Ellipse Orthogonal tangents
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the ellipse x 2 a 2 + y 2 b 2 = 1 {\displaystyle {\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}=1} the intersection points of orthogonal tangents lie...
and 1; the tangents at that point with those slopes are called the left and right tangents. Sometimes the slopes of the left and right tangent lines are...
hyperbolic orthogonal. The notion of hyperbolic orthogonality arose in analytic geometry in consideration of conjugate diameters of ellipses and hyperbolas...
intersection points of orthogonal tangents are points of the circle x 2 + y 2 = a 2 + b 2 {\displaystyle x^{2}+y^{2}=a^{2}+b^{2}} . The ellipse case can be adopted...
pairs of parallel tangent lines have the same distance. For an ellipse, the standard terminology is different. A diameter of an ellipse is any chord passing...
hyperbolas. In the mixture of confocal ellipses and hyperbolas, any ellipse intersects any hyperbola orthogonally (at right angles). Parabolas have only...
Hyperbola (section Orthogonal tangents – orthoptic)x-axis. Every hyperbola in this family is orthogonal to every ellipse that shares the same foci. This orthogonality may be shown by a conformal map of the...- In mathematics, orthogonal coordinates are defined as a set of d coordinates q = ( q 1 , q 2 , … , q d ) {\displaystyle \mathbf {q} =(q^{1},q^{2},\dots...
Steiner inellipse (redirect from Midpoint ellipse)midpoint inellipse, or midpoint ellipse of a triangle is the unique ellipse inscribed in the triangle and tangent to the sides at their midpoints. It...
cross section is either an ellipse, or is empty, or is reduced to a single point (this explains the name, meaning "ellipse-like"). It is bounded, which...
Deming regression (redirect from Orthogonal regression)Steiner inellipse that is tangent to the triangle's sides at their midpoints. The major axis of this ellipse falls on the orthogonal regression line for the...
Perpendicular (category Orthogonality)is that If two tangents to the parabola are perpendicular to each other, then they intersect on the directrix. Conversely, two tangents which intersect...- Conjugate diameters (redirect from Tangent parallelogram)if they are perpendicular. For an ellipse, two diameters are conjugate if and only if the tangent line to the ellipse at an endpoint of one diameter is...
Kepler's laws of planetary motion (redirect from Kepler ellipse)velocities vary. The three laws state that: The orbit of a planet is an ellipse with the Sun at one of the two foci. A line segment joining a planet and...
Cassini oval (redirect from Cassini ellipses)number of cusps = 0, number of double tangents = 8, number of points of inflection = 12, genus = 1. The tangents at the circular points are given by x ± iy = ± a...
from a semi-ellipse. To meet the curvature condition, the semi-ellipse should be bounded by the semi-major axis of its ellipse, and the ellipse should have...
centers lie on an ellipse. This is true for any set of circles that are internally tangent to one given circle and externally tangent to the other; such...
Spherical conic (redirect from Spherical ellipse)every spherical ellipse is also a spherical hyperbola, and vice versa. As a space curve, a spherical conic is a quartic, though its orthogonal projections...
Parabola (section 2-points–2-tangents property)2-points–2-tangents property should not be confused with the following property of a parabola, which also deals with 2 points and 2 tangents, but is not...- Circle (section Tangent lines)tangents can always be drawn to a circle from any point outside the circle, and these tangents are equal in length. If a tangent at A and a tangent at...
- solves such constructions as: given four tangents and one point, three tangents and two points, &c. If a tangent and its point of contact be given, it is
- Chebyshev, Gauss, Jacobi, and Legendre as the main creators of the theory of orthogonal polynomials. However, their contributions were directly influenced by
- Relation between sine x, x, tangent x for small x * Side opposite small angle given * Length of long sides given * Orthogonal projection * Bipolar (biangular)