Lec.02 - 2. EQUATIONS OF LINES AND PLANES I Main...

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Unformatted text preview: 8/25/11 2. EQUATIONS OF LINES AND PLANES I Main Topics A Direc?on cosines B Lines C Planes 8/25/11 GG303 1 Linea?ons Along a Probable Fault MaJerhorn Peak, California 8/25/11 GG303 2 1 8/25/11 Small Fold Rainbow Basin, California hJp://en.wikipedia.org/wiki/File:Rainbow_Basin.JPG 8/25/11 GG303 3 Deforma?on Bands East Rim of Buckskin Gulch, Utah 8/25/11 GG303 4 2 8/25/11 Shee?ng Joints Yosemite Na?onal Park, California 8/25/11 GG303 5 Fractures Austrian Alps 8/25/11 GG303 6 3 8/25/11 2. EQUATIONS OF LINES AND PLANES II Direc?on cosines A The cosines of the angles between a line and the coordinate axes B The coordinates of the endpoint of a vector of unit length C The ordered projec?on lengths of a line of unit length onto the x, y, and z axes D Spherical coordinates of a unit vector 8/25/11 GG303 7 Direc?on Cosines from Geologic Angle Measurements (Spherical coordinates; z up) 8/25/11 GG303 8 4 8/25/11 Direc?on Cosines from Geologic Angle Measurements (Spherical coordinates; z down) 8/25/11 GG303 9 2. EQUATIONS OF LINES AND PLANES MATLAB Cartesian coordinates Spherical coordinates Spherical coordinates Cartesian coordinates >> [TH,PHI,R] = cart2sph(1,0,0) >> [X,Y,Z] = sph2cart(0,0,1) TH = X = 0 PHI = The θ and ϕ values are angles. R is a length. 1 The x,y,z values here are direc?on cosines Y = 0 0 R = Z = 1 0 8/25/11 GG303 10 5 8/25/11 2. EQUATIONS OF LINES AND PLANES A Line defined by 2 points (two ­point form) y − y1 y2 − y1 = x − x1 x2 − x1 where (x1,y1) and (x2,y2) are two known points on the line 8/25/11 GG303 11 2. EQUATIONS OF LINES AND PLANES B Line defined by 1 point (e.g., x0,y0,z0) and a direc?on 1 Slope ­intercept form (2D) y = mx +b 2 General form (2D) Ax + By + C = 0 3 Parametric form (2D or 3D) x = x0 + tα, y = y0 + tβ, z = z0 + tγ, where α, β, and γ are direc?on cosines: α = cos ωx, β = cos ωy, and (for 3D) γ = cos ωz; In 2 ­D, cos ωx = sin ωy 8/25/11 GG303 12 6 8/25/11 2. EQUATIONS OF LINES AND PLANES C Line defined by the intersec?on of two planes 8/25/11 GG303 13 2. EQUATIONS OF LINES AND PLANES IV Plane A Defined by three points B Defined by two intersec?ng lines C Defined by two parallel lines D Defined by a line and a point not on the line 8/25/11 GG303 14 7 8/25/11 2. EQUATIONS OF LINES AND PLANES IV Plane D Defined by a distance and a direc?on (or pole) from a point not on the line 1  General form: Ax + By + Cz + D = 0 2 Normal form: αx + βy + γz = d α= A ± A + B +C 2 2 2 β= d= 8/25/11 B ± A + B +C 2 2 2 γ= C ± A + B2 + C 2 2 −D ± A + B2 + C 2 2 GG303 15 2. EQUATIONS OF LINES AND PLANES 3  n• V = d, where V is any vector from a given point O to plane P; n is the unit normal vector to plane P with direc?on cosines α, β, and γ; n also goes through point O; d is the distance from O to plane P along n; • refers to the dot product: <x1,y1,z1> • <x2,y2,z2> = x1x2 + y1y2 + z1z2 8/25/11 GG303 16 8 8/25/11 2. EQUATIONS OF LINES AND PLANES 3  n• V = d a The distance from a reference point to a plane (as measured along a direc?on perpendicular to the plane) is d. b The projec?on of V onto n has a length d If n points from the reference point to the plane, then d>0. Otherwise, d<0. 8/25/11 GG303 17 9 ...
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