(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}.
3-Geometry-Solid-Cone-Conic Section
Outline of Knowledge Database Home Page
Description of Outline of Knowledge Database
Date Modified: 2022.0225