Three points not on same line define a flat surface {plane, mathematics}. Lines and line points are in a plane. Lines perpendicular to all plane lines are perpendicular to plane. At a plane point, only one line can be perpendicular to a plane. At a line point, only one plane can be perpendicular to a line. If normal to a plane is perpendicular to a line, line and plane are parallel.
Straight line divides plane in half {half-plane}.
Equal congruent arcs can bound plane figure {multifoil}: three arcs {trefoil, figure}, four arcs {quatrefoil}, five arcs {pentafoil}, and six arcs {hexafoil}. Arc centers make regular-polygon vertices.
Planes {osculating plane} can pass through a surface point and two nearby points.
Parabolas and chords, perpendicular to parabola axis, can make plane figures {parabolic segment}. Parabolic-segment area is 2 * c * a / 3, where c is chord length, and a is distance from vertex to chord.
Eliminating spherical-equation second-power terms defines a plane {radical plane}. Radical plane contains circle of sphere pencil.
Plane region rotated around line {axis of revolution} {revolution axis} in the plane makes solid {solid of revolution}. Plane-region perimeter generates surface {surface of revolution}. Volume is integral from x = a to x = b of pi * y^2 * dx, for y = f(x). Area is integral from x = a to x = b of 2 * pi * y * (1 + (dy/dx)^2)^0.5 * dx.
Plane can have equation x/a + y/b + z/c = 1 {intercept form, plane}, where a, b, c are x-axis, y-axis, and z-axis intercepts.
Plane can have equation x * cos(a) + y * cos(b) + z * cos(c) = p {perpendicular form} {normal form, plane}, where p is perpendicular distance from origin to plane, and a, b, c are angles between perpendicular and x y z axes.
Plane can have matrix |x y z 1 / x1 y1 z1 1 / x2 y2 z2 1 / x3 y3 z3 1| {three-point form}, where (xi,yi,zi) are points.
Two planes are either parallel or intersecting {intersecting planes}. Intersecting planes make a wedge.
Plane and solid intersect to makes a plane region {section, plane} {cross-section}|.
Plane sets {sheaf, plane}| can pass through a point {center, sheaf}.
Planes can intersect to make a solid figure {wedge, plane}|.
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Date Modified: 2022.0225