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GR-Tu2-Geometry

GR-Tu2-Geometry - CIVL2002 Engineering Geology and Rock...

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1 CIVL2002 Engineering Geology and Rock Mechanics Review of Analytic Geometry for Lines and Planes in Cartesian Coordinate System Tutorial Q2 This photograph shows many discontinuities in a rock mass in Hong Kong Mathematical Model of Discontinuities For simplicity and to be easy in mathematical representation, a discontinuity is generally assumed to be of a geometry of straight plane. A Rock Mass Block A straight plane to represent a discontinuity 1. In this tutorial, we are going to present a brief review on the Analytic Geometry for mathematical representation of straight lines and planes in two and three-dimensional space. 2. Such knowledge are useful in understanding and examining discontinuities in rock mechanics. Objective Two-dimensional Coordinate Systems 1. Cartesian coordinate system O I x y II III IV M ( x , y ) x-a xis = Horizontal axis - infinity < x < + infinity y-a xis = Vertical axis - infinity < y < + infinity 2. Polar coordinate system O x r M ( r , ϕ ) ϕ = hoop angle 0 ϕ < 360 (2 π ) r-a xis = radial axis 0 r < + infinity ϕ
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2 3. Relationship between Cartesian and Polar coordinate systems O x y II III IV M ( x , y ) = r M ( r , ϕ ) ϕ ϕ ϕ sin cos r y r x = = 2 2 y x r + = < + > = 0 arctan 0 arctan x for x y x for x y π ϕ 4. Parallel Translation of Cartesian coordinate system O x y M ( x , y ) O’ ( g, h ) X Y 1. The coordinates for the point O’ in the original coordinate is ( g, h ). 2. So, we have the relations between the original coordinates ( Oxy ) and the new coordinates (O’XY)are x = X + g y = Y + h Three-dimensional Coordinate Systems 1. Cartesian coordinate system O x z M ( x , y, z ) x-a xis and y-axis = two perpendicular horizontal axes -infinity < x < + infinity; -infinity < y < + infinity z-a xis = Vertical axis - infinity < z < + infinity y M’ ( x , y, 0) M z (0, 0 , z ) M x ( x , 0 , 0) M y (0, y, 0) 2. Cylindrical Polar coordinate system O x z M ( r , ϕ , z ) z-a xis = Vertical axis - infinity < z < + infinity y M’ ( r , ϕ , 0) r ϕ ϕ = angle of longitude 0 ϕ < 360 (2 π ) r = radial length 0 r < + infinity 3. Relationship between Cartesian and Cylindrical polar coordinate systems O x z M ( x , y, z ) y M’ ( x , y, 0) M z (0, 0 , z ) M x ( x , 0 , 0) M y (0, y, 0) = ( r , ϕ , z ) r ϕ = ( r , ϕ , 0) z z r y r x = = = ϕ ϕ sin cos x y y x r arctan 2 2 = + = ϕ 4. Spherical polar coordinate system O x z M ϕ -a xis = angle of longitude; 0 ϕ < 360 0 (2 π ) R-a xis = radial length; 0 R < + infinity y M’ ( x , y, 0) M z (0, 0 , z ) ϕ γ γ -a xis = angle of latitude; 0 γ < 180 0 ( π ) ( R , ϕ , γ ) R (0, ϕ , r )
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3 5. Relationship between Cartesian and Spherical polar coordinate systems O x z M ( x , y , z ) =( R , ϕ , γ ) y M’ ( x , y, 0) M z (0, 0 , z ) M x ( x , 0 , 0) M y
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