Plane can use radius and angle to x-axis as coordinates {polar coordinate}, rather than x and y coordinates.
pole
Origin is reference point {pole, coordinate}.
polar axis
x-axis is reference line {polar axis, coordinate}.
coordinates
Plane points have distance to pole {radius, coordinate} and angle to polar axis.
straight line equation
Straight line has equation r * cos(A) = b, where r is radius, A is angle to polar axis, and b is constant. Straight line can have equation r * cos(a - A) = b, where a is angle to axis perpendicular to polar axis {angle of inclination} {inclination angle}. Slope is tan(a).
circle equation
Circle has equation r^2 - 2 * r * c * cos(A) + c^2 = d^2, where r is radius, A is angle to polar axis, and a, c, and d are constants.
parabola equation
Parabola has equation r = (e * a) / (1 - e * cos(A)), where e is eccentricity and a is constant.
hyperbola equation
With polar coordinates centered at a focus, hyperbola has equation r = (e * a) / (1 + e * sin(A)), where e is eccentricity and a is constant.
ellipse equation
Ellipse has equation r = (e * a) / (1 - e * sin(A)), where e is eccentricity and a is constant.
spiral equation
Polar equations r = A * a, where a is constant, graph to spirals {Archimedes spiral}.
rectangular coordinates
Polar coordinates relate to rectangular coordinates. x = r * cos(A), y = r * sin(A), r = (x^2 + y^2)^0.5, and tan(A) = y/x.
cylindrical coordinates
Space points can use plane polar coordinates and reference line perpendicular to the plane from pole {cylindrical coordinate}. Points have distance to pole, distance along perpendicular axis, and angle to polar axis. Cylindrical coordinates relate to rectangular coordinates. x = p * cos(A), y = p * sin(A), and z = z.
spherical coordinates
Space points can use pole and two perpendicular reference lines through pole {spherical coordinate}. Points have radius to pole and two angles to the reference lines. Spherical coordinates relate to rectangular coordinates. x = r * sin(A) * cos(B), y = r * sin(A) * sin(B), and z = r * cos(B). Spherical coordinates relate to cylindrical coordinates. p = r * sin(A), z = r * cos(A), r = (p^2 + z^2)^0.5, and A = arctan(p/z).
Mathematical Sciences>Geometry>Coordinate System
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Date Modified: 2022.0224