Points in planes can have distances {bipolar coordinate} to two fixed points.
Points in planes have distances to x-axis {abscissa} and distances to y-axis {ordinate} {Cartesian coordinate}. Cartesian coordinates use perpendicular axes {rectangular coordinate}. x-axis and y-axis divide the plane into four parts {quadrant, plane}. Axes intersect at point {origin}. In Cartesian coordinates, distance between two points is ((x2 - x1)^2 + (y2 - y1)^2)^0.5. Space points have distances to x-axis, y-axis, and z-axis. x-axis, y-axis, and z-axis are perpendicular.
System, such as Cartesian space, can have three perpendicular axes {homogeneous coordinate}. Point has coordinates that are perpendicular distances to axes. For homogeneous coordinates u, v, and w {line coordinate}, u*x + v*y + w*z = 0 represents a line.
Using homogeneous coordinates, all curve equations are first-degree homogeneous equations. Equation degree gives curve class. For example, quadratic equations are surfaces.
triangle
In triangle, point has signed perpendicular distances to triangle sides. If triangle side is at infinity, x = x1/x3 and y = x2/x3.
comparison
For point coordinates, equation degree is curve order.
duality
Line coordinates prove the duality principle that points and lines are complementary.
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).
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Date Modified: 2022.0225