1. Anupam Saxena Associate Professor Indian Institute of Technology KANPUR 208016

2. Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1- 2 Transformations and Projections 1-2 Modeling of Curves Representation, Differential Geometry Ferguson Segments Bezier Segments 1-2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite NURBS

3. Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1-2 Transformati ons and Projections 1-2 Modeling of Curves Representati on, Differential Geometry Ferguson Segments Bezier Segments 1- 2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite NURBS

4. r(u, v) c(t)= r(u(t), v(t)) tangent to the curve Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1-2 Transformati ons and Projections 1-2 Modeling of Curves Representati on, Differential Geometry Ferguson Segments Bezier Segments 1- 2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite NURBS

5. r(u, v) c(t) = r(u(t), v(t)) differential arc ds length of the curve Symmetric G is called the first fundamental matrix of the surface Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1-2 Transformati ons and Projections 1-2 Modeling of Curves Representati on, Differential Geometry Ferguson Segments Bezier Segments 1- 2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite NURBS

6. unit tangent t to the curve for t to exist G should be always be positive definite G 11 G 22 – G 12 G 21 > 0 implies that G is always positive definite Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1-2 Transformati ons and Projections 1-2 Modeling of Curves Representati on, Differential Geometry Ferguson Segments Bezier Segments 1- 2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite NURBS

7. length of the curve segment in t 0  t  t 1 c(t 1 ) and c(t 2 ) as two curves on the surface r(u, v) that intersect the angle of intersection  is given by Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1-2 Transformati ons and Projections 1-2 Modeling of Curves Representati on, Differential Geometry Ferguson Segments Bezier Segments 1- 2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite NURBS

8. two curves are orthogonal to each other if If u  t 1 and v  t 2 Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1-2 Transformati ons and Projections 1-2 Modeling of Curves Representati on, Differential Geometry Ferguson Segments Bezier Segments 1- 2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite NURBS

9. u = u 0 u = u 0 + du v = v 0 v = v 0 + dv r(u 0, v 0 ) r(u 0 + du, v 0 ) r(u 0, v 0 + dv) r u du r v dv Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1-2 Transformati ons and Projections 1-2 Modeling of Curves Representati on, Differential Geometry Ferguson Segments Bezier Segments 1- 2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite NURBS

10. n P R d Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1-2 Transformati ons and Projections 1-2 Modeling of Curves Representati on, Differential Geometry Ferguson Segments Bezier Segments 1- 2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite NURBS

11. n P R d n is perpendicular to the tangent plane, r u.n = r v.n = 0 second fundamental matrix D Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1-2 Transformati ons and Projections 1-2 Modeling of Curves Representati on, Differential Geometry Ferguson Segments Bezier Segments 1- 2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite NURBS

12. L, M and N are called the second fundamental form coefficients use

13. r uu = x uu i + y uu j + z uu k r uv = x uv i + y uv j + z uv k r vv = x vv i + y vv j + z vv k

14. tangent plane intersects the surface at all points where d = 0 Case 1:No real value of du P is the only common point between the tangent plane and the surface No other point of intersection P  ELLIPTICAL POINT

15. Case 2: L 2 +M 2 +N 2 > 0 du =  (M/L)dv u – u 0 =  (M/L)(v – v 0 ) tangent plane intersects the surface along this straight line P  PARABOLIC POINT Case 3: two real roots for du tangent plane at P intersects the surface along two lines passing through P P  HYPERBOLIC POINT Case 4: L = M = N = 0 P  FLAT POINT