The document discusses different types of surfaces that can be represented with polygonal meshes, including: 1) Discretely swept surfaces of revolution which are created by rotating a base polygon or profile around an axis. 2) Implicit and parametric representations of surfaces using functions of x, y, and z coordinates. 3) Ruled surfaces where every point lies on a straight line on the surface, such as cones, cylinders, and surfaces swept by a moving line. 4) Quadric surfaces which are 3D analogs of conic sections and have traces that are conic sections when cut by a plane.

1. S U R F A C E S By Abdul Ghaffar

2. Chapter 6: Polygonal Meshes 6.4.5 Discretely Swept Surfaces of Revolution Place all spline points at origin, and use rotation for affine transformation. Base polygon called profile Operation equivalent to circularly sweeping shape about axis Resulting shape called surface of revolution.

3. Chapter 6: Polygonal Meshes 6.5.1 Representation of Surfaces Similar to planar patch P(u,v) = C + a u + b v Generalize: P(u,v) = (X(u,v), Y(u,v), Z(u,v)) (point form). If v constant, u varies: v-contour If u constant, v varies: u-contour Implicit Form of Surface F(x,y,z)=0 iff (x,y,z) is on surface. F(x,y,z)<0 iff (x,y,z) is inside surface F(x,y,z)>0 iff (x,y,z) is outside surface

4. Polygonal Meshes 6.5.6 Rules Surfaces Surface is ruled if, through every one of tis points, there passes at least one line that lies entirely on the surface. Rules surfaces are swept out by moving a straight line along a particular trajectory. Parametric form: P(u,v) = (1-v)P 0 (u) +vP 1 (u). P 0 (u) and P 1 (u) define curves in 3D space, defined by components P 0 (u)=(X 0 (u),Y 0 (u),Z 0 (u)). P 0 (u) and P 1 (u) defined on same interval in u. Ruled surface consists of one straight line joining each pair of points P 0 (u’) and P 1 (u’).

5. Chapter 6: Polygonal Meshes Cones Ruled surface for which P 0 (u) is a single point P(u,v) = (1-v)P 0 +vP 1 (u).

6. Chapter 6: Polygonal Meshes Cylinders Ruled surface for which P 1 (u) is a translated version of P 0 (u): P 1 (u) = P 0 (u) + d =>P(u,v)= P 0 (u) + d v

7. Chapter 6: Polygonal Meshes 6.5.8 The Quadric Surfaces 3D analogs of conic sections

8. Chapter 6: Polygonal Meshes 6.5.8 The Quadric Surfaces

9. Chapter 6: Polygonal Meshes Properties of Quadric Surfaces Trace is curve formed when surface is cut by plane All traces of quadric surfaces are conic sections. Principal traces are curves generated when cutting planes aligned with axes.