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.
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Date Modified: 2022.0225