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Unformatted text preview: 8/25/11 1. ORIENTATIONS OF LINES AND PLANES IN SPACE I Main Topics A Deﬁni?ons of points, lines, and planes B Geologic methods for describing lines and planes C ALtude symbols for geologic maps D Reference frames 8/25/11 GG303 1 1. ORIENTATIONS OF LINES AND PLANES IN SPACE II Deﬁni?ons of points, lines, and planes A Point 1 Deﬁned by one set of coordinates (an ordered triple in 3
D) 2 Deﬁned by distance and direc?on from a reference point 3 Intersec?on of two lines 4 Intersec?on of three planes 8/25/11 GG303 2 1 8/25/11 1. ORIENTATIONS OF LINES AND PLANES IN SPACE B Line 1 Deﬁned by two sets of coordinates 2 Deﬁned by two points 3 Deﬁned by a direc?on from a reference point 4 Intersec?on of two planes 8/25/11 GG303 3 1. ORIENTATIONS OF LINES AND PLANES IN SPACE C Plane 1 Deﬁned by three sets of coordinates 2 Deﬁned by three points 3 Deﬁned by distance and direc?on from a reference point and the direc?on of the line 4 Deﬁned by two intersec?ng or two parallel lines 8/25/11 GG303 4 2 8/25/11 1. ORIENTATIONS OF LINES AND PLANES IN SPACE III Geologic methods for describing lines and planes A Orienta?on of a line 1 Trend & plunge a Trend: Direc?on (azimuth) of a ver?cal plane containing the line of interest. i Azimuth (compass bearing): direc?on of a horizontal line contained in a ver?cal plane. Measured by quadrant or (°). Examples: 90° or N90°E, 270° or N90°W, 270° or S90°W. Ii The trend "points" in the direc?on a line plunges b Plunge: The inclina?on of a line below the horizontal (e.g., 35°) 8/25/11 GG303 5 1. ORIENTATIONS OF LINES AND PLANES IN SPACE III Geologic methods for describing lines and planes A Orienta?on of a line 2 Pitch (or rake): the angle, measured in a plane of speciﬁed orienta?on, between one line and a horizontal line 8/25/11 GG303 6 3 8/25/11 1. ORIENTATIONS OF LINES AND PLANES IN SPACE B Orienta.on of a plane 1 Orienta?on of two intersec?ng lines in the plane Strike & dip a Strike: direc?on of the line of intersec?on between an inclined plane and a horizontal plane (e.g., a lake); b Dip: inclina?on of a plane below the horizontal; 0°≤dip≤90° c The azimuth direc?ons of strike and dip are perpendicular d Good idea to specify the direc?on of dip to eliminate ambiguity, but right hand rule can also be used. 8/25/11 GG303 7 1. ORIENTATIONS OF LINES AND PLANES IN SPACE e Examples: Strike 270°
Dip 45°N right
handed Strike N90°W Dip 45°N right
handed Strike 270°W Dip 45°S
lee
handed: don’t use! f NOTE: Trend and plunge refer to lines; strike and dip refer to planes 8/25/11 GG303 8 4 8/25/11 1. ORIENTATIONS OF LINES AND PLANES IN SPACE 2 Orienta?on of one special line in a plane Dip & dip direc?on (azimuth of dip) a Used mostly in Europe b Water runs down the dip direc?on 8/25/11 GG303 9 1. ORIENTATIONS OF LINES AND PLANES IN SPACE 3 Trend & plunge of pole (unit normal) to plane a Pole is a line tradi?onally taken to point down b The pole trends 180° from the direc?on the plane dips; c Pole trend = strike
90° d Pole plunge + dip = 90°; Pole plunge = 90°
dip; 8/25/11 GG303 Note: In the map
view diagram, the plane dips to the east, and its pole trends to the west 10 5 8/25/11 1. ORIENTATIONS OF LINES AND PLANES IN SPACE IV ALtude symbols (strike and dip of a plane; trend and plunge of a line) The symbols should be ploled on a map with the circled part of the the symbols at the point where the measurements are made 8/25/11 GG303 11 1. ORIENTATIONS OF LINES AND PLANES IN SPACE V Reference frames A Cartesian coordinates 1 Points are described by their x, y, z coordinates 2 The x,y, and z axes are right
handed and mutually perpendicular 3 Direc?on of a line a Given by the coordinates of pairs of points b Given by the diﬀerence in coordinates of pairs of points i Δx = x2
x1 ii Δy = y2
y1 iii Δz = z2
z1 c Given by the angles ωx, ωy, and ωz. These are the angles between a line of unit length and the x, y, and z axes, respec?vely. d The respec?ve cosines (α, β, γ) of these angles are called the direc?on cosines. Angle ωx Direc?on cosine α 8/25/11 GG303 ωy ωz β γ 12 6 8/25/11 1. ORIENTATIONS OF LINES AND PLANES IN SPACE 4 Length of a line segment (distance between two points) 2
2
2
L= ( Δx ) + ( Δy ) + ( Δz ) A line of unit length has a length of one. 8/25/11 GG303 13 1. ORIENTATIONS OF LINES AND PLANES IN SPACE B Cylindrical coordinates (3
D version of polar coordinates) 1 Point P is described by its ρ, θ, z coordinates a ρ = distance from the origin to P', where P' is the projec?on of point P onto x,y plane b θ = angle between x
axis and OP’ c z = distance from origin to projec?on of point P onto z
axis 8/25/11 GG303 14 7 8/25/11 1. ORIENTATIONS OF LINES AND PLANES IN SPACE C Spherical coordinates 1 Points are described by their r, θ, ϕ coordinates a r = distance from the origin to point P Note: r here is diﬀerent than ρ for cylindrical coordinates b θ = angle between x
axis and the OP', where P' is the projec?on of point P onto x,y plane c ϕ = angle between OP' and OP Note: In some spherical schemes, the angle between OP and the z
axis is used as the second angle. 8/25/11 GG303 15 1. ORIENTATIONS OF LINES AND PLANES IN SPACE Cartesian (x,y,z) Spherical (r,θ,ϕ) x = r cosϕ cosθ α = cosϕ cosθ y = r cosϕ sinθ β = cosϕ cosθ z = r sinθ γ = sinθ Spherical (r,θ,ϕ) Cartesian (x,y,z) r = (x2 + y2 + z2)1/2 1 = (α2 + β2 + γ2)1/2 θ = tan
1(y/x) θ = tan
1(β/α) ϕ = tan
1 [z/(y2 + z2)1/2] ϕ = sin
1(γ) Cartesian (x,y,z) Cylindrical (ρ,θ,z) x = ρ cosθ y = ρ sinθ z = z Cylindrical (ρ,θ,z) Cartesian (x,y,z) ρ = (x2 + y2)1/2 θ = tan
1(y/x) z = z Cylindrical (ρ,θ,z) Spherical (r, θ,ϕ) ρ = r cosθ θ = θ z = r sinθ Spherical (r, θ,ϕ) Cylindrical (ρ,θ,z) r = (ρ2 + z2)1/2 θ = θ ϕ = tan
1(z/ρ) Note: some references deﬁne the angles in cylindrical and spherical coordinates diﬀerently. Exercise care, and check to ensure that the equa?ons are consistent with the graphics. The deﬁni?ons here are consistent with Matlab. 8/25/11 GG303 16 8 8/25/11 1. ORIENTATIONS OF LINES AND PLANES IN SPACE 8/25/11 GG303 17 1. ORIENTATIONS OF LINES AND PLANES IN SPACE 8/25/11 GG303 18 9 8/25/11 1. ORIENTATIONS OF LINES AND PLANES IN SPACE 8/25/11 GG303 19 1. ORIENTATIONS OF LINES AND PLANES IN SPACE 8/25/11 GG303 20 10 ...
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 Fall '11
 StephenMartel

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