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 Coordinates System
In the 17th century, René Descartes revolutionized mathematics by creating a link between Euclidean geometry ( http://en.wikipedia.org/wiki/Euclidean_geometry ) and Algebra ( In which letters are used to represent numbers).
"A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin." - http://en.wikipedia.org/wiki/Cartesian_coordinate_system
This coordinate system is widely used in the games industry to create accurate measurements and keep things in shape. The image below shows a grid that uses the Cartesian coordinate system.The creator of this piece has used this system to create the outline of the weapon by joining together points of the grid.
Geometric Theory and polygons
http://en.wikipedia.org/wiki/Polygonal_modeling
In a three-dimensional space, the basic object used is something known as a vertex. This is a point located within a 3D space usually on a coordinate system. Objects are created when vertices join together. Two vertices joining create a straight line, three joining create a triangle. This triangle would be called a polygon, but other shapes such as quads ( 4 vertices which join ) or anything higher can also be called a polygon. A triangular polygon however is more widely used than the others as it is the most stable shape, as it will not bend the 'face' of the shape when a vertex is lifted or moved. A group of polygons is usually known as an element and a single polygon is two dimensional and must be connected at the various vertices by straight lines.
"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, and are a key component of the popular Phong shading model. Some rendering systems use vertex normals instead of face normals to create a better-looking lighting system at the cost of more processing. Note that every triangle has two face normals, which point to opposite directions from each other. In many systems only one of these normals is considered valid – the other side of the polygon is referred to as a backface, and can be made visible or invisible depending on the programmer’s desires." - Extract from the wiki link above. This paragraph mainly describes triangular polygons and their simpler structure.
The images above show just how models are made using polygons. The use of triangular polygons is mainly shown here and you can see that the characters entire bodies have been made using polygonal shapes.
http://www.mathsisfun.com/geometry/polygons.html
Primitives
In 3D modelling programs, 'primitive' shapes are used which are basic shapes such as cones, cubes, spheres and cylinders. These shapes are used as a starting point for modelling, so for example, if you wanted to make a camera, you could use the cube and cylinder primitives as a basic starting point for the model and then use the program to modify these shapes to make the object more realistic. All primitives can be modified in various ways such as point editing, stretch, sizing and you are also able to modify the polygons that make up the shape.
Surfaces
Once a model is created, all the polygons displayed on the exterior of the shape can be known as the surface. This surface can then be modified to add textures, colour or detail. They can also have special photographic maps. The image below shows a baby but on the right of the image is the map. This special map is designed to fit around the model, so once added, the drawings on the map have been coded to wrap around the end model and fit around it, so you can draw on the map and whatever you do will appear on the model.
Geometry
The Cartesian Coordinates System
In the 17th century, René Descartes revolutionized mathematics by creating a link between Euclidean geometry ( http://en.wikipedia.org/wiki/Euclidean_geometry ) and Algebra ( In which letters are used to represent numbers).
"A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin." - http://en.wikipedia.org/wiki/Cartesian_coordinate_system
This coordinate system is widely used in the games industry to create accurate measurements and keep things in shape. The image below shows a grid that uses the Cartesian coordinate system.The creator of this piece has used this system to create the outline of the weapon by joining together points of the grid.
Geometric Theory and polygons
http://en.wikipedia.org/wiki/Polygonal_modeling
In a three-dimensional space, the basic object used is something known as a vertex. This is a point located within a 3D space usually on a coordinate system. Objects are created when vertices join together. Two vertices joining create a straight line, three joining create a triangle. This triangle would be called a polygon, but other shapes such as quads ( 4 vertices which join ) or anything higher can also be called a polygon. A triangular polygon however is more widely used than the others as it is the most stable shape, as it will not bend the 'face' of the shape when a vertex is lifted or moved. A group of polygons is usually known as an element and a single polygon is two dimensional and must be connected at the various vertices by straight lines.
"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, and are a key component of the popular Phong shading model. Some rendering systems use vertex normals instead of face normals to create a better-looking lighting system at the cost of more processing. Note that every triangle has two face normals, which point to opposite directions from each other. In many systems only one of these normals is considered valid – the other side of the polygon is referred to as a backface, and can be made visible or invisible depending on the programmer’s desires." - Extract from the wiki link above. This paragraph mainly describes triangular polygons and their simpler structure.
The images above show just how models are made using polygons. The use of triangular polygons is mainly shown here and you can see that the characters entire bodies have been made using polygonal shapes.
http://www.mathsisfun.com/geometry/polygons.html
Primitives
In 3D modelling programs, 'primitive' shapes are used which are basic shapes such as cones, cubes, spheres and cylinders. These shapes are used as a starting point for modelling, so for example, if you wanted to make a camera, you could use the cube and cylinder primitives as a basic starting point for the model and then use the program to modify these shapes to make the object more realistic. All primitives can be modified in various ways such as point editing, stretch, sizing and you are also able to modify the polygons that make up the shape.
Surfaces
Once a model is created, all the polygons displayed on the exterior of the shape can be known as the surface. This surface can then be modified to add textures, colour or detail. They can also have special photographic maps. The image below shows a baby but on the right of the image is the map. This special map is designed to fit around the model, so once added, the drawings on the map have been coded to wrap around the end model and fit around it, so you can draw on the map and whatever you do will appear on the model.
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