Geometry {solid geometry} can be about three spatial dimensions and solids.
Closed surface can bound figure {solid figure}. Solid can be figure bounded by planes {face, solid}, which meet at points {vertex, solid} and lines {edge, solid}.
Six-sided solids {parallelepiped}| can have six parallelogram faces.
Figure sets {pencil, geometry} can have all figures pass through common points. Lines can pass through one point {vertex, pencil}. Circles can pass through two points. Order-n curves can pass through n^2 points. Parallel figures can have a common point {ideal point} or line {ideal line}. Parallel planes can have a common line {axis, plane}. Spheres can have a common circle. Planes can share a straight line.
Rectangles can transform {torus}|, by connecting top and bottom to make a cylinder and left and right to make a circle. Toruses have two circles, one along cylinder and one around cylinder. Toruses can result when closed plane regions revolve around an axis in a plane that does not intersect surface. If closed plane regions are circles, solid rings form, with volume = 2 * pi^2 * p^2 * r^2 and area = 4 * pi^2 * p * r, where p is torus radius and r is circle radius.
Solid figure has interior space {volume, solid}. Box volume = h*w*l, where h, w, l are side lengths.
Two solids meet at an angle {solid angle}| measured as a wedge. Solid angle measure is in steradians. Solid angle is at vertex.
Two angles {dihedral angle}| are where two planes intersect. The smaller angle is angle size {inclination}. Dihedral angle lies between line in each plane perpendicular to intersection line {vertex, dihedral angle}. Dihedral angle is angle between normals to the planes. Dihedral angle measure is in radians.
Two planes meet at an angle {plane angle} measured between two lines perpendicular to plane-intersection line. Plane angle measure is in radians.
Edges and faces at vertices make solid angles {polyhedral angle}|.
Three lines can meet at one point, defining three planes {trihedral angle}|. Trihedral-angle ray relative directions can be left-handed or right-handed.
Angles can rotate through 360 degrees, around the line {bisector, cone} dividing angle in half, to make solid figure {cone, geometry}. Cones have base, bisector axis, vertex, and vertex angle. Cone lines {element, cone} can go through vertex. Right-circular-cone area is bottom area plus side area: pi * r^2 + pi * r * (r^2 + h^2)^0.5, where r is base radius and h is height. Right-circular-cone volume = (pi * h * r^2) / 3, where r is radius and h is height.
Conical surfaces have two conical halves {nappe}|.
Right circular cones have generating-line length {slant height, cone}.
Planes can intersect cones to make plane curves {conic section}|. Plane can be parallel to base and intersect cone at right angle to axis {circle, cone}. Plane can intersect cone at angle less than vertex angle {ellipse, cone}. Plane can intersect cone at angle equal to vertex angle and parallel to element {parabola, cone}. Plane can intersect cone parallel to axis {hyperbola, cone}. Plane can intersect cone so plane includes vertex and bisector {intersecting lines}. Plane can be tangent to cone {tangent line, cone}.
slope
Circles and ellipses are closed curves and have same slope at diameter ends. Parabolas are not closed curves and approach maximum slope as they go farther from axis. Hyperbolas are not closed curves and approach maximum slope as they go farther from axis.
pole
Two tangents to conic can meet at point {pole, cone}.
conic points
Two conics intersect at four points. Two real conics that do not intersect share two imaginary chords.
generation: point and curve
For conic sections, line goes through fixed point {generator, cone} and closed curve {directrix, cone}.
generation: lines
Conics can be line series and pencils, as can ruled quadrics.
Two non-parallel planes can cut {truncate} cone, cylinder, prism, or pyramid.
(x - h)^2 / a^2 + (y - k)^2 / b^2 = 1, where center is at (h,k), a is longer radius, and b is shorter radius {ellipse, conic}|. x = - h + a^2 / (a^2 + b^2)^0.5, where a > b.
foci
Ellipses have two focuses. Ellipse points have distances to foci. For all ellipse points, distance sum is constant.
Ellipses are symmetric about two lines. Ellipses have four points {vertex, ellipse} intersected by symmetry axes. Longest symmetry axis {major diameter} {major axis} has length = 2*a, where a > b. Shortest symmetry axis {minor diameter} {minor axis} has length = 2*b.
circle
Circle equation is (x - h)^2 + (y - k)^2 = r^2, where r is radius, and center is at (h,k).
auxiliary circle
A circle {auxiliary circle} with diameter equal major axis can surround ellipse.
Curves {helix}| {bolt, helix} can maintain constant angle with cylinder, cone, or sphere generator.
In right circular cylinders, helix {circular helix} has equations x = r * cos(A), y = r * sin(A), z = r * A * cos(B), where A is revolution angle, r is cylinder radius, and B is helix-to-generator inclination angle.
Right-circular-cone helices are like tapered screws. If helices open to become circle sectors, they look like equiangular-spiral pieces.
spherical helix
Loxodromic spirals can be helices {spherical helix}.
In Cartesian coordinates, hyperbolas {hyperbola, conic}| have equation (x - h)^2 / a^2 - (y - k)^2 / b^2 = 1, where center is (h,k), a is length between vertex and center, and b is length between focus and hyperbola point along a line perpendicular to long axis through focus.
Eccentricity e is distance from hyperbola point to focus divided by distance from hyperbola point to directrix and is constant and greater than 1: e = (a^2 + b^2)^0.5 / a. If center is at (0,0), focus is at x = a * e.
In polar coordinates with center at origin, r^2 = a^2 * b^2 / (b^2 * cos^2(A) - a^2 * sin^2(A)). In polar coordinates with center at focus, equation applies to only one branch: r = a * (e^2 - 1) / (1 - e * cos(A)) = a * ((a^2 + b^2)/a^2) / (1 - ((a^2 + b^2)^0.5 / a) * cos(A)), where -1 <= cos(A) <= 1.
directrix
Hyperbolas have two directrixes, a fixed line perpendicular to the long axis, typically between center and vertex, in the same plane as the hyperbola.
foci
Hyperbolas have two focuses, a fixed point on the long axis on the convex side. Hyperbola points have distances to foci. All hyperbola points have the same focal-distance difference, equal to 2*a.
symmetry
Hyperbolas are symmetric about the centers. The symmetry line intersects hyperbola at two points {vertex, hyperbola}.
Hyperbolas can rotate around long axis to make hyperboloid surfaces.
diameters
A line segment {transverse diameter} between vertices has length 2*a. A line segment {conjugate diameter} perpendicular to transverse diameter at focus has length 2*b. Hyperbolas {equilateral hyperbola} can have transverse diameter equal to conjugate diameter. The auxiliary circle, with center at (0,0) and radius a, intersects the vertices.
asymptote
When x is large positive or negative, hyperbola slope approaches straight line {asymptote, hyperbola}.
rectangular hyperbola
If transverse and conjugate axes are equal, hyperbola {rectangular hyperbola} can have asymptotes at right angles. If rectangular hyperbola is symmetric to coordinate axes, equation is x^2 - y^2 = a^2, where a is half axis length. If asymptotes are coordinate axes, equation is x*y = a^2 / 2 = c^2, where a is half axis length and c is constant.
auxiliary rectangle
Conjugate diameter determines rectangle {auxiliary rectangle} between the hyperbola curves.
Conic sections {parabola, conic section} can have U shape.
equation
Parabola equation can be a * (x - h) = (y - k)^2, where h is x-intercept, k is y-intercept, and a is major conic-section diameter. Minor conic-section diameter is zero. For parabola, x = k - a. Parabola equation can be y = a*x^2 + b*x + c.
definition
Distance from any parabola point to parabola center {focus, parabola} equals distance from point to defining line {directrix, parabola}.
axis
A symmetry line {axis, parabola} divides parabolas lengthwise. Axis intersects parabola at point {vertex, parabola}. Distance {focal length, parabola} from focus to vertex is major diameter. Parabolas have no minor diameter.
semicubical parabola
Equation a * y^2 = x^3 or b * y^3 = x^2, where a is focal length, defines parabola {semicubical parabola}.
Distance from conic-section point to focus divided by distance from conic-section point to directrix is constant {eccentricity, conic}: e = (a^2 + b^2)^0.5 / a, where a is major axis and b is minor axis. For circle, e = 2^0.5. For parabola, e = 1, because b = 0. For hyperbola and ellipse, e > 1.
For ellipses, angle A {eccentric angle} in equations x = a * cos(A) and y = b * sin(A), where a is ellipse major axis, and b is ellipse minor axis, determines eccentricity. Hyperbola has eccentric angle A with x = a * sec(A) and y = b * tan(A).
In ellipses, two circles {eccentric circle} can have major and minor axes as diameters. In hyperbola, two eccentric circles have transverse axis and conjugate axis {harmonic conjugate of transverse axis} as diameters.
l / r = 1 - e * cos(A) {polar equation}, where l is latus-rectum length divided by two, r is distance from pole or focus, e is eccentricity, and A is polar angle. Transverse or major-axis positive direction is reference line {initial line} {polar axis, conic}.
For conic sections, a line {directrix, conic} goes through generator point and closed curve.
Distance {focal length, conic} from focus to vertex is major diameter.
Line segments {latus rectum} can go through conic-section focus and two conic-section points, whose distances to focus are equal.
Line segments {polar, conic} can connect conic poles.
Six-sided solids {cube} can have six square faces. Cube volume = s^3, where s is side. Skeletons of n-dimensional cubes correspond to Gray binary codes.
Six-sided solids {cuboid} can have six rectangular faces.
Lines {curve, circles} can vary curvature radius. Curves {regular arc} {regular curve} can be in intervals.
Surfaces have metric elements {arc length, curve}: ds^2 = dx^2 + dy^2 + dz^2. Locally, arc length is invariant. Arc-length curvature and functions {torsion, curve} can determine space-curve properties, except position.
Curve order and class relate to simple singularities {Plucker formula}.
Curvature depends only on surface parameters {Gauss characteristic equation}. Curvature is product of principal curvatures. Linear curvature is tangent-angle change divided by arc length. It is radial-length change times two divided by arc length squared. It is curved-surface solid angle divided by surface area. To measure curvature around point, use regular hexagon and measure angles.
All surface points have a perpendicular {binormal} to osculating plane.
In three dimensions, circles {coaxial circle} can share axis.
Curves {higher curve} {higher plane curve} can have equations with degree greater than two. Defining degree-n curves requires n * (n + 3) / 2 points. Higher plane curves have inflection points, multiple points, cusps, conjugate points, genuses, and branches. One degree-m curve and one degree-n curve can intersect at m*n points.
A point and two points on a curve near the point define a circle with curvature equal to curve {osculating curve}| curvature.
Rotation about, and translation along, line makes space curve {screw curve}.
Non-intersecting curves {simple closed curve} can enclose regions. Simple closed curves {simply connected region} can surround only region points.
Curves {skew curve} {twisted curve} can go outside of plane.
Solid figures {cylinder} can have circle and long axis. Right-circular-cylinder area equals top area plus bottom area plus side area = 2 * pi * r^2 + 2 * pi * r * h, where r is base radius and h is height. Right-circular-cylinder volume = pi * h * r^2, where r is radius and h is height.
Surfaces {ellipsoid}| can have elliptical cross-sections in three coordinate planes. Larger axes can be equal {oblate ellipsoid}. Smaller axes can be equal {prolate ellipsoid}. For sphere, three axes are equal.
Lines {diagonal} can go from one vertex to non-adjacent vertex. Tetrahedron and Csaszar polyhedron have no diagonals.
If two lines are perpendicular {perpendicular lines}, line slopes are negative reciprocals: m2 = -1 / m1, where m1 and m2 are slopes.
Regular pyramid has apex and lateral face, which has median {slant height, pyramid} through apex.
Space can have non-intersecting, non-parallel lines {skew line}.
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}|.
Simple multiple-sided solids {polyhedron, solid}| have genus zero.
Faces can meet at lines {edge, polyhedron}.
Plane polygons {face, polyhedron} can bound solids.
Three or more edges can meet at points {vertex, polyhedron}.
Polyhedrons {Csaszar polyhedron} can model seven-color maps of toruses, finite projective planes, and error-correcting binary codes, when used as Hadamard matrices {Room square}.
Polyhedrons {regular polyhedron} {regular solid} {Platonic solid} can have same regular polygon for all faces: four equilateral triangles {regular tetrahedron}, six squares {cube, Platonic solid}, six regular hexagons {regular hexahedron}, eight equilateral triangles {regular octahedron}, twelve regular pentagons {regular dodecahedron}, and twenty equilateral triangles {regular icosahedron}. All vertices are on circumscribed sphere. Concave regular polyhedrons are small stellated dodecahedron, great dodecahedron, or great icosahedron.
Congruent parallel faces {base, prism} and congruent parallelograms {lateral face}, joining corresponding base vertices, can make solids {prism}|. Lateral faces can be rectangles {right prism}. Prisms have adjacent lateral faces {prismatic surface}.
Polyhedrons {prismatoid} can have two faces that are parallel planes, with no vertices outside the faces. Lateral faces are triangles or quadrilaterals.
Prisms {prismoid} can have quadrilaterals for all lateral faces. Bases have same number of sides and vertices.
Prismatoids {pyramid} can have polygon bases, which contain all vertices except apex. Lateral faces are triangles. For tetrahedrons, bases are triangles. In regular pyramids, regular polygons can be bases and isosceles triangles can be lateral faces.
Polyhedrons {tetrahedron}| can have four faces.
Polyhedrons {pentahedron}| can have five faces.
Polyhedrons {diamond} can have six equal equilateral triangle faces. Diamonds are two tetrahedrons that share a face.
Polyhedrons {hexahedron} can have six faces.
Polyhedrons can have six rhombus faces {rhombohedron}|.
Polyhedrons {octahedron}| can have eight faces.
Polyhedrons {dodecahedron}| can have 12 faces.
Polyhedrons {icosahedron}| can have 20 faces.
Concave regular polyhedrons {Kepler-Poinsot concave solid} can be small stellated dodecahedron, great dodecahedron, or great icosahedron.
Concave regular polyhedrons can have 12 regular pentagons {great dodecahedron}.
Concave regular polyhedrons can have 20 equilateral triangles {great icosahedron}.
Concave regular polyhedrons can have 12 regular pentagons {small stellated dodecahedron}.
Solids {sphere} can result when semicircle rotates around its diameter. Equation is x^2 + y^2 + z^2 <= r^2, where r is radius.
area
Area is 4 * pi * r^2, where r is radius.
volume
Volume is 4 * pi * r^3 / 3, where r is radius.
imaginary circle
Two spheres share imaginary circle.
secondaries
Great circles can pass through poles.
spherical distance
Geodesic has length {spherical distance}.
spherical polygon
Great-circle arcs can bound spherical surface region {spherical polygon}.
spherical surface
x^2 + y^2 + z^2 = r^2 defines spherical surface. Area is 4 * pi * r^2, where r is radius.
diameter
Diameter is perpendicular to sphere at both endpoints.
coordinates
Sphere coordinates are longitude (360 degrees) and latitude (180 degrees), because they define unique points. Longitudes are perpendicular to latitudes. For spinning spheres, longitudes are along general direction of spherical axis, and latitudes are perpendicular to spherical axis.
Spherical coordinates can be vertical and horizontal latitude, so axes are perpendicular, but two latitudes define two different points, so points must have one more coordinate, such as north or south. Spherical coordinates can be vertical and horizontal longitudes, with axes not always perpendicular, but two longitudes can define the same great circle, so points must have one more coordinate. Therefore, only longitude and latitude define sphere points using two numbers.
Diameter intersects sphere at two points {antipodes, geometry}|.
Spheres have points {pole, sphere} where meridians meet.
Radii can make an angle {azimuth}| {polar angle} with polar axis.
Dihedral angles {spherical angle} are at diameter where great-circle planes intersect.
1/720 {spherical degree}| of sphere surface is solid-angle unit. One spherical degree is the birectangular spherical triangle whose third angle is one degree.
Difference {spherical excess} between spherical-polygon angle sum and plane angle sum is (n - 2) * (180 degrees), where n is number of angles.
Sphere has solid angle 4 * pi {steradian, solid angle}|.
Spherical areas {cap, sphere} can go from pole down to circle where plane intersects sphere. Cap area is pi * d * h, where d is diameter, and h is cap height from center.
Sphere halves {hemisphere}| are solids.
Two great circles not in perpendicular planes make two major and two minor spherical-surface regions {lune}.
Sectors {segment, sphere} {major sector, sphere} {major segment, sphere} can be greater than hemisphere. Sectors {minor sector, sphere} {minor segment, sphere} can be less than hemisphere.
Planes of two great circles intersect at diameter {spherical wedge} and divide sphere into four parts. Spherical-wedge volume is (A / (3 * pi / 2)) * pi * r^2, where A is angle between planes, and r is radius.
Sphere and two parallel planes intersect to make a solid figure {zone, sphere region}. Sphere and plane intersection has area pi * d * h, where d is diameter, and h is height from center.
Solid surfaces can unfold or unroll so all faces or surfaces lie in one plane {development, solid}.
Surfaces {developable surface} can flatten onto a plane without distortion, so surface line elements become plane line elements.
Continuous transformations of n-simplexes onto themselves have at least one fixed point {fixed point theorem}.
Processes {quadrature} can try to find squares equal in area to surfaces. If plane figure has only straight lines, compass and straightedge can perform quadrature.
Spaces can have simplest geometric figure {simplex, space}| {space, cell}. Number of space dimensions defines simplex: 0 is point, 1 is line, 2 is triangle, 3 is tetrahedron, and n is n-simplex. Simplexes are manifolds. Simplexes have orientation. Even numbers of permutations make same orientation. Odd numbers of permutations make opposite orientation. Simplex boundaries are next-lower-dimension simplexes and have orientation.
Surface areas can use measures {square measure}.
To find area under curve, replace curve with chords, to make equal-width orthogonal projections onto independent-variable axis, and add trapezoid areas {trapezoidal rule}|.
Surfaces {continuous surface} can have tangent plane and normal line at all points.
Truncated solids can have parallel plane sections {frustum}.
Surfaces {hyperboloid} can have cross-sections that are hyperbolas.
If surfaces {isometric surface} bend without stretching and keep one-to-one correspondence, curvature and all other properties stay the same.
Focal surfaces of systems of second-order rays are fourth-degree and class-four surfaces {Kummer surface}. Fresnel wave surfaces are special cases.
Solids can have plane parallel faces {lamina} that are small distance apart compared to face length.
Surfaces {paraboloid} can have sections that are parabolas. x^2 / a^2 + y^2 / b^2 = 2*c*z {elliptic paraboloid}, where a b c are axes. Elliptic paraboloid with a = b is parabola rotated about its z-axis. x^2 / a^2 - y^2 / b^2 = 2*c*z {hyperbolic paraboloid}, where a b c are axes.
Straight lines {rectilinear generator} can generate surfaces {ruled surface}| by translation in one direction, making lines at equal intervals. If there are two distinct generators, surfaces are doubly ruled. No generator can make skew surfaces.
Surfaces {smooth surface} can have no irregularities. Objects on smooth surfaces move only in direction tangent to surface.
Surfaces {unilateral surface} can have one side.
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Date Modified: 2022.0225