Orthoptic (geometry)
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

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 geometries ...
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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 ...

Rigid Motion
Rigid or rigidity may refer to: Mathematics and physics *Stiffness, the property of a solid body to resist deformation, which is sometimes referred to as rigidity *Structural rigidity, a mathematical theory of the stiffness of ensembles of rigid objects connected by hinges *Rigidity (electromagnetism), the resistance of a charged particle to deflection by a magnetic field *Rigidity (mathematics), a property of a collection of mathematical objects (for instance sets or functions) *Rigid body, in physics, a simplification of the concept of an object to allow for modelling * Rigid transformation, in mathematics, a rigid transformation preserves distances between every pair of points *Rigidity (chemistry), the tendency of a substance to retain/maintain their shape when subjected to outside force *(Modulus of) rigidity or shear modulus (material science), the tendency of a substance to retain/maintain their shape when subjected to outside force Medicine *Rigidity (neurology), an ...
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Chord (geometry)
A chord of a circle is a straight line segment whose endpoints both lie on a circular arc. The infinite line extension of a chord is a secant line, or just ''secant''. More generally, a chord is a line segment joining two points on any curve, for instance, an ellipse. A chord that passes through a circle's center point is the circle's diameter. The word ''chord'' is from the Latin ''chorda'' meaning '' bowstring''. In circles Among properties of chords of a circle are the following: # Chords are equidistant from the center if and only if their lengths are equal. # Equal chords are subtended by equal angles from the center of the circle. # A chord that passes through the center of a circle is called a diameter and is the longest chord of that specific circle. # If the line extensions (secant lines) of chords AB and CD intersect at a point P, then their lengths satisfy AP·PB = CP·PD (power of a point theorem). In conics The midpoints of a set of parallel chords of a coni ...
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