Introduction to Computer Graphics. Farhana Bandukwala, PhD Lecture 11: Surfaces

Transcription

1 Introduction to Computer Graphics Farhana Bandukwala, PhD Lecture 11: Surfaces

2 Outline Linear approximation Parametric bicubic surfaces Subdividin surfaces Drawin surfaces

3 Implicit Representation f(x,y,z)0 describes a surface Plane: ax+by+cz+d0 Sphere: x 2 +y 2 +z 2 +r 2 0 Divides 3-D space into points on or off surface 3-D curve is represented as the intersection of two surfaces f(x,y,z)0 and (x,y,z)0

4 Linear approximation Connected 3D polyons Polyon defined by tanent planes at sampled points Good linear approximation needs several polyons at hih curvature reions

5 Parametric form Each spatial variable expressed in terms of 2 independent variables x x(u,v), y y(u,v), z z(u,v) Locus of points p[x(u,v) y(u,v) z(u,v)] for u[u min,u max ] & v[v min,v max ] Derivative (w/ respect to u) [dx(u,v)/du dy(u,v)/du dz(u,v)/du] (w/ respect to v) [dx(u,v)/dv dy(u,v)/dv dz(u,v)/dv]

6 Parametric bicubic surfaces Generalization of cubic curves Geometry vector is dependent on parameter v For a specific v 1, Q x (u,v 1 ) is a curve Matrix form: Q x (u,v 1 )U*M*G(v 1 ) u 3 u 2 u 1 * M * G x0 (v) G x1 (v) G x2 (v) G x3 (v)

7 Example: Curves alon parameter u G x G y G z

8 Example: Curves alon parameter u&v G x G y G z

9 Example: Curves alon parameter u&v G x G y G z

10 Example: Curves alon parameter u&v G x G y G z

11 Parametric bicubic surfaces (contd) G i (v) V*M*G where G [ i1 i2 i3 i4 ] : constraints for cubic curve Usin identity : (V*M*G) T G T *M T *V T Q x (u,v) U * M * * M T * V T

12 Bezier Surfaces v Q x (u,v) U * M B * G * M BT * V T 4x4 Geometry matrix consists of respective component of 16 control points C0 and G0 continuity by makin 4 common control points between patches equal G1 continuity when 2 sets of 4 control points on either side of ede are collinear Parameter space P 30 P 31 P 32 P 33 P 20 P21 P 22 P 23 3-D Geometry space P 03 P 13 P 23 P 02 P 12 P 22 P 01 P 11 P 21 P 32 P 33 y P 10 P 11 P 12 P 13 P 10 P 20 P 31 P 00 P 01 P 02 P 03 u P 00 P 30 x z

13 Example: Bezier surface

14 Subdividin Surfaces Split surface alon one parameter, u Curve subdivision method applied to each set of four control points alon u Stop subdividin when flatness test positive Then subdivide over other parameter, v New surface will consist of 4 patches Problem: cracks can appear because different levels of subdivision at adjoinin patches Solution: subdivide over fixed depth at all patches reardless of flatness test

15 Use recursive subdivision Generalize curve subdivision to 2d 1 st subdivide curves alon u 2 nd subdivide (new) curves alon v Draw polyons usin : lbein(gl_quad_strip) for all vertices in v i [vmin-vmax-1] Drawin Surfaces for all vertices in u j [umin-umax] lvertex3f(xval(v i,u j ),yval(v i,u j ),zval(v i,u j )) lend() lvertex3f(xval(v j+1,u j ),yval(v i +1,u j ),zval(v i +1,u j )) P 1.5(P 1 +P 2 ) P 2 L 1 L 2 R 1 R 2 L 3 R 0 L 0 P 0 P 3 R 3

16 Cross product The cross product of 2 non-parallel vectors, is a vector orthoonal to the oriinal 2. To compute normal to plane containin oriinal 2 vectors Use cross product to compute normals to polyons Use dot product to compute distance from plane/polyon w u x v u 2 v 3 u 3 v 2 u 3 v 1 u 1 v 3 u 1 v 2 u 2 v 1 w u v

17 Usin evaluators in OpenGL Beziers can be evaluated usin OpenGL API One dimensional evaluator Setup curve mappin (lmap1f(gl_map1_vertex_3,u min,u max,stride,order,&ctrl_vector)) Enable evaluator (lenable(gl_map1_vertex_3)) Evaluate vertices at appropriate parameter value (levalcoord1f(u i )) Two dimensional evaluator Setup surface mappin (lmap2f(gl_map2_vertex_3,u min,u max,stride u,order u, v min,v max,stride v,order v,&ctrl_vector)) Enable evaluator (lenable(gl_map2_vertex_3)) Evaluate vertices at appropriate parameter value (levalcoord2f(u i,v i ))

18 Hermite Surfaces Completely defined by 4x4 eometry matrix G H P x0 (v) and P 3x (v) define x components of startin and endin points of curve in parameter u R x0 (v) and R x3 (v) are tanent vectors at these points v1.0 u 3 u 2 u 1 * M H * P x0 (v) P x3 (v) R x0 (v) R x3 (v) P 0 (v) v v0.75 v0.5 v0.25 v0.0 P 3 (v) u

19 Hermite Surfaces (contd) Suppose: P 0x (v) T * M H * x coordinates of 4 corners of patch x (0,0) x (0,1) x(0,0) x(0,1) v v x coordinates of tanent vector alon v P 3x (v) T * M H * x G Hx u x (1,0) x (1,1 ) x(1,0) x(1,1 ) v v x(0,0) u x(0,1) 2 x(0,0) u v 2 x(0,1) u v R 0x (v) T * M H * x x x coordinates of u x(1,0) tanent vector alon u u x(1,1) 2 x(1,0) u v 2 x(1,1) u v x coordinates of partial derivative w/ respect to both parameters R 3x (v) T * M H * x

20 B-splines Advantaes over Beziers: Local control of shape: one control point affects a portion of curve not the whole curve Deree of resultin curve more independent of number of eometric constraints Context: curve composed of several sements Need not pass throuh control points Sharin control points between sements provides continuity m+1 control points (P0 Pm) m>3 (atleast 4 points) m-2 curve sements (Q3 Qm) (atleast 1 sement) each sement defined over t i <t<t i+1, for 3<i<m t i between sements and at ends are called knots

21 Bsplines, contd P 0 t 3 0 P 1 P P 4 2 t 7 4 t 4 1 t 6 3 P 5 P 3 P 6 t 8 5 P 7 Q Q 7 Q 6 3 t Q8 5 2 Q4 Q 5 t 4 6 B-spline basis are nonneative Normalized basis Each curve sement lies in the convex hull of its control points Geometry vector for i th sement P i-3 P i-2 P i-1 P i 1/6 Basis matrix

22 B-Spline Surfaces Q x (u,v) U * M Bs * G Bs * M BsT * V T 4x4 Geometry matrix consists of respective component of 16 control points