geometric theory
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Geometric Theory. By Eugene Adams. Vertices Edges Faces Polygons Meshes 2D co-ordinates 3D co-ordinates. Vertices. - PowerPoint PPT PresentationTRANSCRIPT
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By Eugene Adams
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Vertices Edges Faces Polygons Meshes 2D co-ordinates 3D co-ordinates
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A vertex is the singular form of vertices. The term is used in geometry to describe the corner point of a shape: In laymans terms, the point at which a line either terminates or is intersected is a vertex.
In computer graphics a vertex is associated not only with the three spatial coordinates which dictate its location, but also with any other graphical information necessary to render the object correctly.
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On the Tetrahedron to the left, I have circled all of the vertices on the shape. Being a point on a shape, a vertex has no standardised form and changes according to the shape it is a part of.
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In geometrical terms, an edge is a 1 dimensional line that connects two 0 dimensional vertices.
When a minimum of 3 vertices are combined with a minimum of 3 edges, a 2D polygon (a Triangle in this case) or ‘face’ is created. Adding in a minimum of 1 more vertex and 3 more edges will transform the 2D polygon into a 3D polyhedron (in this case a Tetrahedron).
This means that to create a 3D shape, a minimum of 4 vertices and 6 edges are required.
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2 vertices connected by a single edge.
3 vertices and 3 edges creating a face or triangular polygon.
4 vertices and 6 edges creating a tetrahedron.
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The term polyhedron is a greek term meaning “many faces”. Any 2D shape can be a face, combining several to create a polyhedron (e.g. 6 squares can combine to create a cube), thus making the term very appropriate.
A face is created using 3 or more vertices and 3 or more edges in a closed pattern.
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A single face (or ‘hedra’), in this case a square.
A combination of 6 sqares to create a cube. Being made of 6 faces, the cube is thus a hexahedron.
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The word “polygon” is derived from the Greek words “poly” meaning “many”, and “gonia”, meaning “knee” or “angle”. In moodern geometry, however, a polygon is usually defined according to its amount of sides rather than its angles.
A polygon is always 2D, and many polygons can be used as faces to create 3D Polyhedrons. A polygon is composed of a closed circuit of vertices and edges, with a minimum of 3 vertices and 3 edges required.
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A polygon mesh is a combination of vertices, edges and faces that form a Polyhedron. The faces used are usually triangles to simplify rendering, though (in theory) any shape can be used.
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A dolphin made using a triangle mesh.
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