Geometry
The Cartesian Coordinates System
The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry (from the Classical Greek mathematician Euclid of Alexandria) and algebra (where letters and other general symbols are used to represent numbers and quantities in formulae and equations).
BBC Coordinate Introduction
In 2D vector artwork, but use a further axis. When creating 2D vector artwork, the computer draws the image by plotting points on X and Y axes (creating coordinates) and joining these points with paths (lines). The subsequent shapes can be filled with colour and the lines stroked with colour and thickness if required.
3D programs operate on a grid of 3D co-ordinates. 3D co-ordinates are pretty much the same as 2D co-ordinates except there’s a third axis known as the Z or ‘depth’ axis.
MathFortress.com Introduction
The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry (from the Classical Greek mathematician Euclid of Alexandria) and algebra (where letters and other general symbols are used to represent numbers and quantities in formulae and equations).
BBC Coordinate Introduction
In 2D vector artwork, but use a further axis. When creating 2D vector artwork, the computer draws the image by plotting points on X and Y axes (creating coordinates) and joining these points with paths (lines). The subsequent shapes can be filled with colour and the lines stroked with colour and thickness if required.
3D programs operate on a grid of 3D co-ordinates. 3D co-ordinates are pretty much the same as 2D co-ordinates except there’s a third axis known as the Z or ‘depth’ axis.
MathFortress.com Introduction
Geometric Theory and Polygons
The basic
object used in mesh modeling is a vertex, a point in three dimensional space.
Two vertices connected by a straight line become an edge. Three vertices,
connected to each other by three edges, define a triangle, which is the
simplest polygon in Euclidean space. More complex polygons can be created out
of multiple triangles, or as a single object with more than 3 vertices. Four
sided polygons (generally referred to as quads) and triangles are the most
common shapes used in polygonal modeling. A group of polygons, connected to
each other by shared vertices, is generally referred to as an element. Each of
the polygons making up an element is called a face.
In
Euclidean geometry, any three non-collinear points determine a plane. For this
reason, triangles always inhabit a single plane. This is not necessarily true
of more complex polygons, however. The flat nature of triangles makes it simple
to determine their surface normal, a three-dimensional vector perpendicular to
the triangle's surface. Surface normals are useful for determining light
transport in ray tracing.
A group of
polygons which are connected by shared vertices is referred to as a mesh, often ferred to as a wireframe model.
In
order for a mesh to appear attractive when rendered, it is desirable that it be
non-self-intersecting, meaning that no edge passes through a polygon. Another
way of looking at this is that the mesh cannot pierce itself. It is also
desirable that the mesh not contain any errors such as doubled vertices, edges,
or faces. For some purposes it is important that the mesh be a manifold – that
is, that it does not contain holes or singularities (locations where 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 of this shapes are the Standard Primitive Objects, or
the Common Primitives, these shapes vary from the basic cube or box to spheres,
cylinders, pyramids (both triangular and square based) and cones. They are used as the starting point for modelling. They can be edited once created.
Surfaces
Polygons can be defined as specific surfaces and then have colour, texture or photographic maps added to them to create the desired look. The example below shows how a map is displayed as if the object has been unwrapped.
Sources
http://goanna.cs.rmit.edu.au/~gl/teaching/Interactive3D/2012/images/uv-unwrap.jpg
