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COORDINATE SYSTEM.pdf
- 1. COORDINATE SYSTEM Manju T. Kurian AP, ECE BM II CE
- 2. COORDINATE SYSTEM & TRANSFORMATION • Coordinate systems are defined as a system used to represent a point in space • Classified a Orthogonal and Nonorthogonal Coordinate system • For orthogonal coordinate system, the coordinates are mutually perpendicular. The orthogonal coordinate systems include • Rectangular or Cartesian coordinate system • Cylindrical or circular coordinate system • Spherical coordinate system
- 3. Rectangular Coordinate System • This system is formed by three mutually orthogonal straight lines. The three straight lines are called x,y,z axis. The point of intersection of these lines are called the origin • We will use the unit vectors ෞ 𝑎𝑥, ෞ 𝑎𝑦, ෞ 𝑎𝑧 to indicate the direction of components along x,y,z axis respectively • Range of coordinates x,y,z are −∞ < 𝑥 < ∞ −∞ < 𝑦 < ∞ −∞ < 𝑧 < ∞ • Vector Ԧ 𝐴 in cartesian coordinate can be written as • Ԧ 𝐴=(Ax , Ay , A𝑧) =Ax ෞ 𝑎𝑥+ Ay ෞ 𝑎𝑦 + A𝑧ෞ 𝑎𝑧
- 4. • The position vector ത 𝒓 , a vector directed from the origin 0 to point P, P(x,y,x) is given by 𝑂𝑃 = 𝑂𝐴 + 𝑂𝐵 + 𝑂𝐶 ҧ 𝑟 = 𝑥ෞ 𝑎𝑥 + 𝑦ෞ 𝑎𝑦 + 𝑧ෞ 𝑎𝑧 • x,y,z are scalar projections of ҧ 𝑟 𝑜𝑛 x,y and z axis • Magnitude of ത 𝒓 is given by ҧ 𝑟 = 𝑥2 + 𝑦2 + 𝑧2 • Orientation of ҧ 𝑟 on the x,y,z axis is given by 𝛼, 𝛽, 𝛾 𝑎𝑟𝑒 called Direction Angles ҧ 𝑟 𝑟 = 𝑥 𝑟 ෞ 𝑎𝑥 + 𝑦 𝑟 ෞ 𝑎𝑦 + 𝑧 𝑟 ෞ 𝑎𝑧 =Cos 𝛼 ෞ 𝑎𝑥+ Cos𝛽ෞ 𝑎𝑦+ Cos𝛾ෞ 𝑎𝑧 =𝑙𝑥 ෞ 𝑎𝑥+𝑙𝑦 ෞ 𝑎𝑦+ 𝑙𝑧ෞ 𝑎𝑧 𝑙𝑥, 𝑙𝑦, 𝑙𝑧 are known as Direction Cosines
- 5. • The unit vector along the direction of the position vector can be expressed in terms of the direction cosines. • Direction cosines are nothing but the magnitude of the position vector along the coordinate axis • Also Cos 𝛼 + Cos𝛽 + Cos𝛾 = 1 • Components in a specific direction Consider a unit vector in a specific direction. The projection of Ԧ 𝐴 along the desired direction is A.ො 𝑛 =A.Cos𝜃 This gives the magnitude of projection. Scalar Component of Ԧ 𝐴 on 𝐵 = Ԧ 𝐴. ෞ 𝑎𝐵 Vector Component of Ԧ 𝐴 on 𝐵 =( Ԧ 𝐴. ෞ 𝑎𝐵) ෞ 𝑎𝐵
- 6. Relative Position Vector (Rij) • Vector expression for Coulombs law, electric field intensity etc contains the position vectors relative to the points other than the origin. • Using the law of addition of vectors, 𝑟1 + 𝑟12 = 𝑟2 𝑟12 = 𝑅 = 𝑟2 − 𝑟1 • This can be considered as the position vector of (2) w.r.t position vector(1), hence the name relative position vector • The Unit vector of a relative position vector is 𝑎12= 𝑟12 𝑟12 = 𝑟2−𝑟1 𝑟2−𝑟1 , 𝑟2 − 𝑟1 = 𝑟1 − 𝑟2 Similarly, 𝑎𝑖𝑗 = 𝑟𝑗−𝑟𝑖 𝑟𝑗−𝑟𝑖
- 7. Cylindrical Coordinate System • A point P in the cylindrical coordinate system is represented as P(𝜌, ∅, 𝑧) • Range of Coordinates • Unit vectors 0≤ 𝜌 ≤ ∞ 0≤ ∅ ≤ 2𝜋 -∞ ≤ 𝑧 ≤ ∞ 𝑎𝜌 → 𝑎𝑙𝑜𝑛𝑔 𝜌 →Radius of the cylinder 𝑎∅ → 𝑎𝑙𝑜𝑛𝑔 ∅ →Azimuthal angle 𝑎𝑧 → 𝑎𝑙𝑜𝑛𝑔 𝑧 → 𝑠𝑎𝑚𝑒 𝑎𝑠 𝑖𝑛 𝑐𝑎𝑟𝑡𝑒𝑠𝑖𝑎𝑛 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 𝑠𝑦𝑠𝑡𝑒𝑚
- 8. Spherical Coordinate System • A point P in the Spherical coordinate system is represented as P(r, 𝜃, ∅) • Range of Coordinates • Unit vectors 0≤ 𝑟 ≤ ∞ 0≤ 𝜃 ≤ 𝜋 0≤ ∅ ≤ 2𝜋 𝑎𝑟 → 𝑎𝑙𝑜𝑛𝑔 r →Radius of the sphere(meters) 𝑎𝜃 → 𝑎𝑙𝑜𝑛𝑔 𝜃 →Angle of elevation measured from z-axis (radians) 𝑎∅ → 𝑎𝑙𝑜𝑛𝑔 ∅ → Azimuthal Angle(𝑟𝑎𝑑𝑖𝑎𝑛𝑠)
- 9. Rectangular to Cylindrical Coordinate system • 𝜌 = 𝑥2 + 𝑦2 • ∅ = 𝑡𝑎𝑛−1 (𝑦 ⁄ 𝑥) • 𝑧 = 𝑧 • Sin ∅= 𝑦 𝜌 , Cos ∅= 𝑥 𝜌 , z=z • X= 𝜌. Cos ∅ • y= 𝜌. Sin ∅ • Z=z Cylindrical to Rectangular Coordinate system
- 10. Relationship between various coordinate system • (x,y,z)↔ 𝜌, ∅, 𝑧 • (x,y,z)↔ (𝑟, 𝜃, ∅) 𝜌 = 𝑥2 + 𝑦2 X= 𝜌. Cos ∅ ∅ = 𝑡𝑎𝑛−1(𝑦 ⁄ 𝑥) y= 𝜌. Sin ∅ z=z z=z 𝑆𝑖𝑛𝜃 = 𝐶𝑃 𝑟 CP=r.Sin𝜃 Cos𝜃 = 𝑂𝐶 𝑟 = 𝑧 𝑟 z=r.Cos𝜃 = 𝑧 𝑆𝑖𝑛∅ = 𝐸𝐷 𝐶𝑃 = 𝑦 r.Sin𝜃 y= r.Sin𝜃. 𝑆𝑖𝑛∅ Cos∅ = 𝑂𝐸 𝐶𝑃 = 𝑥 r.Sin𝜃 x= r.Sin𝜃. Cos ∅ x= r.Sin𝜃. Cos ∅ r= 𝑥2 + 𝑦2 + 𝑧2 y= r.Sin𝜃. 𝑆𝑖𝑛∅ 𝜃=𝑡𝑎𝑛−1 𝑥2+𝑦2 𝑧 Z=r.Cos 𝜃 ∅=𝑡𝑎𝑛−1 𝑦 𝑥
- 11. • (𝜌, ∅, 𝑧)↔ 𝑟, 𝜃, ∅ 𝜌=r.Sin 𝜃 r2= 𝜌2+z2 ∅ = ∅ tan 𝜃 = 𝜌 𝑧 Z=r.Cos 𝜃 ∅ = ∅
- 12. Transformation Matrix • Rectangular to Cylindrical Coordinate System 𝑎𝑥 𝑎𝑦 𝑎𝑧 = 𝐶𝑜𝑠∅ −𝑆𝑖𝑛∅ 0 𝑆𝑖𝑛∅ 𝐶𝑜𝑠∅ 0 0 0 1 𝑎𝜌 𝑎∅ 𝑎𝑧 • Cylindrical to Rectangular Coordinate System 𝑎𝜌 𝑎∅ 𝑎𝑧 = 𝐶𝑜𝑠∅ 𝑆𝑖𝑛∅ 0 −𝑆𝑖𝑛∅ 𝐶𝑜𝑠∅ 0 0 0 1 𝑎𝑥 𝑎𝑦 𝑎𝑧
- 13. Transformation Matrix • Rectangular to Spherical Coordinate System 𝑎𝑥 𝑎𝑦 𝑎𝑧 = 𝑆𝑖𝑛𝜃. 𝐶𝑜𝑠∅ 𝐶𝑜𝑠𝜃. 𝐶𝑜𝑠∅ −𝑆𝑖𝑛∅ 𝑆𝑖𝑛𝜃. 𝑆𝑖𝑛∅ 𝐶𝑜𝑠𝜃. 𝑆𝑖𝑛∅ 𝐶𝑜𝑠∅ 𝐶𝑜𝑠𝜃 −𝑆𝑖𝑛𝜃 0 𝑎𝑟 𝑎𝜃 𝑎∅ • Spherical to Rectangular Coordinate System 𝑎𝑟 𝑎𝜃 𝑎∅ = 𝑆𝑖𝑛𝜃. 𝐶𝑜𝑠∅ 𝑆𝑖𝑛𝜃. 𝑆𝑖𝑛∅ 𝐶𝑜𝑠𝜃 𝐶𝑜𝑠𝜃. 𝐶𝑜𝑠∅ 𝐶𝑜𝑠𝜃. 𝑆𝑖𝑛∅ −𝑆𝑖𝑛𝜃 −𝑆𝑖𝑛∅ 𝐶𝑜𝑠∅ 0 𝑎𝑥 𝑎𝑦 𝑎𝑧
- 14. DIFFERENTIAL VECTOR • A differential vector or line element dl in Rectangular coordinate system is given by • ഥ 𝑑𝑙 = 𝑑𝑥ෞ 𝑎𝑥+ dyෞ 𝑎𝑦+ dzෞ 𝑎𝑧 𝑑𝑙 = 𝑑𝑥2 + 𝑑𝑦2 + 𝑑𝑧2 • A differential volume element is given by dv=dxdydz
- 15. Differential vector in Cylindrical coordinate system • A differential vector or line element dl in Cylindrical coordinate system is given by ഥ 𝑑𝑙 = 𝑑𝑙𝜌ෞ 𝑎𝜌+ 𝑑𝑙∅ෞ 𝑎∅+ 𝑑𝑙𝑧ෞ 𝑎𝑧 = 𝑑ρෞ 𝑎𝜌+ ρ𝑑∅ෞ 𝑎∅+ 𝑑𝑧ෞ 𝑎𝑧 • Differential arc length 𝑑𝑙 = (𝑑ρ)2+ (ρ𝑑∅)2+ (𝑑𝑧)2 • Differential area in the lateral surface of a cylinder ρ𝑑∅𝑑𝑧. ෞ 𝑎𝜌 Differential in Coordinate Corresponding change in length 𝑑𝜌 𝑑∅ 𝑑𝑧 𝑑𝑙𝜌=𝑑𝜌 𝑑𝑙∅=𝜌𝑑∅ 𝑑𝑙𝑧=dz
- 16. Differential vector in Spherical coordinate system • A differential vector or line element dl in Spherical coordinate system is given by ഥ 𝑑𝑙 = 𝑑𝑙𝑟ෞ 𝑎𝑟+ 𝑑𝑙𝜃 ෞ 𝑎𝜃 + 𝑑𝑙∅ෞ 𝑎∅ = 𝑑𝑟ෞ 𝑎𝑟+ 𝑟𝑑𝜃ෞ 𝑎𝜃+ 𝑟. 𝑆𝑖𝑛𝜃. 𝑑∅ෞ 𝑎∅ • Differential arc length 𝑑𝑙 = (𝑑𝑟)2+ (𝑟𝑑𝜃)2+ (𝑟. 𝑆𝑖𝑛𝜃. 𝑑∅)2 • Differential area on the Spherical Surface 𝑟𝑑𝜃𝑟.𝑆𝑖𝑛𝜃𝑑∅ • Differential Volume (𝑑𝑟)(𝑟𝑑𝜃)(𝑟.𝑆𝑖𝑛𝜃𝑑∅) Differential in Coordinate Corresponding change in length 𝑑𝑟 𝑑𝜃 𝑑∅ 𝑑𝑙𝑟=𝑑𝑟 𝑑𝑙𝜃 =r𝑑𝜃 𝑑𝑙∅=𝑟. 𝑆𝑖𝑛𝜃. 𝑑∅
- 18. Curvilinear, Cartesian, Cylindrical, Spherical Curvilinear Cartesian Cylindrical Spherical Coordinate 𝑈1 x 𝜌 r 𝑈2 y ∅ 𝜃 𝑈3 z z ∅ Scale factor ℎ1 1 1 1 ℎ2 1 𝜌 r ℎ3 1 1 r. Sin𝜃 Unit Vector 𝑒1 ෞ 𝑎𝑥 ෞ 𝑎𝜌 ෞ 𝑎𝑟 𝑒2 ෞ 𝑎𝑦 ෞ 𝑎∅ ෞ 𝑎𝜃 𝑒3 ෞ 𝑎𝑧 ෞ 𝑎𝑧 ෞ 𝑎∅ Differential length 𝑑𝑙1=ℎ1𝑈1 𝑑1 𝑑𝜌 𝑑𝑟 𝑑𝑙2=ℎ2𝑈2 𝑑2 𝜌 𝑑∅ r𝑑𝜃 𝑑𝑙3=ℎ3𝑈3 𝑑3 𝑑𝑧 r. Sin𝜃 𝑑∅