# Variable Thickness Surfaces and Their applications Project Report

Variable thickness has a wide role to play in the field of computer graphics and designing many mechanical objects. There are many wide ranges of applications that basically run on the principle of variable thickness and most of these applications run on the formula of two dimensional coordinate space. A two dimensional coordinate space can be defined as the space that is represented by two points geometrically and the thickness and surface area of any object can be defined in the terms of these two points.

Different applications related to physics, computer graphics and engineering subjects has implications over the surface related issues in terms of both the longitude and latitude aspects of the coordinate space. A simple example for this case is aerodynamics, where the airplane central properties are mainly dependent on the variable thickness and surface dynamics. Animation and interface design fields also include their place across the variable thickness and surface aspects. Main aim of this project is to critically review importance of variable thickness surfaces and its applications in different areas with respect to few examples.

Technical aspects of the surface designed are reviewed and a research analysis section is covered to understand the in depth implementation and design aspects of coordinate system of Variable thickness space. Applications and technical implementations of variable thickness surface across the computer graphics filed is studied in this project with detailed analysis of the design aspects. MATLAB simulation is used to calculate the thickness of a surface using the algorithm to generated shape of the surface mesh using a finite element method (FEM). A mathematical approach is used to determine the 3D surface from a 2D surface using a free-form surface generation.

## Surface

The term surface can be defined as the outer boundary of an object or artifact. Surface can be used to estimate the geometrical attributes of an object and in general there are different definitions that define the surface in different perspectives. If a three dimensional object is considered, a surface can be defined as the extended two dimensional outer boundary of the corresponding object. In general a surface can be explained as the boundary between an object and non object and can be considered as the limit for a visual perception.

A surface of an object is the boundary that can be represented in the form of three dimensional aspects in the Euclidean space R3 and the best example for this case is the surface of a ball or cube. Even there are some cases, where the surfaces can’t be represented as the three dimensional manner and the best example that can be considered related to this case is the Klein bottle, where the surface of this bottle cant be embedded in the three dimensional Euclidean space and a 3D approach can be achieved with the implementation of self interaction points across the boundaries. There are different applications that run with the perfect implementation of this surface concept and these include physics, computer graphics, mechanical engineering and other engineering areas.

The historical definition of surfaces defines surface as a subspace of Euclidean spaces. Surface can be stated as the topological space that holds certain properties like every point across this space has a non empty neighbor homeomorphic to an open subset of Euclidean plane E2. In general surfaces can be of two types like closed surfaces and non-closed surfaces. A closed surface is compact in nature and has no boundary and a non-closed surface always holds a boundary and the example that can be considered here is a punctured cube.

The complete visual perception of an object can be estimated with the surface characteristics of that particular object and this is the main reason why there is lot of scope across the study of surfaces in computer graphics. In the field of computer graphics the surface of an object is represented as rigid and explicit triangular meshes. Thickness of a surface plays a very important role in identifying the physical characteristics of an object and in general this thickness can be either constant or variable. Variable thickness surface of objects has a wide range of applications and lot of scope to research. Deriving the variable thickness surfaces has wide scope to study across the fields like computer graphics and aerodynamics.

Variable thickness surfaces include objects like curves and rigid surface objects. Even bending surfaces can be considered as an example of variable thickness surfaces and there are several forces that act on the bending surfaces and make the thickness attribute as variable. Surface tracking of variable thickness surfaces can be considered as an important problem across the research and computer graphics is the best example of this case. In general in the field of computer graphics, variable thickness surface and rigid surface objects are represented as triangular meshes.

## Curved Surfaces

Curved surface are in general of variable thickness and can be represented in several forms. A good curve surface representation should include few important aspects as discussed below

Accuracy: The curved surface should be represented in a manner such that the required dimensions and attributes can be measured accurately. Parameters to measure the variable thickness of the surfaces should be focused more in the process of representation.

Concise: Representation of the curved surface should be as simple as possible and the complex functions and algorithm should be avoided in the process of curved surfaces. In general this curved surface has variable thickness and to measure and represent these aspects, the mathematical functions should be as simple as possible

Affine Invariance: Curve representation should follow basic principle of geometry like Affine Invariance and a curve is said to be affine invariance if the relative coordinate system is changed without affecting the geometry of the curve. Variable thickness can be identified with the coordinates of the curve and if the required curve is affine invariant, representation and identification this thickness would be much easier.

Arbitrary Topology: Topology should be flexible enough while representing the curved surfaces. There should not be any standard functions and rules while representing the required curved surfaces and the thickness of these surfaces will vary across the curve and thus a perfect arbitrary topology should be implemented across the representation of the curved surfaces.