Greenwood press
Download 1.81 Mb. Pdf ko'rish
|
book-20600
- Bu sahifa navigatsiya:
- Complex Numbers
POLAR COORDINATES
73 Graph of r = 4 cos 5θ for θ = 0 to 2π. Some graphs are easier to represent in polar rather than rectangular form, especially if they have rotational symmetry. logarithmic spiral r = 5(1.3) θ shell of a chambered nautilus A nautilus resembles the polar graph r = ab θ . The chambered nautilus picture from the previous figure after being distorted with the polar coordinate filter in Adobe Photoshop. Log-polar transformations have been developed to embed copyright infor- mation in computer-graphic files to preserve the copyright notice from deletion. Spherical polar coordinates are useful in simplifying physics equations such as Schrodinger’s equation and the Maxwell speed equation. In many cases, writ- ing expressions in polar form simplifies the application of calculus and differen- tial equation techniques. Polar coordinates have another important application in mathematics: They simplify some operations with complex numbers. Multiplication of the complex numbers z = a + bi and w = c + di gives (ac − bd) + (ad + bc)i. The corre- sponding multiplication in polar form would be of the numbers z = [r, θ] and w = [s, φ]. Then zw = [rs, θ + φ]. The polar form simplifies the powers and roots of complex numbers. In polar form, z n = [r n , nθ], which is known as De- Moivre’s theorem. (See Complex Numbers and Vectors.) online sources for further exploration Polar plotting and graphing Spherical and cylindrical coordinates Logarithmic spirals Sun position in polar coordinates Azimuth and elevation Computer graphics Polar coordinates in robotics Polar distortion filter ▲ ▼ ▲ Download 1.81 Mb. Do'stlaringiz bilan baham: |
Ma'lumotlar bazasi mualliflik huquqi bilan himoyalangan ©fayllar.org 2024
ma'muriyatiga murojaat qiling
ma'muriyatiga murojaat qiling