Isoptic
In the geometry of curves, an orthoptic is the set of points for which two tangents of a given curve meet at a right angle. Examples: # The orthoptic of a parabola is its directrix (proof: see below), # The orthoptic of an ellipse \tfrac + \tfrac = 1 is the director circle x^2 + y^2 = a^2 + b^2 (see below), # The orthoptic of a hyperbola \tfrac - \tfrac = 1,\ a > b is the director circle x^2 + y^2 = a^2 - b^2 (in case of there are no orthogonal tangents, see below), # The orthoptic of an astroid x^ + y^ = 1 is a quadrifolium with the polar equation r=\tfrac\cos(2\varphi), \ 0\le \varphi < 2\pi (see below). Generalizations: # An isoptic is the set of points for which two tangents of a given curve meet at a ''fixed angle'' (see
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Isoptic Of A Parabola, An Ellipse And A Hyperbola
In the geometry of curves, an orthoptic is the set of points for which two tangents of a given curve meet at a right angle. Examples: # The orthoptic of a parabola is its directrix (proof: see below), # The orthoptic of an ellipse \tfrac + \tfrac = 1 is the director circle x^2 + y^2 = a^2 + b^2 (see below), # The orthoptic of a hyperbola \tfrac - \tfrac = 1,\ a > b is the director circle x^2 + y^2 = a^2 - b^2 (in case of there are no orthogonal tangents, see below), # The orthoptic of an astroid x^ + y^ = 1 is a quadrifolium with the polar equation r=\tfrac\cos(2\varphi), \ 0\le \varphi < 2\pi (see below). Generalizations: # An isoptic is the set of points for which two tangents of a given curve meet at a ''fixed angle'' (see
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Director Circle
In geometry, the director circle of an ellipse or hyperbola (also called the orthoptic circle or Fermat–Apollonius circle) is a circle consisting of all points where two perpendicular tangent lines to the ellipse or hyperbola cross each other. Properties The director circle of an ellipse circumscribes the minimum bounding box of the ellipse. It has the same center as the ellipse, with radius \sqrt, where a and b are the semi-major axis and semi-minor axis of the ellipse. Additionally, it has the property that, when viewed from any point on the circle, the ellipse spans a right angle. The director circle of a hyperbola has radius , and so, may not exist in the Euclidean plane, but could be a circle with imaginary radius in the complex plane. Generalization More generally, for any collection of points , weights , and constant , one can define a circle as the locus of points such that :\sum w_i \, d^2(X,P_i) = C. The director circle of an ellipse is a special case of th ...
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Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geome ...
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Orthoptic Of An Astroid
Orthopic may refer to: * Orthoptic (geometry), the set of points for which two tangents of a given curve meet at a right angle, a type of isoptic * Orthoptics, the diagnosis and treatment of defective eye movement and coordination * A form of eye exercise designed to correct vision See also *Orthotopic Orthotopic procedures (from Greek ''orthos'', straight + ''topos'', place) are those occurring at the normal place. Examples include: * Orthotopic liver transplantation, in which the previous liver is removed and the transplant is placed at that lo ...

Orthoptic Of An Ellipse And A Hyperbola
Orthopic may refer to: * Orthoptic (geometry), the set of points for which two tangents of a given curve meet at a right angle, a type of isoptic * Orthoptics, the diagnosis and treatment of defective eye movement and coordination * A form of eye exercise designed to correct vision See also *Orthotopic Orthotopic procedures (from Greek ''orthos'', straight + ''topos'', place) are those occurring at the normal place. Examples include: * Orthotopic liver transplantation, in which the previous liver is removed and the transplant is placed at that lo ...

Proofs
Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a construct in proof theory * Mathematical proof, a convincing demonstration that some mathematical statement is necessarily true * Proof complexity, computational resources required to prove statements * Proof procedure, method for producing proofs in proof theory * Proof theory, a branch of mathematical logic that represents proofs as formal mathematical objects * Statistical proof, demonstration of degree of certainty for a hypothesis Law and philosophy * Evidence, information which tends to determine or demonstrate the truth of a proposition * Evidence (law), tested evidence or a legal proof * Legal burden of proof, duty to establish the truth of facts in a trial * Philosophic burden of proof, obligation on a party in a dispute to pro ...
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Angle Sum And Difference Identities
In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles. They are distinct from triangle identities, which are identities potentially involving angles but also involving side lengths or other lengths of a triangle. These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity. Pythagorean identities The basic relationship between the sine and cosine is given by the Pythagorean identity: :\sin^2\theta + \cos^2\theta = 1, where \sin^2 \theta means (\sin \theta)^2 ...
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Orthoptic Locus Of A Circle, Ellipses And Hyperbolas
Orthopic may refer to: * Orthoptic (geometry), the set of points for which two tangents of a given curve meet at a right angle, a type of isoptic * Orthoptics, the diagnosis and treatment of defective eye movement and coordination * A form of eye exercise designed to correct vision See also *Orthotopic Orthotopic procedures (from Greek ''orthos'', straight + ''topos'', place) are those occurring at the normal place. Examples include: * Orthotopic liver transplantation, in which the previous liver is removed and the transplant is placed at that lo ...