points (2D graph) –Degree = k-1 (u direction) and l–1 (v direction) –w i,j – weights (homogenous coordinates of the control points) i=0 j=0 n m i=0 n j=0 m

14. disediakan oleh Suriati bte Sadimon GMM, FSKSM, UTM 2004 NURBS Surface P 0,0 P 0,1 P 0,2 P 0,3 P 1,0 P 2,0 P 3,0 P 3,3 u v

15. disediakan oleh Suriati bte Sadimon GMM, FSKSM, UTM 2004 Normal Vector Perpendicular to the surface a  Tangent vector in u direction. b  tangent vector in v direction. Normal vector, n = a x b (cross product) a = dP(u,v)b = dP(u, v) dudv N a b

16. disediakan oleh Suriati bte Sadimon GMM, FSKSM, UTM 2004 NURBS surface generated by sweeping a curve Example – sweep along a vector/ line NURB curve, P has degree l-1, knot value (0,1,…m) and control points Pj Sweep along a line  translate the curve in u direction. direction  linear  degree = 1  2 control points  knot value = 0,0,1,1 Pj u v d a

17. disediakan oleh Suriati bte Sadimon GMM, FSKSM, UTM 2004 NURBS surface generated by sweeping a curve Example – sweep along a vector/ line P 0, j = P j, P 1, j = P j + da, h 0, j = h 1, j = h j NURBS equation –P(u,v) =   w i,j N i,2 (u) j N j,l (v) p i,j   w i,j N i,2 (u) j N j,l (v) Pj u v d a 1 i=0 m j=0

18. disediakan oleh Suriati bte Sadimon GMM, FSKSM, UTM 2004 NURBS surface generated by revolving a curve x z v y x P 0, j = P 8, j = P j P 1, j P 2, j P 3, j P 4, j P 5, j P 6, j P 7, j PjPj v direction  NURBS curve, P has degree = l-1, control points P j, Revolution axis = z axis u direction  circle  9 control points  degree = 2  knot vector (0,0,0,1,1,2,2,3,3,4,4,4) P 0, j = P j, h 0, j = h j P 1, j = P 0,j + x j j, h 1, j = h j.1/  2 P 2, j = P 1,j - x j i, h 2, j = h j P 3, j = P 2,j - x j i, h 3, j = h j.1/  2 u

19. disediakan oleh Suriati bte Sadimon GMM, FSKSM, UTM 2004 NURBS surface generated by revolving a curve x z v y x P 0, j = P 8, j = P j P 1, j P 2, j P 3, j P 4, j P 5, j P 6, j P 7, j PjPj P 4, j = P 3,j - x j j, h 4, j = h j P 5, j = P 4,j - x j j, h 5, j = h j 1/  2 P 6, j = P 5,j - x j i, h 6, j = h j P 7, j = P 6,j - x j i, h 7, j = h j 1/  2 P 8, j = P 0,j, h 8, j = h j NURBS equation P(u,v) =   w i,j N i,3 (u) j N j,l (v) p i,j   w i,j N i,3 (u) j N j,l (v) u 8 i=0 m i=0

20. disediakan oleh Suriati bte Sadimon GMM, FSKSM, UTM 2004 NURBS surface display Use simple basic surface –Mesh polygon – flat faces  triangle / rectangle Patches –A patch is a curve-bounded collection of points whose coordinates are given by continuous, two parameter, single-valued mathematical functions of the form –x = x(u,v) y= y(u,v) z = z(u,v)

21. disediakan oleh Suriati bte Sadimon GMM, FSKSM, UTM 2004 Idea of subdivision Subdivision defines a smooth curve or surface as the limit of a sequence of successive refinements. The geometric domain is piecewise linear objects, usually polygons or polyhedra..

22. disediakan oleh Suriati bte Sadimon GMM, FSKSM, UTM 2004 Example- curve subdivision for curve(bezier) in the plane

23. disediakan oleh Suriati bte Sadimon GMM, FSKSM, UTM 2004 Example - surface

24. disediakan oleh Suriati bte Sadimon GMM, FSKSM, UTM 2004

25. Benefit of subdivision The benefit – simplicity and power Simple – only polyhedral modeling needed, can be produced to any desired tolerance, topology correct Power – produce a hierarchy of polyhedra that approximate the final limit object