HA4 Task 3 - Geometric Theory

What are the fundamental principles behind the creation of virtual 3D space? Describe and explain 3D geometry.

Remember that you are trying to comprehensively explain the theory and applications of 3D with elucidated examples and consistently using subject terminology correctly.


Geometry - The Cartesian co-ordinates system is a great example of geometry. The invention of Cartesian co-ordinates was created in the 17th century by Rene Descartes. He basically revolutionized mathematics by providing the first systematic link between Euclidean geometry. Apparently, It took Rene over a year to provide the systematic link and many times before he failed to find it.  The amazing math magician Euclid of Alexandria simply made co-ordinates in which he manipulated algebra and geometry together. Through this method he came up with Euclidean geometry in which Rene Descartes then found the first systematic link.

Apart from Euclidean geometry, 3D computer graphics portray the same principles found in 2D vector art work. However, They use a further axis. When creating 2D vector artwork, the computer draws the image by plotting points on X and Y. From this co-ordinates are created. Furthermore by joining these paths or more commonly known as lines; the subsequent shapes can be filled with color. Also, the lines can be stroked with color and thickness if required. All 3D programs operate on a grid of 3D co-ordinates. 3D co-ordinates are pretty much the same as 2D co-ordinates however, there's a third axis in which is called the depth axis or more commonly known as "Z".





Geometry theory and polygons - In mesh modelling the most basic and frequent object used is a Vertex. A vertex is basically a point in 3D space. When two vertices are connected by a straight line it will then be classed as an edge. This then means that if 3 vertices are connected together by three edges; it creates a triangle. In addition, a triangle is the simplest polygon in Euclidean space. However, more polygons can be created out of a bunch of triangles; or as a single object with more than 3 vertices. Four sided polygons are in fact the most common shapes used in polygonal modelling. 4 sided polygons are more commonly known as quads. However, a group of polygons e.g. more that 4 sides; in which are connected to each other by shared vertices are are known as an element. In this case each of the polygons making an element is called a face.


Refering to Euclidean geometry any three non collier points determine a plain. For this reason, triangles always inhabit a single plain. However, this is not necessarily true when it comes to more complex polygons. In addition, the flat nature of triangles makes it more simple to determine there surface normal, a 3D vector perpendicular to the triangles surface. Surface normal's are commonly known for the usefulness of the effect to determine light transport in ray tracing. Furthermore, a group of polygons in which are connected by shared vertices are known as a mesh. A mesh is more commonly known as a wire frame model.


In some occasions it is hard to keep a mesh attractive when rendered. In order the keep a mesh attractive when rendered, it is desirable that it be non-intersecting; in other words, this means that no edge will pass through a polygon. Also it is important that the mesh does not pierce itself. It is also desirable that the mesh does not contain any errors such as a double vertices. The same applies for double edges, faces.  Lastly, it is also important that the mesh is able to be a manifold. ( this means that it does not contain holes or singularities in which are mainly caused when two distinct sections of the mesh are connected by a single vertex.)



Primitives - In 3D applications, pre made objects can be used to make models out of various shapes. The most basic from these shapes are the standard primitive objects, or the common primitives. These shapes vary from the basic cube, cylinders  spheres to pyramids. The cone shape is also included. These shapes are used at the starting point of modelling; then they can be edited once created.





Surfaces - polygons can be defined as specific surfaces. Then they can have color, texture and even photographic maps added to them in which will create the desired look.



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