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 ). Generalizations: # An isoptic is the set of points for which two tangents of a given curve meet at a ''fixed angle'' (see

Isoptic Of A Parabola, An Ellipse And A Hyperbola
In the geometry of curves, an orthoptic is the Set (mathematics), 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 #Orthoptic of a parabola, below), # The orthoptic of an ellipse \tfrac + \tfrac = 1 is the director circle x^2 + y^2 = a^2 + b^2 (see #Ellipse, 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 #Hyperbola, 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 #Orthoptic of an astroid, below). Generalizations: # An isoptic is the set of points for which two tangents of a given curve meet at a ''fixed angle'' (see #Isoptic of a parabola, an ellipse and a hyperbola, below). # An isoptic of ''two'' plane curves is the set of points for which two tangents meet at a ...
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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 \sqrt, and so, may not exist in the Euclidean plane, but could be a circle with imaginary radius in the complex plane. The director circle of a circle is a concentric circle having radius \sqrt times the radius of the original circle. Generalization More generally, for any collection of points , weights , and constant , one can define a circle as th ...
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Geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a ''List of geometers, geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point (geometry), point, line (geometry), line, plane (geometry), plane, distance, angle, surface (mathematics), surface, and curve, as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, Wiles's proof of Fermat's ...
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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 Greek may refer to: Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group *Greek language, a branch of the Indo-European language family **Proto-Greek language, the assu ...

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 provide s ...
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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 ^2 and \cos^2 \ ...
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Parameter Space
The parameter space is the space of all possible parameter values that define a particular mathematical model. It is also sometimes called weight space, and is often a subset of finite-dimensional Euclidean space. In statistics, parameter spaces are particularly useful for describing parametric families of probability distributions. They also form the background for parameter estimation. In the case of extremum estimators for parametric models, a certain objective function is maximized or minimized over the parameter space. Theorems of existence and consistency of such estimators require some assumptions about the topology of the parameter space. For instance, compactness of the parameter space, together with continuity of the objective function, suffices for the existence of an extremum estimator. Sometimes, parameters are analyzed to view how they affect their statistical model. In that context, they can be viewed as inputs of a function, in which case the technical term for ...
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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 Greek may refer to: Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group *Greek language, a branch of the Indo-European language family **Proto-Greek language, the assu ...