Polars

Polar math uses a 2D coordinate system where each point is determined by a distance r from a central point (pole) and an angle, 0, from a reference direction

Polar Coordinates

  • [r,0] is how a point is defined in a polar graph where r is the distance from the origin (pole) and 0 is the angle meausre

  • to go from rectangular (x,y) to polar coordinates x² + y² = r² and arctan(y/x)=0

  • to go from polar to rectangual coordiantes x = rsin0 and y = rcos0

  • a negative r results in a reflection over the pole

Polar Lines

  • r = a sec0 is a vertical line at x = a

  • r = a csc0 is a horizontal line at y = a

  • 0 = k is an oblique line passing through the origin and having a slope of tan (k)

  • the equation r = d/(bcos 0 + csin 0) is an oblique line with a slope of -b/c, a y intercept of (0, d/c) and an x-intercept of (d/b, 0)

Polar Circles, Roses, and Limacons

  • r = k is a circle centered at the pole with a radius of k

  • r = dcos0 is a circle on the x-axis, where d is the diameter of the circle

  • r = dsin0 is a circle on the y-axis, where d is the diameter of the circle

  • r = dcos(0a) is a rose with a petals if its odd or 2a if its even, a radius of d and starts with the tip on the positive x axis

  • r = dsin(0a) is a rose with a petals if its odd or 2a if its even, a radius of d and starts with the tip on the y axis if its an odd number of petals or at pi over 4 if its an even number

  • r = a + bcos(0) where a < b is a limacon with an inner loop where the distance from the outer loop to the inner loop is a + b and b - a

  • r = a + bcos(0) where a = b is a cardioid, where the distance from the pole to the outer circle is a + b

  • r = a + bcos(0) where a > b is a limacon with a dent

Conics in Polar Form

  • conics in polar form follow the equation r = ed/ 1 ± ecos(0 - 01) where e is the eccentricity of the conic, d is the distance to the directrix, and 01 is the orientation angle

Polar Spirals

  • r = a + b0 is an Archimedean spiral with a constant distance between the arms of the spiral

  • r = ab^0 is a logarithmic spiral with a constant ratio between the arms of the spiral