Lec2Defs - Definitions Coordinate systems and dimensions...

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Definitions Coordinate systems and dimensions The objects considered in Computational Geometry are points, lines, line segments, polygons, polyhedron, hyper-rctacgles etc. A coordinate system provides a means to specify positions or points in space. The Cartesian coordinate system labels a d -dimensional space with d mutually perpendicular (orthogonal) coordinate axes, one per dimension. d -dimensional space ( d -space) Notation d defined as number of dimensions of space or a geometric object. As a prefix on the name of an object, d - denotes the number of dimensions of the object, e.g. d -rectangle or 2-rectangle. We will most often work in d = 2 (plane), which is the default, sometimes in d = 1 or d > 2. 0 x y 0 x y z d = 2 d = 3 Right-handed coordinate system
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Definitions Point Object with d dimensions and 0 extent. Location in d -space. Given as an ordered sequence of d coordinates. d = 1 ( x ) or x d = 2 ( x , y ) d = 3 ( x , y , z ) d 4 ( x 1 , x 2 , . .., x d ) or ( x 0 , x 1 , . .., x d -1 ) p 0 Line Infinite “straight” 1-dimensional set of points, determined by two points p 0 , p 1 220d p 0 p 1 . p 0 p 1
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Definitions Ray Infinite 1-dimensional subset of a line determined by two points p 0 , p 1 220d p 0 p 1 , where one point is denoted as the endpoint. p 0 Segment Finite 1-dimensional subset of a line, determined by two endpoints p 0 , p 1 . p 0 p 1
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Preliminaries Point-Line classification We now consider the geometric primitive operation of classifying a point w.r.t. a line (both in the plane). A directed line segment partitions the plane into 7 non-overlapping regions. The possibilities are shown below. The problem, given p 0 , p 1 , and p 2 , is to determine which region p 2 lies in. p 0 p 1 terminus origin beyond right left behind between
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Preliminaries Parametric equation of a line We use the following equation of a line: line = { α ( p 0 ) + (1 - α29 ( p 1 ) }, where α ∈ ℜ (real numbers) where p 0 and p 1 as usual are the points determining the line. p 0 = ( x 0 , y 0 ) p 1 = ( x 1 , y 1 ) Substituting gives { α ( x 0 , y 0 ) + (1 - ( x 1 , y 1 ) } Multiplying through gives the coordinates { α x 0 + (1 - x 1 , α y 0 + (1 - y 1 } Work out an example with points (4,3) and (7,5) as the two end points with values of α as 0, 1, 0.5, 2 and -3. For example, when α equal to 2 , (x,y)=(2 x 0 x 1 , 2 y 0 - y 1 ) = (1,1). p 1 p 0 α > 1 α = 1 α = 0 α < 0 0< α < 1
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Line Segment A line segment is a closed subset of a line contained between two points which are called the end points. The subset is closed in the sense that it includes the end points. The equation of the line segment is the same as the parametric equation of a line with the restriction that α has the value 0 <= α<= 1. This is also called the convex combination of the two end points.
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Explicit Form of Line Equation y= mx +c m=slope=tan θ where θ is the angle made by the line with positive x-axis c=intercept of the line with the y-axis. Vertical line with x=k cannot be represented since these lines have infinite slopes.
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This note was uploaded on 07/14/2011 for the course COT 5520 taught by Professor Mukherjee during the Summer '11 term at University of Central Florida.

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Lec2Defs - Definitions Coordinate systems and dimensions...

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