Elliptic curveIn mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point O. An elliptic curve is defined over a field K and describes points in K^2, the Cartesian product of K with itself. If the field's characteristic is different from 2 and 3, then the curve can be described as a plane algebraic curve which consists of solutions (x, y) for: for some coefficients a and b in K. The curve is required to be non-singular, which means that the curve has no cusps or self-intersections.
Parallel postulateIn geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: If a line segment intersects two straight lines forming two interior angles on the same side that are less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.
Tangential angleIn geometry, the tangential angle of a curve in the Cartesian plane, at a specific point, is the angle between the tangent line to the curve at the given point and the x-axis. (Some authors define the angle as the deviation from the direction of the curve at some fixed starting point. This is equivalent to the definition given here by the addition of a constant to the angle or by rotating the curve.) If a curve is given parametrically by (x(t), y(t)), then the tangential angle φ at t is defined (up to a multiple of 2π) by Here, the prime symbol denotes the derivative with respect to t.
Algebraic curveIn mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenizing its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equation h(x, y, t) = 0 can be restricted to the affine algebraic plane curve of equation h(x, y, 1) = 0.
Industrial RevolutionThe Industrial Revolution, also known as the First Industrial Revolution, was a period of global transition of human economy towards more efficient and stable manufacturing processes that succeeded the Agricultural Revolution, starting from Great Britain, continental Europe, and the United States, that occurred during the period from around 1760 to about 1820–1840. This transition included going from hand production methods to machines; new chemical manufacturing and iron production processes; the increasing use of water power and steam power; the development of machine tools; and the rise of the mechanized factory system.
American RevolutionThe American Revolution was an ideological and political revolution that generally occurred in British America between 1765 and 1783. In the American Revolutionary War, which lasted from 1775 to 1783, the Thirteen Colonies secured their independence from the British Crown and established the United States as the first sovereign nation state founded on Enlightenment principles of constitutionalism and liberal democracy. American colonists objected to being taxed by the British Parliament, a body in which they had no direct representation.
Iranian RevolutionThe Iranian Revolution (انقلاب ایران, Enqelâb-e Irân ʔeɴɢeˌlɒːbe ʔiːɾɒːn), or the Islamic Revolution (انقلاب اسلامی, Enqelâb-e Eslâmī), was a series of events that culminated in the overthrow of the Pahlavi dynasty in 1979. The revolution also led to the replacement of the Imperial State of Iran by the present-day Islamic Republic of Iran, as the monarchical government of Mohammed Reza Pahlavi was superseded by the theocratic government of Ruhollah Khomeini, a religious cleric who had headed one of the rebel factions.
Russian RevolutionThe Russian Revolution was a period of political and social change in the Russian Empire, starting in 1917. This period saw Russia abolish its monarchy and adopt a socialist form of government following two successive revolutions and a bloody civil war. The Russian Revolution can also be seen as the precursor for the other European revolutions that occurred during or in the aftermath of World War I, such as the German Revolution of 1918–1919. The Russian Revolution was inaugurated with the February Revolution in early 1917.
French RevolutionThe French Revolution (Révolution française ʁevɔlysjɔ̃ fʁɑ̃sɛːz) was a period of radical political and societal change in France that began with the Estates General of 1789 and ended with the formation of the French Consulate in November 1799. Many of its ideas are considered fundamental principles of liberal democracy, while the values and institutions it created remain central to French political discourse. Its causes are generally agreed to be a combination of social, political and economic factors, which the Ancien Régime proved unable to manage.
Tangent vectorIn mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in Rn. More generally, tangent vectors are elements of a tangent space of a differentiable manifold. Tangent vectors can also be described in terms of germs. Formally, a tangent vector at the point is a linear derivation of the algebra defined by the set of germs at .
October RevolutionThe October Revolution, known in Soviet historiography as the Great October Socialist Revolution, was a revolution in Russia led by the Bolshevik Party of Vladimir Lenin that was a key moment in the larger Russian Revolution of 1917–1923. It was the second revolutionary change of government in Russia in 1917. It took place through an armed insurrection in Petrograd (now Saint Petersburg) on . It was the precipitating event of the Russian Civil War.
Rotation matrixIn linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix rotates points in the xy plane counterclockwise through an angle θ about the origin of a two-dimensional Cartesian coordinate system. To perform the rotation on a plane point with standard coordinates v = (x, y), it should be written as a column vector, and multiplied by the matrix R: If x and y are the endpoint coordinates of a vector, where x is cosine and y is sine, then the above equations become the trigonometric summation angle formulae.
