**3D objects representation methods:-**

Many real world objects are inherently smooth and involve curves to represent them some natural objects are neither perfectly flat nor smoothly curved but often have rough jagged contours. These objects have different characteristics and different materials. So there is no one method to describe such objects.

Polygon and quadric surfaces provide precise description for simple object such as polyhedrons and ellipsoids.The spline surfaces and construction techniques are useful for designing aircraft wings, gears and other engineering structures with curved surfaces.The procedural methods such as fractal constructions and particle system allow us represent mountains, clouds, clumps of grass etc. The physically based modeling methods are useful for describing nonrigid behavior of objects such as cloth. The octree encoding is used to represent internal features of object such as obtained from CT scans.There are number schemes to represent solid objects, these schemes are broadly categorized into two categories :

**Boundary representations (B-reps)** and **Space-partitioning representations**. The boundary representations describe a three-dimensional object as a set of surfaces that separates the object interior from the environment.On the otherhand, Space-partitioning describes interior properties by partitioning the spatial region containing an object into a small, non overlapping, contiguous solids.

**Sweep Representations:-**

Sweep representations are used to construct three dimensional objects from two dimensional shape .There are two ways to achieve sweep: **Translational sweep** and **Rotational sweep**. In translational sweeps, the 2D shape is swept along a linear path normal to the plane of the area to construct three dimensional object. To obtain the wireframe representation we have to replicate the 2D shape and draw a set of connecting lines in the direction of shape, as shown in the Fig. (8).

In rotational sweeps, the 2D shape is rotated about an a axis of rotation specified in the plane of 2D shape to produce three dimensional object. This is illustrated in Fig. (9).

**Constructive Solid Geometry (CSG):-**

Another technique for solid modeling is to combine the volumes occupied by overlapping three-dimensional objects using Boolean set operations.This modeling technique is called **Constructive Solid Geometry (CSG)**. It creates a new volume by applying Boolean operators such as union. intersection, or difference to two specified objects.

The Fig. (10), Fig. (11), Fig. (12) show the example for forming new shapes using Boolean set operations The Fig. 10 (a) shows that two rectangular blocks are placed adjacent to each other. We can obtain the combined object with the union operation as shown in Fig. 10 (b).

The Fig.(11) shows the result of intersection operation obtained by overlapping cylinder and cube. With the difference operation, we can obtain the resulting solid as shown in Fig. (12). The CSG method Uses three dimensional objects such as blocks, pyramids, cylinders, cones, spheres, and closed spline surfaces to generate other solid objects In this method, an object is stored as a tree with operators at the internal nodes and simple primitives at the leaves. Some nodes represent Boolean operators, whereas others represent operations such as translation, rotation, and scaling. It is important to note that Boolean operations are not, in general, communicative Therefore the edges of the trees must be in proper order.

**CSG Tree:-**

In fact,every solid constructed using CSG technique has a corresponding CSG expression which in turn has an associated CSG tree. The expression of the CSG tree is a representation of the final design.

**Boundary Representations (B - reps):-**

Boundary Representation, B-rep in short, can be considered as an extension to the wireframe model. The merit of a B-rep is that a solid is bounded by its surface and has its interior and exterior. The surface of a solid consists of a set of well-organized faces each of which is a piece of some surface (a surface patch). Faces may share vertices and edge. that are curve segments. Therefore, a B-rep is an extension to the wireframe model by adding face information to the latter.

There are two types of information in a B-rep: **topological** and **geometric**. Topological information provides the relationships among vertices, edges and faces similar to that used in a wireframe model. In addition to connectivity, topological information also includes orientation of edges and faces. Geometric information are usually equations of the edges and faces.

The orientation of each face is important. Normally, a face is surrounded by a set of vertices. Using the right-handed rule, the ordering of these vertices for describing a particular face must guarantee that the normal vector of that face is pointing to the exterior of the solid. Normally, the order is counter clockwise. If that face is given by an equation, the equation must be rewritten so that the normal vector at every point on the part that is being used as a face Points to the exterior of the solid. Therefore, by inspecting normal vectors one can immediately tell the inside/outside of a solid under B-rep.

The orientation must be done for all faces. The Fig. (14) shows three faces and their pointing normal vectors. To describe the **top surface**, the vertices should be **6,7,2,1**, or **7, 2, 1, 6** or **2, 1, 6, 7** or **1, 6, 7, 2**. To describe the **left face**, the order should be **1,2, 3, 4** or **2, 3, 4, 1** or **3, 4, 1, 2** or **4, 1, 2, 3**.

Unfortunately, not all surfaces can he oriented this way. If the surface or a solid can be oriented this way, it is called orientable, otherwise, it is non-orientable. The Fig. (14) shows the well-known Mobius band which is one-sided and non-orientable